Selmer Complexes 9782856292273

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SELMER COMPLEXES (Final version, March 2007, to appear in Ast´ erisque) Jan Nekov´ aˇ r 0. Introduction

(0.0) Big Galois representations In this work we study cohomological invariants of “big Galois representations” ρ : G −→ AutR (T ), where (i) G is a suitable Galois group. (ii) R is a complete local noetherian ring, with a finite residue field of characteristic p. (iii) T is an R-module of finite type. (iv) ρ is a continuous homomorphism of pro-finite groups. We develop a general machinery that covers duality theory, Iwasawa theory, generalized Cassels-Tate pairings and generalized height pairings. (0.1) Examples (0.1.0) An archetypal example of a big Galois representation arises as follows. Let K be a field of characteristic char(K) 6= p. For every K-scheme X −→ Spec(K) put X = X ⊗K K sep , where K sep is a fixed separable closure of K. Given a projective system X∞ = (Xα )α∈I (indexed by some directed set I) of separated K-schemes of finite type with finite transition morphisms Xβ −→ Xα , put i i n H i (X∞ ) := lim ←− Het (X α , Zp ) = lim ←− lim ←− Het (X α , Z/p Z), α

α

n

where the transition morphisms are given by trace maps. This is a representation of GK = Gal(K sep /K), linear over the Zp -algebra generated by “endomorphisms” of the tower X∞ . In practice, H i (X∞ ) is often too big and must be first decomposed into smaller constituents. One can also use more general coefficient sheaves, not just Z/pn Z. sep (0.1.1) Iwasawa theory. Let K∞ /K be a Galois S extension (contained in K ) with Γ = Gal(K∞ /K) r isomorphic to Zp for some r ≥ 1. Write K∞ = Kα as a union of finite extensions of K. For a fixed separated K-scheme of finite type X −→ Spec(K), consider the projective system Xα = X ⊗K Kα . In this case ∼

i H i (X∞ ) −→ Het (X, Zp ) ⊗Zp Λ,

where ∼

Λ = Zp [[Γ]] −→ Zp [[X1 , . . . , Xr ]] is the Iwasawa algebra of Γ and GK acts on Λ by the tautological character χΓ : GK  Γ ,→ Λ∗ (or its inverse, depending on the sign conventions). Thus T = H i (X∞ ) is a big Galois representation of G = GK over R = Λ. (0.1.2) Hida theory. Let N ≥ 1 be an integer not divisible by p (and such that N p > 4). Let X∞ be the projective system of modular curves (1) (1)

There are two choices of transition morphisms; see e.g. [N-P] for more details. 1

X1 (N p) ←− · · · ←− X1 (N pr ) ←− X1 (N pr+1 ) ←− · · · over K = Q. The tower X∞ has many endomorphisms, namely Hecke correspondences. The Galois module H 1 (X∞ ) is too big, but its ordinary part H 1 (X∞ )ord , defined as the maximal Zp submodule on which the Hecke operator (2) T (p) is invertible, is of finite type over the ordinary Hecke algebra hord , defined as the projective limit of the ordinary parts of the Zp -Hecke algebras acting on S2 (Γ1 (N pr )) (for variable r). ∼ The ring hord is semilocal, in fact finite and free over Λ = Zp [[Γ]] −→ Zp [[X]], where Γ = 1 + pZp (resp., Γ = 1 + 4Z2 , if p = 2) acts on X∞ by diamond operators. If we fix a maximal ideal m ⊂ hord , then ord T = H 1 (X∞ )ord m is a big Galois representation of G = GQ over R = hm . (0.1.3) One can, of course, combine the constructions in 0.1.1-2. (0.2) Selmer groups In the case when K is a number field and the projective system X∞ has good reduction (i.e. each Xα has) outside a finite set S of places of K containing all places above p, then H i (X∞ ) is a representation of the Galois group with restricted ramification GK,S = Gal(KS /K), where KS is the maximal extension of K which is unramified outside S. In general, given a big Galois representation T of GK,S , the main objects of interest are the following: i (i) (continuous) Galois cohomology groups Hcont (GK,S , T ). (ii) Selmer groups 1 Sel(GK,S , T ) ⊂ Hcont (GK,S , T ), 1 consisting of elements satisfying suitable local conditions in Hcont (Gv , T ) for v ∈ S (where Gv = Gal(K v /Kv )). Similar objects were first considered by R. Greenberg [Gre 4] as a natural generalization of Iwasawa theory. Greenberg expressed hope that there should be a variant of the Main Conjecture of Iwasawa theory in this context, i.e. a relation between the “characteristic power series” of a big Selmer group and an appropriate p-adic L-function. A big Galois representation ρ can be viewed as a family of “usual” Galois representations ρλ : GK,S −→ GLn (Zp ), which depends analytically on the parameter λ. One of the main motivations of the present work was to develop a homological machinery that would control the variation of the Selmer groups attached to the individual ρλ ’s as a function of λ. A statement such as the Main Conjecture for T should then imply a relation between the Selmer group of ρλ and the special value at λ of the p-adic L-function in question. (0.3) Big vs. finite Galois representations Every big Galois representation ρ : G −→ AutR (T ) is the projective limit of Galois representations ρn : G −→ AutR (T /mn T ) with finite targets. Using known properties of ρn one can sometimes pass to the limit and deduce results valid for ρ. Consider, for example, a representation ρ : GK −→ AutR (T ) of GK = Gal(K sep /K) for a local field K (with finite residue field) of characteristic char(K) 6= p. Writing D for the Pontrjagin dual functor

D(−) = Homcont (−, R/Z), Tate’s local duality states that the (finite) cohomology groups H i (GK , T /mn T ) o

D

/ H 2−i (GK , D(T /mn T )(1))

are Pontrjagin duals of each other. Taking projective limit one obtains Pontrjagin duality between a compact and a discrete R-module H i (GK , T ) o

D

/ H 2−i (GK , D(T )(1)),

where (2)

Again, there are two choices of T (p); which one is correct depends on the choice made in 2

(1)

.

i i n H i (GK , T ) = Hcont (GK , T ) = ← lim − H (GK , T /m T ). n

(0.4) Compact vs. discrete modules Attentive readers will have noticed that Greenberg [Gre 2-4] considers Selmer groups for discrete Galois modules, while our T is compact. Let us investigate the relationship between discrete and compact Galois representations more closely. In fact, understanding the interplay between discrete and compact modules is at the basis of the whole theory developed in this work. Let us first consider the “classical” case of R = Zp . Given a representation ρ : G −→ AutZp (T ), where T is free of finite rank over Zp , there are three more representations of G attached to T , namely A = T ⊗Zp Qp /Zp ,

T ∗ = HomZp (T, Zp ) = D(A),

A∗ = T ∗ ⊗Zp Qp /Zp = D(T ).

(0.4.1)

They can be arranged into the following diagram: D

T ←−−−−−−−−→  ←−  −− D −−→ −−−−− Φ  −−−−−−− y − − → − ← A

T∗   Φ  y

(0.4.2)

A∗

Here D(−) = HomZp (−, Zp ) and Φ(−) = (−) ⊗Zp Qp /Zp . What is the analogue of this construction for general R (or even for R = Zp if T is not free over Zp )? Let us temporarily ignore the action of G and consider this question only for R-modules. For R = Zp , the tensor product T ⊗Zp Qp /Zp loses any information about the torsion submodule Ttors ⊂ T . On the other hand, ∼

Z

Ttors −→ Tor1 p (T, Qp /Zp ). This suggests that one should consider the derived tensor product L

A = T ⊗Zp Qp /Zp as a correct version of (0.4.1). In concrete terms, Qp /Zp has a natural flat resolution [Zp −→ Qp ]

(0.4.3)

(in degrees −1, 0) and A is represented by the complex   T ⊗Zp [Zp −→ Qp ] = T −→ T ⊗Zp Qp , again in degrees −1, 0. What is the analogue of (0.4.3) for general R? Fix a system of parameters of R, i.e. elements x1 , . . . , xd ∈ m (where d = dim(R)) such that dim(R/(x1 , . . . , xd )) = 0. An analogue of (0.4.3) is then given by the complex   M M C • = C • (R, (xi )) = R −→ Rxi −→ Rxi xj −→ · · · −→ Rx1 ···xd  i

i and M ι by requiring that (i) M < n >= M as an R-module; g ∈ G acts by gM = χΓ (g)n gM . (ii) M ι = M as an R[G]-module; γ ∈ Γ acts by γM ι = ι(γ)M = (γ −1 )M . (0.12) Duality for Galois cohomology in Iwasawa theory The main point is that cohomology of R-representations over K∞ can be expressed in terms of cohomology of R-representations over K (cf. [Gre 4], Prop. 3.2; [Col], Prop. 2). Let T, T ∗ (resp., A, A∗ ) be bounded complexes of admissible R[G]-modules, of finite (resp., co-finite) type over R. In Iwasawa theory one is often interested in the cohomology groups i i HIw (K∞ /K, T ) = ← lim − Hcont (Gal(KS /Kα ), T ) α

i H i (KS /K∞ , A) = − lim → Hcont (Gal(KS /Kα ), A) α

(and their counterparts with compact support). An easy application of Shapiro’s Lemma (cf. Sect. 8.3 and 8.4) shows that these are the cohomology groups of the following objects of D(R Mod): 10

RΓIw (K∞ /K, T ) = RΓcont (GK,S , FΓ (T )) RΓ(KS /K∞ , A) = RΓcont (GK,S , FΓ (A)), where FΓ (T ) = (T ⊗R R) < −1 > FΓ (A) = HomR,cont (R, A) < −1 > (and similarly for cohomology with compact support). The crucial observation (cf. 8.4.6.6) is the following: if T, T ∗ , A, A∗ are related (over R) as in (0.5.1), then FΓ (T ), FΓ (T ∗ ), FΓ (A), FΓ (A∗ ) are related by the duality diagram D

FΓ (T ) ←−−−−−−−−→ FΓ (T ∗ )ι   ←−   −− D −−→   −−−−− Φ Φ − − − − − y y − −→ −− − ← FΓ (A) FΓ (A∗ )ι

(0.12.1)

over R (here we use the notation D(−) = RHomR (−, ωR ), and similarly for Φ and D). Applying the Poitou-Tate duality 0.7.1 (over R) to (0.12.1), we obtain a duality diagram in D(R Mod) D

RΓIw (K∞ /K, T ) ←−−−−−−−−→ RΓc,Iw (K∞ /K, T ∗(1))ι [3]   ←−   −− D −−→   −−−−− Φ Φ − − − − − y y − −→ −− − ← ι RΓ(KS /K∞ , A) RΓc (KS /K∞ , A∗ (1)) [3]

(0.12.2)

and spectral sequences 3−j i+j E2i,j = ExtiR (D(H j (KS /K∞ , A)), ωR ) = ExtiR (Hc,Iw (K∞ /K, T ∗(1)), ωR )ι =⇒ HIw (K∞ /K, T ) 0

3−j i+j E2i,j = ExtiR (D(Hcj (KS /K∞ , A)), ωR ) = ExtiR (HIw (K∞ /K, T ∗(1)), ωR )ι =⇒ Hc,Iw (K∞ /K, T ).

(0.12.3)

In the classical case R = Zp the ring R = Zp [[Γ]] = Λ is the usual Iwasawa algebra. The spectral sequence i+j E2i,j = ExtiR (D(H j (KS /K∞ , A)), Λ) =⇒ HIw (K∞ /K, T )

was in this case constructed in an unpublished note of Jannsen [Ja 3] (who also considered the case of non-commutative Γ). Back to the general case, recall that an R-module M is pseudo-null if it is finitely generated and its support supp(M ) has codimension ≥ 2 in Spec(R). As codimR (supp(E2i,j )) ≥ i, the spectral sequence Er degenerates in the quotient category (R Mod)/(pseudo − null) into short exact sequences n 0 −→ E21,n−1 −→ HIw (K∞ /K, T ) −→ E20,n −→ 0,

in which E20,n has no R-torsion and E21,n−1 has support in codimension ≥ 1. (0.13) Duality for Selmer complexes in Iwasawa theory Given suitably compatible systems of local conditions Uv+ (X) along the tower of fields {Kα }, one can define Selmer complexes gf,Iw (K∞ /K, X), RΓ

gf (KS /K∞ , Y ) RΓ

(X = T, T ∗ (1), Y = A, A∗ (1)).

Although over each finite layer Kα the diagram (0.9.2) may involve non-zero error terms, the limit of these error terms over K∞ is very often pseudo-null (or co-pseudo-null). 11

For example, Greenberg’s local conditions 0.8.1 induce similar local conditions over each Kα . If we assume that no prime v ∈ Σ0 splits completely in K∞ /K, then (cf. 8.9.9) D gf,Iw (K∞ /K, T ) ←−−−− gf,Iw (K∞ /K, T ∗ (1))ι [3] RΓ −−−−→ RΓ   ←−   −− D −−→   −−−−− Φ Φ − − −− −−− y y − − → − ← gf (KS /K∞ , A) gf (KS /K∞ , A∗ (1))ι [3] RΓ RΓ

(0.13.1)

becomes a duality diagram without any error terms, if we consider the top (resp., bottom) two objects in Df t ((R Mod)/(pseudo − null)) (resp., in Dcof t ((R Mod)/(co − pseudo − null))). Equivalently, for every prime ideal p ∈ Spec(R) of height ht(p) = 1, the localization of (0.13.1) at p is a duality diagram in D(R Mod). p

As in 0.12, this leads to exact sequences of Rp -modules e 4−q (K∞ /K, T ∗(1)), ω )ι −→ H e q (K∞ /K, T ) −→ Hom (H e 3−q (K∞ /K, T ∗ (1)), ω )ι −→ 0, 0 −→ Ext1R (H f,Iw f,Iw f,Iw R p R R p p (0.13.2) in which Xpι is a shorthand for (X ι )p and ∼ e q (K∞ /K, T ∗ (1)) −→ e 3−q (KS /K∞ , A)ι ) H D(H f,Iw f

(and similarly for T and A∗ (1)). If R has no embedded primes, then we obtain isomorphisms ∼ e q (K∞ /K, T ) e 4−q (K∞ /K, T ∗(1)), ω )ι H −→ Ext1R (H f,Iw f,Iw R R−tors

in (R Mod)/(pseudo − null). If there is a prime v ∈ Σ0 that splits completely in K∞ /K, then the above statements hold for those prime ideals p ∈ Spec(R) of height ht(p) = 1 which are not of the form p = pR, where p ∈ Spec(R) has ht(p) = 1 (cf. 8.9.8). In particular, if R is regular and no prime v ∈ Σ0 splits completely in K∞ /K, then the R-modules e f1 (KS /K∞ , A)))tors (D(H

and

e f1 (KS /K∞ , A∗ (1))))ιtors (D(H

are pseudo-isomorphic. This is a generalization of Greenberg’s results ([Gre 2], Thm. 2; [Gre 3], Thm. 1), according to which for R = Zp and K∞ /K the cyclotomic Zp -extension (5) the two Λ-modules in question have the same characteristic power series (more precisely, Greenberg works with his “strict Selmer groups” str str e1 SA (K∞ ) and SA ∗ (1) (K∞ ); their relation to our Hf is explained in 9.6). A similar result for Selmer groups of abelian varieties defined in terms of flat cohomology was proved by Wingberg [Win]. If the complex T = σ≤0 T is concentrated in non-positive degrees, then one can say much more: the horizontal arrow in (0.13.1) becomes an isomorphism after tensoring with Rq , for any minimal prime q of R (8.9.11-12). str Greenberg [Gre 2] also defined “non-strict” Selmer groups SA ⊇ SA . One of their interesting features is the fact that a trivial zero over K can sometimes be detected by the Λ-module SA (K∞ ) (but not by the Selmer e 1 (KS /K∞ , A) may often have the group SA (K) over K). Although the Pontrjagin duals of SA (K∞ ) and H f str same characteristic power series, they need not be isomorphic as Λ-modules, as SA (K∞ ) is a subgroup of 1 e SA (K∞ ), but a quotient of Hf (KS /K∞ , A) (cf. 9.6.2-6). It seems that in the presence of a trivial zero e 1 (KS /K∞ , A) has better semi-simplicity properties than SA (K∞ ). H f

(0.14) Classical Iwasawa theory (0.14.0) Traditionally, the main objects of interest in Iwasawa theory have been the following: e 1 (KS /K∞ , A)) and D(H e 1 (KS /K∞ , A∗ (1))) are torsion over And assuming, in addition, that both D(H f f Λ = Zp [[Γ]]. (5)

12

(i) The Galois group Gal(M∞ /K∞ ) of the maximal abelian pro-p-extension of K∞ , unramified outside primes above S. (ii) The projective (resp., inductive) limit X∞ (resp., A∞ ) of the p-primary parts of the ideal class groups of OKα . 0 (iii) The projective (resp., inductive) limit X∞ (resp., A0∞ ) of the p-primary parts of the ideal class groups of OKα ,Sα , where Sα is the set of primes of Kα above S. These are closely related to RΓIw (K∞ /K, T ) and RΓc,Iw (K∞ /K, T ) for T = Zp , Zp (1) (with R = Zp , R = Λ = Zp [[Γ]]). Can one obtain anything interesting from the general machinery (0.12-13) in this classical setup? (0.14.1) First of all, the spectral sequence 0 Er in (0.12.3) for T = Zp and T = Zp (1) gives very short proofs of the following well-known results (cf. 9.3): (i) The Pontrjagin dual of A0∞ contains no non-zero pseudo-null Λ-submodules. (ii) If the weak Leopoldt conjecture holds for K∞ (i.e. H 2 (KS /K∞ , Qp /Zp ) = 0), then Gal(M∞ /K∞ ) contains no non-zero pseudo-null Λ-submodules. ∼ (0.14.2) For Γ −→ Zp , Iwasawa [Iw] constructed canonical isomorphisms in (Λ Mod)/(pseudo − null) ∼

∼

Ext1Λ (X∞ , Λ) −→ D(A∞ ),

0 Ext1Λ (X∞ , Λ) −→ D(A0∞ ). ∼

One expects analogous statements to hold for arbitrary Γ −→ Zrp (cf. [McCa 2]). Our machinery gives only partial results in this direction, such as the following (cf. 9.4-5): (i) There is a canonical morphism of Λ-modules 0 α0 : X∞ −→ Ext1Λ (D(A0∞ ), Λ).

(ii) Coker(α0 ) is almost pseudo-null in the sense that there is an explicit finite set P of height one prime ideals p ∈ Spec(Λ) such that Coker(α0 )p = 0 for all p ∈ Spec(Λ), ht(p) = 1, p 6∈ P . 0 (iii) The characteristic power series of D(A0∞ ) divides that of X∞ . Slightly weaker results can be proved for X∞ and A∞ . In 1998 the author announced a proof of the fact that Coker(α0 ) is pseudo-null. Unfortunately, the argument for exceptional p ∈ P turned out to be flawed, which means that the claim has to be retracted. (0.14.3) Greenberg’s local conditions have built into them a fundamental base change property (cf. 8.10.1) L

∼ g gf,Iw (K∞ /K, T )⊗ R −→ RΓ RΓf (T ) R

with respect to the augmentation map R −→ R. This can be interpreted as a derived version of Mazur’s “control theorem” for Selmer groups, according to which the canonical map Sel(GK,S , E[p∞ ]) −→ Sel(GK∞ ,S , E[p∞ ])Γ ∼

has finite kernel and cokernel (assuming that Γ −→ Zp and E has good ordinary reduction at all primes dividing p). More generally, there are canonical isomorphisms L

∼ g gf,Iw (K∞ /K, T )⊗ R[[Gal(L/K)]] −→ RΓ RΓf,Iw (L/K, T ) R

(0.14.3.1)

for arbitrary intermediate fields K ⊂ L ⊂ K∞ . (0.14.4) A typical situation in which Mazur’s control theorem fails for the clasical Selmer groups but holds e 1 (−, E[p∞ ]) is the following: E is an elliptic curve defined over Q with for the extended Selmer groups H f multiplicative reduction at p and K∞ /K is the anti-cyclotomic Zp -extension of an imaginary quadratic field K in which p is inert. Note that a trivial zero is again lurking behind this example. (0.14.5) Our duality results also show that, in the classical case when R = Zp and T = T 0 is concentrated gf,Iw (K∞ /K, T ) can often be represented by relatively simple complexes (see in degree zero, the objects RΓ 13

gf,Iw (L/K, T ) and the corresponding height pairings for all subextensions L/K 9.7), which then control RΓ of K∞ /K, thanks to (0.14.3.1). These results were used, in the context of classical Selmer groups, in the work of Mazur and Rubin [M-R 2] on “organizing complexes” in Iwasawa theory of elliptic curves. (0.15) Generalized Cassels-Tate pairings One of the main applications of the duality theory for Selmer complexes is a construction of higherdimensional generalizations of Cassels-Tate pairings (cf. Chapter 10). These pairings are used in 10.7 to prove several versions of the following general principle (for Greenberg’s local conditions): the parity of ranks e 1 (Tλ ) attached to a one-parameter family Tλ of self-dual Galois representations of extended Selmer groups H f ∼ (with respect to a family of skew-symmetric isomorphisms Tλ −→ Tλ∗ (1) respecting the local conditions) is constant. In [N-P], [Ne 3], [Ne 5] and in Chapter 12 we deduce from this principle parity results for ranks of Selmer groups attached to modular forms, elliptic curves and Hilbert modular forms, respectively. On should keep in mind the following topological analogue (see 10.1): if X is a compact oriented topological manifold of (real) dimension 3, then Poincar´e duality with finite coefficients induces a non-degenerate symmetric pairing H 2 (X, Z)tors × H 2 (X, Z)tors −→ Q/Z. (0.15.0) Let us return to the exact sequence (0.13.2), assuming in addition that depth(Rp ) = dim(Rp ) = 1. Under this assumption the first term in (0.13.2) is canonically isomorphic to   e 4−q (K∞ /K, T ∗(1))ι HomR ( H , IR ), f,Iw p (Rp )−tors

p

p

∼

where IR −→ H 0 (ωR )p ⊗R (Frac(Rp )/Rp ) denotes the injective hull of the (Rp )-module Rp /pRp (e.g. p

p

IR = Frac(Rp )/Rp , if Rp is a discrete valuation ring). As a result, one obtains a non-degenerate bilinear p form (cf. 10.3.3, 10.5.5)     e q (K∞ /K, T ) e 4−q (K∞ /K, T ∗ (1))ι H × H −→ IR . f,Iw f,Iw p p (Rp )−tors

(Rp )−tors

p

This pairing is of particular interest for q = 2. In the self-dual case, i.e. when there is a skew-symmetric ∼ ∼ isomorphism Tp −→ T ∗ (1)p compatible with isomorphisms (Tv+ )p −→ (T ∗ (1)+ v )p for all v ∈ Σ, then the induced pairing     e 2 (K∞ /K, T ) e 2 (K∞ /K, T )ι h, i: H × H −→ IR (0.15.0.1) f,Iw f,Iw p p (Rp )−tors

(Rp )−tors

p

is skew-hermitian (cf. 10.3.4.2). (0.15.1) All of the above makes sense in the absence of Γ (i.e. for R = R and p = p), when we obtain bilinear forms (cf. 10.3.2, 10.5.3)     e q (T ) e 4−q (T ∗ (1)) H × H −→ IRp , f f p p Rp −tors

Rp −tors

which can be degenerate (because of the presence of error terms in (0.9.2)). In the self-dual case, the induced pairing     e f2 (T ) e f2 (T ) H × H −→ IRp p p Rp −tors

Rp −tors

is skew-symmetric (cf. 10.2.5). For R = Zp , p = (p) and T = Tp (E) (where E is an elliptic curve with ordinary reduction at all primes above p), we recover essentially the classical Cassels-Tate pairing on the quotient of Sel(GK,S , E[p∞ ]) by its maximal divisible subgroup (combining 10.8.7 with 9.6.7.3 and 9.6.3). Perhaps the simplest non-classical example comes from Hida theory (cf. [N-P]). For simplicity, let us begin with an elliptic curve E over Q, with good ordinary reduction at p (if p = 2, then the following discussion has 14

to be slightly modified). It is known that E is modular [B-C-D-T], hence L(E, s) = L(fE , s) for a newform fE ∈ S2 (Γ0 (N ), Z), where N is the conductor of E. Our assumptions imply that p - N and fE = f2 is a member of a Hida family of ordinary eigenforms (6) fk ∈ Sk (Γ0 (N ), Zp ), for weights k ∈ Z≥2 sufficiently close to 2 in the p-adic space of weights Zp × Z/(p− 1)Z. The Galois representations attached to various fk can be interpolated by a big Galois representation ρ : GQ,S −→ AutR (T ), where S consists of all primes dividing N p (and infinity) and R, T are as in (0.1.2). Replacing T by a suitable twist ([N-P], 3.2.3), one obtains a representation (also denoted by T ) with the following properties: (i) There is a prime ideal P ∈ Spec(R) with ht(P) = 1 such that ∼

(T /PT ) ⊗Zp Qp −→ Vp (E). Moreover, RP is a discrete valuation ring, unramified over ΛP∩Λ . (ii) T is self-dual, i.e. there is a skew-symmetric isomorphism ∼

T −→ T ∗ (1) = HomΛ (T, Λ)(1). (iii) There is a self-dual exact sequence of RP [GQp ]-modules 0 −→ TP+ −→ TP −→ TP− −→ 0, ∼

in which TP± is free of rank one over RP and there is an isomorphism TP+ /PTP+ −→ Vp (E)+ (compatible with that in (i)). gf (GQ,S , TP ) satisfies the base change property 12.7.13.4(i) The corresponding big Selmer complex RΓ L

∼ g gf (TP )⊗RP RP /P −→ RΓ RΓf (Vp (E)),

(0.15.1.1)

which gives an exact cohomology sequence e f1 (TP )/P −→ H e f1 (Vp (E)) −→ H e f2 (TP )[P ] −→ 0, 0 −→ H in which the middle term is equal to the classical Selmer group Sel(GQ,S , Vp (E)). The existence of a canonical non-degenerate skew-symmetric pairing e 2 (TP )RP −tors × H e 2 (TP )RP −tors −→ Frac(RP )/RP H f f then implies the following result (see 12.7.13.5). (0.15.2) Let E be an elliptic curve over Q with a good ordinary reduction at p. Then: (i) There exists a canonical decreasing filtration by Qp -vector spaces on S = Sel(GQ,S , Vp (E)): S = S1 ⊇ S2 ⊇ · · · . (ii) There exist non-degenerate skew-symmetric pairings S i /S i+1 × S i /S i+1 −→ Qp

(i ≥ 1) ∼

depending on the choice of an isomorphism Γ = 1 + qZp −→ Zp , and otherwise canonical (where q = p (resp., q = 4) if p 6= 2 (resp., p = 2)). (iii) The common kernel \ S∞ = Si i≥1 (6)

In this introduction we ignore the phenomenon of p-stabilization. 15

is equal to the “generic subspace of S”   e f1 (TP ) −→ H e f1 (Vp (E)) . S gen = Im H In particular,

dimQp (S) ≡ dimQp (S gen ) (mod 2).

e 1 (V ), where L = Frac(RP ), V = T ⊗R L and H e 1 (V ) = H e 1 (TP ) ⊗RP L is Above, dimQp (S gen ) = dimL H f f f the Selmer group attached to the whole Hida family passing through fE . e 1 of self-dual Galois representations attached to Hilbert A similar result holds for extended Selmer groups H f modular forms (see 12.7.13.5). The results of [N-P] show that certain non-vanishing conjectures for the two-variable p-adic L-function of E would imply (at least for p > 3) that ( 0, if 2 | ords=1 L(E, s) gen dimQp (S ) = 1, if 2 - ords=1 L(E, s) (in the language of 12.7.13.5, e h1f (Q, V 0 ) = 0 (resp., = 1) for infinitely many P 0 , hence dimQp (S gen ) = dim F ∞ is also equal to 0 (resp., to 1)). (0.15.3) Self-duality in Iwasawa theory is more complicated; because of the presence of the involution ι, we obtain skew-hermitian pairings (10.3.4.2(ii)). In an important dihedral case it is possible to get rid of the involution and obtain, as in 0.15.2, skew-symmetric pairings. √ Consider, for example, the following situation. Let K = Q( D), D < 0, be an imaginary quadratic field and K∞ /K the anti-cyclotomic Zp -extension of K. This is a dihedral extension of Q, i.e. Γ+ = Gal(K∞ /Q) = Γ o {1, τ }, where τ 2 = 1,

∼

τ γτ −1 = γ −1

(γ ∈ Γ −→ Zp ).

This implies that, for every R[Γ+ ]-module M , the action of τ ∈ Γ+ induces an isomorphism of R[Γ]-modules ∼

τ : M −→ M ι . e 2 (K∞ /K, Tp (E)), where E is an elliptic curve over Q with good ordinary Applying this remark to M = H f,Iw reduction at p, the non-degenerate skew-hermitian pairing   2 e f,Iw h, i: H (K∞ /K, Tp (E))p

Λp −tors

  2 e f,Iw × H (K∞ /K, Tp (E))p

Λp −tors

−→ Frac(Λp )/Λp

from (0.15.0.1) (where R = Zp , R = Λ = Zp [[Γ]], T = Tp (E), p ∈ Spec(Λ) = Spec(Zp [[Γ]]), ht(p) = 1, p 6= (p)) induces a non-degenerate skew-symmetric pairing   e 2 (K∞ /K, Tp (E))p (, ): H f,Iw

Λp −tors

  e 2 (K∞ /K, Tp (E))p × H f,Iw

Λp −tors

(x, y) = hx, τ yi. This implies that, for each prime ideal p as above, we have
∼ 2 e f,Iw H (K∞ /K, Tp (E))ιp −→ D(Sel(GK∞ ,S , E[p∞ ])) p , with 16

−→ Frac(Λp )/Λp ,

D(Sel(GK∞ ,S , E[p∞ ]))p


Λp −tors

∼

−→ Y ⊕ Y

for some Λp -module Y of finite length. The control theorem 0.14.3 for Selmer groups then gives a congruence analogous to that in (0.15.2)(iii) dimQp (Sel(GK,S , Vp (E))) ≡ rkΛ D(Sel(GK∞ ,S , E[p∞ ])) (mod 2)

(0.15.3.1)

(which holds in a general “dihedral” context; see 10.7.19). If p 6= 2 and K satisfies the following “Heegner condition” (Heeg)

Every prime dividing NE splits in K,

then everything works even for p = (p) (see 10.7.18). A recently proved ([Cor], [Va]) conjecture of Mazur [Maz 2] implies that, assuming (Heeg), the R.H.S. in (0.15.3.1) is equal to 1. As shown in [Ne 3], the congruence (0.15.3.1) for suitably chosen K implies that dimQp (Sel(GQ,S , Vp (E))) ≡ ords=1 L(E, s) (mod 2) (still assuming that E has good ordinary reduction at p). A generalization of this parity result to Hilbert modular forms is proved in Chapter 12. If dimQp (Sel(GK,S , Vp (E))) ≡ 1 (mod 2), the congruence (0.15.3.1) together with the control theorem 0.14.3 imply that dimQp (Sel(GK 0 ,S , Vp (E))) ≥ [K 0 : K], for all finite subextensions K 0 /K of K∞ /K (cf. 10.7.19). The phenomenon of systematic growth of Selmer groups in dihedral extensions was systematically investigated by Mazur and Rubin [M-R 3,4] (cf. 12.12). (0.16) Generalized height pairings (0.16.0) Our formalism also gives a new approach to p-adic (in fact, R-valued) height pairings, which greatly simplifies all previous constructions (due to many people, including Zarhin, Schneider, Perrin-Riou, Mazur and Tate, Rubin, and the author; see the references in [Ne 2] and in Sect. 11.3 below). Let J = Ker(R −→ R) be the augmentation ideal of R; there is a canonical isomorphism ∼

J/J 2 −→ ΓR = Γ ⊗Zp R γ − 1 (mod J 2 ) 7→ γ ⊗ 1

(γ ∈ Γ).

Assume that ΓR is flat over R. For T as in (0.12.1), denote FΓ (T ) by T . The exact triangle T ⊗R J/J 2 −→ T /J 2 T −→ T /J T −→ T ⊗R J/J 2 [1] is canonically isomorphic to T ⊗R ΓR −→ T /J 2 T −→ T −→ T ⊗R ΓR [1].

(0.16.0.1)

Greenberg’s local conditions for T induce similar local conditions for each term in (0.16.0.1); the corresponding Selmer complexes also form an exact triangle in Dfb t (R Mod) gf (T ) ⊗R ΓR −→ RΓ gf (T /J 2 T ) −→ RΓ gf (T ) −→ RΓ gf (T ) ⊗R ΓR [1], RΓ 17

L

gf (T )⊗ (−) to the exact triangle which can also be obtained by applying RΓ R J/J 2 −→ R/J 2 −→ R −→ J/J 2 [1]. The cup product (0.9.3) and the “Bockstein map” gf (T ) −→ RΓ gf (T ) ⊗R ΓR [1] β : RΓ induce a morphism in Dfb t (R Mod) L

gf (T )⊗R RΓ gf (T ∗ (1)) −→ ω ⊗R ΓR [−2], RΓ which is a derived version of the height pairing. In practice, the only interesting component of this pairing is given by e e f1 (T ) ⊗R H e f1 (T ∗ (1)) −→ H 0 (ω) ⊗R ΓR , h:H which can be written as e h(x ⊗ y) = Tr(β(x) ∪ y). This construction makes sense also in the case when Γ is a finite abelian group of exponent pm and R is an Z/pm Z-algebra. It is very likely that there is a similar cohomological formalism behind real-valued heights. There is also a more general version of this construction, which yields pairings L

gf,Iw (L/K, T )⊗R[[Gal(L/K)]] RΓ gf,Iw (L/K, T ∗ (1))ι −→ ω ⊗R R[[Gal(L/K)]] ⊗R Gal(K∞ /L)R [−2] RΓ for arbitrary subextensions L/K of K∞ /K (assuming that Gal(K∞ /L)R is flat over R). (0.16.1) This approach to height pairings has many advantages over traditional treatments even in the simplest case when R = Zp and T = Tp (E), where E is an elliptic curve over Q with ordinary reduction at p. For example, (i) The pairing e e 1 (Tp (E)) ⊗R H e 1 (Tp (E)) −→ Zp h:H f f has values in Zp ; there are no denominators involved. (ii) If E has split multiplicative reduction at p, then e h is a natural height pairing on the extended Selmer group (Mazur, Tate and Teitelbaum [M-T-T] and their followers used an ad-hoc definition). e 1 (Tp (E)) (i.e. the image of H e 1 (K∞ /K, Tp (E))) lie in Ker(β), hence are auto(iii) Universal norms in H f f,Iw matically contained in the kernel of the height pairing. (0.16.2) The definition of e h in terms of the Bockstein map β also sheds new light on the formulas of the Birch and Swinnerton-Dyer type proved by Perrin-Riou [PR 2-5] and Schneider [Sch 2,3]. These formulas ∼ express (for Γ −→ Zp ) the leading term of the “arithmetic p-adic L-function” (i.e. of the characteristic power gf,Iw (K∞ /K, T )) as a product of the determinant of the height pairing e series of detR RΓ h with, essentially, the p-part of the various rational terms appearing in the conjecture of Birch and Swinnerton-Dyer. In our approach, such formulas boil down to the additivity of Euler characteristics in a suitable exact triangle (see 11.7.11). For example, in the classical case R = Zp , the leading term in question is equal, up to a p-adic unit, to the product of det(e h)

3 Y i=1

e i (T ) |(−1) |H f tors

i

with a certain fudge factor. As in the classical case, all this works under the following assumptions: e i (K∞ /K, X) (X = T, T ∗(1)) satisfy suitable finiteness properties. (i) The R-modules H f,Iw 18

(ii) e h is non-degenerate. If (i) holds but (ii) fails, then it is necessary to consider also higher order terms Er in the Bockstein spectral sequence and suitable higher order height pairings that generalize the “derived heights” of Bertolini and Darmon [B-D 1,2]. In [PR 3], Perrin-Riou considered the case of anti-cyclotomic Zp -extensions and proved a suitable Λ-valued version of the formulas alluded to above. This result is also covered by our machinery. Burns and Venjakob [Bu-Ve] combined our approach to the formulas of the Birch and Swinnerton-Dyer type with the formalism of non-commutative Iwasawa theory. (0.17) Parity results The symplectic pairings constructed in Chapter 10 can be used to generalize the parity results proved in [Ne 3] to Hilbert modular forms and abelian varieties of GL(2)-type over totally real number fields. We refer the reader to Sections 12.1-2 for a detailed description of our results, which include, for example, the following generalization of [Ne 3, Thm. A] (see Corollary 12.2.10 below): Theorem. Let F be a totally real number field, F0 /F an abelian 2-extension, E an elliptic curve over F which is potentially modular in the sense of 12.11.3(1) below (7) and p a prime number such that E has potentially ordinary (= potentially good ordinary or potentially multiplicative) reduction at each prime of F above p. Assume that at least one of the following conditions holds: (1) j(E) 6∈ OF . (2) E is modular over F and 2 - [F : Q]. (3) j(E) ∈ OF , E has good ordinary reduction at each prime of F above p, the prime number p is unramified in F0 /Q and p > 3. If E does not have CM, assume, in addition, that Im(GF −→ Aut(E[p])) ⊇ SL2 (Fp ). Then: for each finite Galois extension of odd degree F1 /F0 , rkZ E(F1 ) + corkZp X(E/F1 )[p∞ ] ≡ ords=1 L(E/F1 , s) (mod 2). These parity results can be combined with (generalizations of) (0.15.3.1), giving rise to many situations in which Selmer groups “grow systematically” in the sense of [M-R 3]. See Sect. 12.12 for more details. (0.18) Contents Let us give a brief description of the contents of each chapter of this work. In Chapter 1 we collect the necessary background material from homological algebra. We pay particular attention to signs, as one of our main goals is to construct higher-dimensional generalizations of the Cassels-Tate pairing, and verify that they are skew-symmetric. The reader is strongly advised to skip this chapter and return to it only when necessary. In Chapter 2 we recall the formalism of Grothendieck’s duality theory over (complete) local rings. In Chapter 3 we develop from scratch the formalism of continuous cohomology for what we call (ind)-admissible R[G]-modules. Chapter 4 deals with finiteness results for continuous cohomology of pro-finite groups. In Chapter 5 we deduce from the classical duality results for Galois cohomology of finite Galois modules over local and global fields (due to Tate and Poitou) the corresponding results for big Galois representations. In Chapter 6 we introduce Selmer complexes in an axiomatic setting and prove a duality theorem for them (as a consequence of the Poitou-Tate duality in our formalism). In Chapter 7 we investigate a generalization of unramified cohomology (over local fields) in our set-up. In Chapter 8 we apply Shapiro’s Lemma to deduce duality results in Iwasawa theory from those over number fields. Chapter 9 is devoted to applications to classical Iwasawa theory, namely to p-parts of ideal class groups (resp., of S-ideal class e 1 , Greenberg’s (strict) groups). It also includes comparison results between the extended Selmer groups H f Selmer groups and classical Selmer groups for abelian varieties. In Chapter 10 we construct and study various incarnations of generalized Cassels-Tate pairings. We pay particular attention to the self-dual case, which is important for arithmetic applications. In Chapter 11 we construct generalized p-adic height pairings and relate them to the formulas of the Birch and Swinnerton-Dyer type. In Chapter 12, we apply the results from Chapter 10 to big Galois representations arising from Hida families of Hilbert modular forms of parallel (7)

Potential modularity of E seems to be well-known to the experts [Tay 5]; a proof is expected to appear in a forthcoming thesis of a student of R. Taylor. 19

weight, and also to anticyclotomic Iwasawa theory of CM points on Shimura curves. This allows us to deduce a far-reaching generalization of the parity results from [Ne 3]. (0.19) Directions for further research The fact that Selmer complexes ‘see’ trivial zeros of p-adic L-functions and satisfy the base change properties 0.14.3 and (0.15.1.1) indicates that they – and not the usual Selmer groups – are the correct algebraic counterparts of p-adic L-functions. It would be of some interest, therefore, to reformulate all aspects of Iwasawa theory from this perspective. (0.19.1) Non-commutative Iwasawa theory. It seems very likely that the results discussed in 0.11-13 can be generalized to a fairly large class of non-commutative p-adic Lie groups Γ. One would expect the duality diagram (0.12.1) to hold over R = R[[Γ]] again with ωR = ωR ⊗R R (this time as a complex of R-bimodules) and FΓ , FΓ defined as in 8.3.1. Note that both FΓ (M ) and FΓ (M ) are R-bimodules equipped with an involution compatible with the bimodule structure, and the action of G = GK,S commutes with one of the R-module structures. It seems that this extra structure can be used to generalize the cohomological theory of admissible R[G]-modules to the non-commutative setting. (0.19.2) Local Iwasawa theory. It would be highly desirable to develop a theory of local conditions at primes dividing p that would go beyond Greenberg’s local conditions. One should view Perrin-Riou’s theory [PR 6] interpolating the Bloch-Kato exponential in the local cyclotomic Zp -extension as a first step in this direction. Another challenge is posed by the families of Galois representations arising from Coleman’s theory of rigid analytic families of modular forms. It is clear that in this generality one would have to work with more general coefficient rings R. In fact, there should be a common generalization of 0.19.1 and 0.19.2, perhaps in the context of “Fr´echet-Stein algebras” introduced by Schneider-Teitelbaum [Sch-Te]. One can also envisage a version of the theory involving directly ´etale cohomology of towers of varieties, rather than Galois cohomology. (0.19.3) Euler systems. The machinery of Euler systems is a powerful tool for obtaining upper bounds for the size of (dual) Selmer groups. It would seem natural to incorporate Selmer complexes into this theory, which would allow for the treatment of trivial zeros. In practice, elements of an Euler system are obtained from suitable elements of motivic cohomology to which one applies the p-adic regulator or the p-adic Abel-Jacobi map. It is a natural question whether one can, in the presence of a trivial zero, canonically lift an Euler system from the Selmer group to its extended e 1 . This can be done, for example, for the Euler system of Heegner points in the presence of an version H f “anticyclotomic trivial zero” ([B-D 3], 2.6). (0.20) Miscellaneous (0.20.1) An embryonic version of Selmer complexes appears in [Fo 1] (following a suggestion of Deligne). The first consistent use of derived categories in Iwasawa theory is due to Kato [Ka 1]; his approach has been incorporated into the general formalism of Equivariant Tamagawa Number Conjecture [Bu-Fl 1,2]. Recent articles of Burns-Greither [Bu-Gr], Burns-Venjakob [Bu-Ve], Fukaya-Kato [Fu-Ka] and Mazur-Rubin [M-R 1-3] are also closely related to our framework. (0.20.2) To our great embarrassment, it has proved impossible to keep the length of this work under control, even though much of what we do is just an exercise in linear algebra. This is a consequence of our early decision not to use any homotopical machinery (such as infinite hierarchies of higher-order homotopies) in our treatment of Selmer complexes, only brute force. (0.20.3) The idea of a ‘Selmer complex’ occurred to the author while he was staying at Institut Henri Poincar´e in Paris in spring 1997. It was further developed during stays at the Isaac Newton Institute in Cambridge in spring 1998 and (again) at Institut Henri Poincar´e in spring 2000. During the visits at IHP the author was partially supported by a grant from EPSRC and by the EU research network “Arithmetic Algebraic Geometry”. Main results of this theory were presented in a series of lectures at University of Tokyo in spring 2001; this visit was supported by a fellowship from JSPS. The first version of this work was completed during the author’s visit at Institut de Math´ematiques de Jussieu in Paris in October 2001. The author is grateful to all these institutions for their support. He would also like to thank D. Blasius, D. Burns, O. Gabber, R. Greenberg, H. Hida, U. Jannsen, B. Mazur, K. Rubin, P. Schneider, A.J. Scholl, C. Skinner, 20

R. Taylor and A. Wiles for helpful discussions and inspiring questions, and to C. Cornut, D. Mauger, J. Oesterl´e, L. Orton, J. Pottharst and the referee for pointing out several inaccuracies in the text. (0.20.4) To sum up, this work gives a unified treatment of much of (commutative) Iwasawa theory (8) organized around a small number of simple, but sufficiently general principles. We hope that our attempt to Grothendieckify the subject will help integrate it into a wider landscape of arithmetic geometry.

Selmer groups are dead. Long live Selmer complexes!

(8)

We consider only the algebraic side of the subject. The relation of Selmer complexes to p-adic Lgf,Iw (K∞ /K, T ), whenever defined, functions remains to be explored, but it is natural to expect that detR RΓ should be closely related to a suitable analytic p-adic L-function. 21

1. Homological Algebra: Products and Signs This chapter should be skipped at first reading. Sect. 1.1 and 1.2 collect basic conventions involving signs, tensor products and Hom’s in derived categories. In Sect. 1.3 we define and study abstract pairings between cones. Such pairings will be used in Chapter 6 as a fundamental tool for developing duality theory for Selmer complexes. (1.1) Standard notation and conventions We follow the sign conventions of [B-B-M] (with one important correction; see 1.2.8 below). We fix an abelian category C and work with the corresponding category of complexes C(C). (1.1.1) Translations (= shifts). For n ∈ Z, the translation by n of a complex X (resp., of a morphism of complexes f : X −→ Y ) is given by diX[n] = (−1)n di+n X ,

X[n]i = X n+i ,

f [n]i = f i+n .

(1.1.2) Cones. The cone of a morphism of complexes f : X −→ Y is equal to Cone(f ) = Y ⊕ X[1] with differential diCone(f )

=

!

diY

f i+1

0

−di+1 X

: Y i ⊕ X i+1 −→ Y i+1 ⊕ X i+2 .

There is an exact sequence of complexes j

p

0 −→ Y −→Cone(f )−→X[1] −→ 0, in which j and p are the canonical inclusion and projection, respectively; the corresponding boundary map ∂ : H i (X[1]) = H i+1 (X) −→ H i+1 (Y ) is induced by f i+1 . (1.1.3) Exact (= distinguished) triangles. triangles of the form f

These are isomorphic (in the derived category D(C)) to j

−p

X −→Y −→Cone(f )−→X[1], or, equivalently, to p[−1]

f

j

Cone(f )[−1] −→ X −→Y −→Cone(f ). The translation of an exact triangle f

g

h

X −→Y −→Z −→X[1] is equal to f [1]

g[1]

−h[1]

X[1]−→Y [1]−→Z[1]−−→X[2]. (1.1.4) Exact sequences. For every exact sequence of complexes f

g

0 −→ X −→Y −→Z −→ 0, the morphism of complexes 22

(1.1.4.1)

q : Cone(f ) −→ Z equal to g (resp., to zero) on Y (resp., on X[1]) is a quasi-isomorphism (Qis). The corresponding map in the derived category q

−p

h : Z ←−Cone(f )−→X[1] defines an exact triangle f

g

h

X −→Y −→Z −→X[1] such that the map H i (h) : H i (Z) −→ H i (X[1]) = H i+1 (X) is equal to the coboundary map arising from the original exact sequence (1.1.4.1). Assume that, for each i ∈ Z, the epimorphism in C gi : Y i −→ Z i admits a section si : Z i −→ Y i ,

gi si = id.

Then there is a unique collection of morphisms in C βi : Z i −→ X i+1 characterized by diY si − si diZ = fi+1 βi . As i di+1 X βi = −βi+1 dZ ,

the collection of maps β = (βi ) is a morphism of complexes β : Z −→ X[1]. The morphism of complexes r = (s, −β) : Z −→ Cone(f ) is a section of q, which implies that the morphism of complexes β = −p ◦ r : Z −→ X[1] represents the ‘boundary’ map h in the derived category D(C). (1.1.5) Homotopies. A homotopy a between morphisms of complexes f, g : X −→ Y (i.e. a collection of maps a = (ai : X i+1 −→ Y i ) such that da + ad = g − f ) will be denoted by a : f − g. If u : X 0 −→ X (resp., v : Y −→ Y 0 ) is a morphism of complexes, then a ? u = (ai ◦ ui+1 : (X 0 )i+1 −→ Y i ) (resp., v ? a = (v i ◦ ai : X i+1 −→ (Y 0 )i )) is a homotopy a ? u : f u − gu (resp., v ? a : vf − vg). A second order homotopy α between homotopies a, b : f − g (i.e. a collection of maps α = (αi : X i+2 −→ Y i ) such that αd − dα = b − a) will be denoted by α : a − b. Assume that we are given complexes X • , Y • , Z • and collections of maps h = (hi : X i+1 −→ Y i ), h0 = ((h0 )i : Y i+1 −→ Z i ). Then dh0 + h0 d : Y • −→ Z •

dh + hd : X • −→ Y • , 23

are morphisms of complexes, (dh0 + h0 d) ? h, h0 ? (dh + hd) : 0 −

(dh0 + h0 d) ◦ (dh + hd)

are homotopies and H = h0 h : (dh0 + h0 d) ? h −

h0 ? (dh + hd)

is a 2-homotopy between these homotopies. (1.1.6) Functoriality of cones. ([Ve 2], 3.1) Let tr1 (C) be the category with objects f : X −→ Y (morphisms of complexes in C) and morphisms (g, h, a) f

X

/Y

g

a

 X0

f0

2*  h / Y 0,

where g : X −→ X 0 , h : Y −→ Y 0 are morphisms of complexes and a : f 0 g − composition of (g0 ,h0 ,a0 )

f0

(g,h,a)

f

hf is a homotopy. The

f 00

(X −→Y ) −−→(X 0 −→Y 0 ) −−→ (X 00 −→Y 00 ) is defined as (g 0 g, h0 h, a0 ? g + h0 ? a), where h0 ? a : h0 f 0 g − A morphism

h0 hf , a0 ? g : f 00 g 0 g − f0

f

(g, h, a) : (X −→Y ) −→ (X 0 −→Y 0 ) in tr1 (C) defines a morphism of complexes Cone(g, h, a) : Cone(f ) −→ Cone(f 0 ) given by i

Cone(g, h, a) =

hi

ai

0

i+1

g

! : Y i ⊕ X i+1 −→ Y 0i ⊕ X 0i+1 .

In other words, “Cone” is a functor Cone : tr1 (C) −→ C(C). (1.1.7) Homotopies in tr1 (C). By definition, a homotopy (b, b0 , α) : (g, h, a) −

(g 0 , h0 , a0 )

between two morphisms f0

f

(g, h, a), (g 0 , h0 , a0 ) : (X −→Y ) ⇒ (X 0 −→Y 0 ) in tr1 (C) consists of homotopies b:g−

g0,

b0 : h −

h0

and a second order homotopy α : f 0 ? b + a0 − Such a homotopy in tr1 (C) induces a homotopy 24

b0 ? f + a.

h0 f 0 g.

b0

α

0

−b

! Cone(g 0 , h0 , a0 ).

: Cone(g, h, a) −

(1.1.8) Assume that we are given the following cubic diagram of complexes: A1 E EE EE f1 EE EE E"

α1

 A01

u

/ A2 EE EE f EE 2 α2 E h E *2 EE " / B2

v

B1

β1

DD DD k1 2* DD D f10 DDD !  B10

3;

*2

`

 / A02

u0

m

v0

β

2 DD f 0 DD2 k2 *2 DD DD h0 *2 DD !  / B20

In other words, A1 , . . . , B20 are complexes in C; u, . . . , β2 are morphisms of complexes and h : v ◦ f1 − . . ., k2 : f20 ◦ α2 − β2 ◦ f2 are homotopies.

f2 ◦ u,

Assume, in addition, that the boundary of the cube is trivialized by a 2-homotopy H = (H i : Ai+2 −→ (B20 )i ), 1 i.e. H : v 0 ? k1 + m ? f1 + β2 ? h −

k2 ? u + h0 ? α1 + f20 ? `.

Then the triple (k1 , k2 , H) defines a homotopy (k1 , k2 , H) : (f10 , f20 , h0 ) ◦ (α1 , α2 , `) = (f10 α1 , f20 α2 , h0 ? α1 + f20 ? `) − − (β1 f1 , β2 f2 , m ? f1 + β2 ? h) = (β1 , β2 , m) ◦ (f1 , f2 , h), i.e. the diagram (f1 ,f2 ,h)

Cone(u) −−−−→  (α ,α ,`) y 1 2 (f 0 ,f 0 ,h0 )

1 2 Cone(u0 ) −−−−→

Cone(v)  (β ,β ,m) y 1 2 Cone(v 0 )

is commutative up to homotopy. (1.1.9) A covariant additive functor F : C −→ C 0 induces a functor on complexes F : C(C) −→ C(C 0 ) given by diF (X) = F (diX ). The identity morphisms define (for all n ∈ Z) canonical isomorphisms of complexes ∼

F (X[n]) −→ F (X)[n]. For a contravariant additive functor F : C op −→ C 0 we define F : C(C)op −→ C(C 0 ) by
diF (X) = (−1)i+1 F d−i−1 . X (1.1.10) If G : (C 0 )op −→ C 00 is another contravariant additive functor, then dG(F (X)) = −G(F (dX )). We define an isomorphism of complexes 25

∼

G(F (X)) −→ (G ◦ F )(X) to be equal to (−1)i times the identity morphism in degree i. (1.1.11) Truncations. If X is a complex, we use the usual notation for the truncations σ≤i X

= [· · · X i−2

−→

X i−1

−→

Xi

−→

0

−→

0

· · ·]

τ≤i X

= [· · · X i−2

−→

X i−1

−→

Ker(diX )

−→

0

−→

0

· · ·]

σ≥i X

= [· · ·

0

−→

0

−→

Xi

−→

X i+1

−→

X i+2

· · ·]

τ≥i X

= [· · ·

0

−→

0

−→

Coker(di−1 X )

−→

X i+1

−→

X i+2

· · ·]

(1.2) Tensor products and Hom In the rest of Chapter 1, C = (R Mod) will be the category of modules over a commutative ring R. If X • is a complex and x ∈ X i , we denote the degree of x by x = i. • (1.2.1) For complexes X = X • , Y = Y • we define the complexes HomR (X, Y ) and X ⊗R Y by Y HomnR (X, Y ) = HomR (X i , Y i+n )

i∈Z n

(X ⊗R Y ) =

M

(X i ⊗R Y n−i ),

i∈Z

with differentials df = d ◦ f + (−1)f−1 f ◦ d d(x ⊗ y) = dx ⊗ y + (−1)x x ⊗ dy. If Y = Y 0 is concentrated in degree zero, then Hom•R (X, Y ) = F (X) for F (−) = HomR (−, Y 0 ) (with the sign conventions of 1.1.9). If Y is a bounded (resp., bounded below) complex of injective R-modules and X is any (resp., bounded above) complex of R-modules, then Hom•R (X, Y ) represents RHomR (X, Y ). An element f ∈ Hom0R (X, Y ) satisfies df = 0 ⇐⇒ f is a morphism of complexes f : X −→ Y f = dg ⇐⇒ g is a homotopy g : 0 −

f.

(1.2.1.1)

•,naive There is also a “naive” version HomR (X, Y ) of Hom•R (X, Y ), in which the differential of f : X i −→ Y j is equal to

dnaive f = d ◦ f + (−1)j f ◦ d. (1.2.2) Morphisms of complexes u : X −→ X 0 , v : Y −→ Y 0 induce morphisms • Hom• (u, v) : HomR (X 0 , Y ) −→

f

7→

• HomR (X, Y 0 )

v◦f ◦u

and u ⊗ v : X ⊗R Y

−→

X 0 ⊗R Y 0

7→

u(x) ⊗ v(y)

x⊗y 26

(1.2.3) Tensor products of complexes admit various symmetries, such as Associativity isomorphism (X ⊗R Y ) ⊗R Z

−→

∼

X ⊗R (Y ⊗R Z)

(x ⊗ y) ⊗ z

7→

x ⊗ (y ⊗ z)

Transposition isomorphism s12 : X ⊗R Y

−→

∼

Y ⊗R X

x⊗y

7→

(−1)xy y ⊗ x

Another transposition isomorphism ∼

(X ⊗R Y ) ⊗R (A ⊗R B)

7→

(−1)ay (x ⊗ y) ⊗ (a ⊗ b)

s23 : (X ⊗R A) ⊗R (Y ⊗R B) −→ (x ⊗ a) ⊗ (y ⊗ b)

(1.2.4) Lemma. The following diagram is commutative: s

(X ⊗R Y ) ⊗R (A ⊗R B)  s ⊗s y 12 12

s

(Y ⊗R X) ⊗R (B ⊗R A)

23 (X ⊗R A) ⊗R (Y ⊗R B) −→  s12 y 23 (Y ⊗R B) ⊗R (X ⊗R A) −→

Proof. (−1)(x+a)(y+b) (−1)xb = (−1)ay (−1)xy (−1)ab . (1.2.5) With the sign conventions of 1.2.1, the canonical isomorphism ∼

• Hom•R (X, Y )[n] −→ HomR (X, Y [n])

does not involve any signs, i.e. it is given in all degrees by the identity maps. (1.2.6) The adjunction morphism on the level of complexes • adj : HomR (X ⊗R Y, Z) −→

f

7→

• • HomR (X, HomR (Y, Z))

(x 7→ (y 7→ f (x ⊗ y)))

induces a morphism of R-modules • HomC(R Mod) (X ⊗R Y, Z) −→ HomC(R Mod) (X, HomR (Y, Z)),

which preserves homotopy classes (by (1.2.1.1)). Both of these maps are monomorphisms; they are isomorphisms, provided X and Y are bounded above and Z is bounded below. (1.2.7) The evaluation maps • ev1 : HomR (X, Y ) ⊗R X

−→

Y

f ⊗x

7→

f (x)

and ev2 : X ⊗R Hom•R (X, Y )

−→

Y

x⊗f

7→

(−1)xf f (x)

are morphisms of complexes making the following diagram commutative: 27

ev

Hom•R (X, Y ) ⊗R X  s y 12

1 −→

X ⊗R Hom•R (X, Y )

2 −→

Y k

ev

(1.2.7.1)

Y

We have adj(ev1 ) = id under the adjunction morphism • • • adj : HomC(R Mod) (HomR (X, Y ) ⊗R X, Y ) −→ HomC(R Mod) (HomR (X, Y ), HomR (X, Y )).

More generally, there are evaluation morphisms • ev1 : HomR (X, Y ) ⊗R Hom•R (W, X) −→

Hom•R (W, Y )

g⊗f

7→

g◦f

• • ev2 : HomR (W, X) ⊗R HomR (X, Y )

−→

Hom•R (W, Y )

f ⊗g

7→

(−1)fg g ◦ f

satisfying ev1 = ev2 ◦ s12 . Another generalization of ev1 is given by the morphism Hom•R (X, Y ) ⊗R (X ⊗R Z) −→ f ⊗ (x ⊗ z)

7→

Y ⊗R Z f (x) ⊗ z,

which corresponds to

• • • HomR (HomR (X, Y ), HomR (X ⊗R Z, Y ⊗R Z))

f 7→ f ⊗ idZ under the adjunction map. (1.2.8) The biduality morphism • εY : X −→ Hom•R (HomR (X, Y ), Y )

is given on x ∈ X i by x 7→ (−1)ik x∗∗ k


k∈Z

,

where Xi

−→

HomR (HomR (X i , Y i+k ), Y i+k )

x

7→

(x∗∗ k : f 7→ f (x))

is the usual biduality map. This corrects a sign error in ([B-B-M], 0.3.4.2) - also discovered by B. Conrad [Con] - where the authors give an erroneous sign (−1)i instead of (−1)ik ; that would not make εY a morphism of complexes. We have adj(ev2 ) = εY under the adjunction morphism • adj : HomC(R Mod) (X ⊗R Hom•R (X, Y ), Y ) −→ HomC(R Mod) (X, Hom•R (HomR (X, Y ), Y )).

The statements of Lemmas 1.2.9-13,16 below follow immediately from the definitions; we leave the details to the reader (for the homotopy versions of 1.2.11-13 it is sufficient to recall that adj preserves homotopy classes). 28

(1.2.9) Lemma. The following diagram (and a symmetric diagram, in which the roles of ev1 and ev2 are interchanged) is commutative: ev

2 −→

X ⊗R Hom•R (X, Y )  ε ⊗id yY

Y k

ev1

• • HomR (Hom•R (X, Y ), Y ) ⊗R HomR (X, Y ) −→

Y.

(1.2.10) Lemma. Assume we are given a morphism of complexes f : X −→ Hom•R (X, Y ); denote the composite morphism of complexes Hom• (f,idY )

ε

Y • • • X −→Hom R (HomR (X, Y ), Y )−−−−−−→HomR (X, Y )

by g. Then the following diagram is commutative: idX ⊗g

X ⊗R X  f ⊗id Y y

−−−−→

• X ⊗R HomR (X, Y )  ev y 2

ev

1 −→

Hom•R (X, Y ) ⊗R X

Y.

(1.2.11) Lemma. If the following diagram of morphisms of complexes is commutative (resp., commutative up to homotopy) id⊗f 0 X ⊗R Y 0 −−−−→ X ⊗R X 0   f ⊗id g y y h

Y ⊗R Y 0 so is

−→

adj(g)

Hom•R (X 0 , Z)   yHom• (f 0 ,id)

X −−→  f y Y

adj(h)

−−→

Z,

• HomR (Y 0 , Z).

(1.2.12) Lemma. If the following diagram of morphisms of complexes is commutative (resp., commutative up to homotopy) u X ⊗R Y −→ Z   f ⊗g  y yh X 0 ⊗R Y 0 so is

adj(u)

X −−→  f y X0

adj(u0 )

−−→

u0

−→

Z 0,

Hom• (id,h)

• HomR (Y, Z)

• HomR (Y, Z 0 )

−−−−−−→

k Hom•R (Y 0 , Z 0 )

Hom• (g,id)

Hom•R (Y, Z 0 ).

−−−−−−→

(1.2.13) Lemma. Let A, B, B 0 , U, U 0 , C be complexes of R-modules and f

A ⊗R B −→U,

f0

b

A ⊗R B 0 −→U 0 ,

B 0 ⊗R C −→B,

u

U 0 ⊗R C −→U

morphisms of complexes. If the diagram (A ⊗R B 0 ) ⊗R C

∼

/ A ⊗R (B 0 ⊗R C)

id⊗b

/ A ⊗R B

f 0 ⊗id

f

 U 0 ⊗R C

u

29

 /U

is commutative (resp., commutative up to homotopy), so is adj(f )

A

• / HomR (B, U )

adj(f 0 )

 Hom•R (B 0 , U 0 )

Hom• R (b,id)

  Hom• R (id,u) • / HomR Hom•R (B 0 ⊗R C, U 0 ⊗R C) (B 0 ⊗R C, U ) (1.2.14) Lemma. If

λ

Z   yh

µ

Z

X ⊗R Y −→  s ◦(f ⊗g) y 12 Y ⊗R X

−→

is a commutative diagram of morphisms of complexes, so are adj(µ)◦g

Y  ε yZ

−−−−→ Hom• (adj(λ),h)

Hom•R (Hom•R (Y, Z), Z) −−−−−−−−→ and

• HomR (X, Z)  Hom• (f,id) y

Hom•R (X, Z).

adj(λ)

X  ε ◦f yZ

−−−−→ Hom• (adj(µ)◦g,id)

Hom•R (Hom•R (X, Z), Z) −−−−−−−−−−→

• HomR (Y, Z)  Hom• (id,h) y

Hom•R (Y, Z).

Proof. Let y ∈ Y j , x ∈ X i . Then Hom• (f, id) ◦ adj(µ) ◦ g(y) sends x to µ(g(y) ⊗ f (x)) = (−1)ij h(λ(x ⊗ y)). In the notation of 1.2.8, the only component of εZ (y) = ((−1)jk yk∗∗ )k∈Z contributing to Hom• (adj(λ), h) ◦ εZ (y)(x) is (−1)ji yi∗∗ ; its image in Hom•R (X, Z) also sends x to (−1)ij h(λ(x ⊗ y)). A similar argument works for the second diagram: Hom• (id, h) ◦ adj(λ)(x) sends y to h(λ(x ⊗ y)) = (−1)ij µ(g(y) ⊗ f (x)), while the only • ij ∗∗ component of εZ ◦ f (x) = ((−1)ik f (x)∗∗ k )k∈Z contributing to Hom (adj(µ) ◦ g, id) ◦ εZ (f (x)) is (−1) f (x)j , • ij the image of which in HomR (Y, Z) sends y to (−1) µ(g(y) ⊗ f (x)). (1.2.15) For each n ∈ Z, the following formulas define isomorphisms of complexes: ∼

sn : X • [n] ⊗R Y • −→ (X • ⊗R Y • ) [n] x ⊗ y 7→ x ⊗ y ∼

s0n : X • ⊗R (Y • [n]) −→ (X • ⊗R Y • ) [n] x ⊗ y 7→ (−1)nx x ⊗ y ∼

• tn : Hom•R (X • , Y • ) −→ HomR (X • [n], Y • [n])

f 7→ (−1)nf f. (1.2.16) Lemma. Given a morphism of complexes u : A• ⊗R B • −→ C • and n ∈ Z, put s0

u[n]

n v = u[n] ◦ s0n : A• ⊗R (B • [n])−−→ (A• ⊗R B • ) [n]−−→C • [n].

Then adj(v) is equal to the composite map adj(u)

t

n • • • A• −−→Hom•R (B • , C • )−→Hom R (B [n], C [n]).

30

(1.2.17) Tensor product of homotopies Assume that fi : X −→ X 0 ,

gi : Y −→ Y 0

(i = 1, 2)

are morphisms of complexes and u : f1 −

f2 ,

v : g1 −

g2

homotopies between them. Then the formulas (u ⊗ v)1 (x ⊗ y) = u(x) ⊗ g1 (y) + (−1)x f2 (x) ⊗ v(y) (u ⊗ v)2 (x ⊗ y) = u(x) ⊗ g2 (y) + (−1)x f1 (x) ⊗ v(y) define two homotopies (u ⊗ v)j : f1 ⊗ g1 −

f2 ⊗ g2

(j = 1, 2)

between the morphisms fi ⊗ gi : X ⊗R Y −→ X 0 ⊗R Y 0 . The formula α(x ⊗ y) = (−1)x u(x) ⊗ v(y) defines a second order homotopy α : (u ⊗ v)1 −

(u ⊗ v)2 .

(1.2.18) If X, Y are complexes of R-modules, then the maps x ⊗ y 7→ (−1)x x ⊗ y,

y ⊗ x 7→ y ⊗ x

define isomorphisms of complexes ∼

∼

X ⊗R (Y [1]) −→ (X ⊗R Y )[1],

(Y [1]) ⊗R X −→ (Y ⊗R X)[1],

which make the following diagram commutative: X ⊗R (Y [1])  s y 12

−→

∼

(X ⊗R Y )[1]  s [1] y 12

(Y [1]) ⊗R X

−→

∼

(Y ⊗R X)[1].

(1.2.19) Lemma. Assume that the following commutative diagram of morphisms of complexes has exact rows and columns: 0 0 0       y y y ρA

0

−→

A00 −→   00 yi

0

−→

B 00 −→   00 yj

0

−→

C 00   y

ρB

0

ρC

−→

σ

A −→   yi

0

σ

B −→  j y

0

σ

C   y

0

A A0 −→  0 yi B B 0 −→  0 yj

C0   y 0 31

C −→

0

−→

Then the diagram H q−1 (C)   y∂ H q (A)   yi

σ

A H q (A0 ) −→

∂

H q−1 (C)  −∂ y

C −→

H q (A)

A −→

∂

H q (C 00 )   00 y∂

ρC

−→ ρA

H q+1 (A00 ) −→

σ

H q (B)  j y

σ

H q (C)   y∂

σ

H q+1 (A)

H q (B 0 )  0 yj

B −→

H q (C 0 )   0 y∂

C −→

A H q+1 (A0 ) −→

is also commutative, and if [c00 ] ∈ H q (C 00 ), [b0 ] ∈ H q (B 0 ) are cohomology classes satisfying ρC [c00 ] = j 0 [b0 ], then there is a unique coset [a] + Im(∂) ∈ H q (A) + Im(∂) such that i[a] = σB [b0 ]. This coset satisfies ∂A [a] + ∂ 00 [c00 ] ∈ Im(∂A ∂) = Im(∂ 00 ∂C ). Above, ∂, ∂ 0 , ∂ 00 , ∂A and ∂C denote coboundary maps attached to the original diagram. Proof. This is a well-known fact, which can be verified by an explicit calculation. (1.3) Products In this section we construct products in a slightly more general context than considered by Niziol([Ni], Prop. 3.1). The main difference is that we allow certain diagrams to commute only up to homotopy. In what follows, R is a commutative ring and all complexes are complexes of R-modules. (1.3.1) Assume we are given the following data: (1.3.1.1) Complexes Aj , Bj , Cj (j = 1, 2, 3). (1.3.1.2) Morphisms of complexes fj gj Aj −→Cj ←−Bj (1.3.1.3) Morphisms of complexes

(j = 1, 2, 3).

∪A : A1 ⊗R A2 −→ A3 ∪B : B1 ⊗R B2 −→ B3 ∪C : C1 ⊗R C2 −→ C3

(1.3.1.4) A pair h = (hf , hg ) of homotopies hf : ∪C ◦(f1 ⊗ f2 ) −

f3 ◦ ∪A

hg : ∪C ◦(g1 ⊗ g2 ) −

g3 ◦ ∪B

We define new complexes Ej by fj −gj

Ej = Cone (Aj ⊕ Bj −−→Cj ) [−1],

(j = 1, 2, 3)

i.e. Ejn = Anj ⊕ Bjn ⊕ Cjn−1 , d(aj , bj , cj ) = (daj , dbj , −fj (aj ) + gj (bj ) − dcj ). An element ej = (aj , bj , cj ) has degree ej = aj = bj = 1 + cj . 32

(1.3.2) Proposition. Given the data 1.3.1.1–4, then (i) For every r ∈ R the formula (a1 , b1 , c1 ) ∪r,h (a2 , b2 , c2 ) = (a1 ∪A a2 , b1 ∪B b2 , c1 ∪C (rf2 (a2 ) + (1 − r)g2 (b2 ))+ +(−1)a1 ((1 − r)f1 (a1 ) + rg1 (b1 )) ∪C c2 − (hf (a1 ⊗ a2 ) − hg (b1 ⊗ b2 ))) defines a morphism of complexes ∪r,h : E1 ⊗R E2 −→ E3 . (ii) For r1 , r2 ∈ R, the formula k ((a1 , b1 , c1 ) ⊗ (a2 , b2 , c2 )) = (0, 0, (−1)a1 (r1 − r2 ) c1 ∪C c2 ) defines a homotopy k : ∪r1 ,h − ∪r2 ,h . (iii) If h0 = (h0f , h0g ) is another pair of homotopies as in (1.3.1.4), then ∪r,h0 − ∪r,h : (a1 , b1 , c1 ) ⊗ (a2 , b2 , c2 ) 7→ (0, 0, (hf − h0f )(a1 ⊗ a2 ) − (hg − h0g )(b1 ⊗ b2 )). If α : hf −

h0f , β : hg −

h0g is a pair of second order homotopies, then the formula

k ((a1 , b1 , c1 ) ⊗ (a2 , b2 , c2 )) = (0, 0, α(a1 ⊗ a2 ) − β(b1 ⊗ b2 )) defines a homotopy k : ∪r,h −

∪r,h0 .

Proof. Explicit calculation (cf. [Ni], Prop. 3.1). (1.3.3) Functoriality of products Assume we are given another piece of data as in (1.3.1–4), namely morphisms of complexes e fej gj ej −→ ej ←− ej , A C B ej . A morphism between e ∗ (for ∗ = A, B, C) and homotopies e products ∪ h = (e hfe, e he ), yielding complexes E g the data ej , B ej , C ej , fej , gej , ∪ e∗, e (Aj , Bj , Cj , fj , gj , ∪∗ , h) −→ (A h) consists of the following: (1.3.3.1) Morphisms of complexes ej αj : Aj −→ A ej βj : Bj −→ B

(j = 1, 2, 3)

ej γj : Cj −→ C (1.3.3.2) Homotopies uj : fej ◦ αj −

γj ◦ f j

vj : e g j ◦ βj −

γj ◦ g j

(j = 1, 2, 3)

(1.3.3.3) Homotopies e A ◦ (α1 ⊗ α2 ) − α3 ◦ ∪A kα : ∪ e B ◦ (β1 ⊗ β2 ) − β3 ◦ ∪B kβ : ∪ e C ◦ (γ1 ⊗ γ2 ) − γ3 ◦ ∪C kγ : ∪ 33

(1.3.3.4) A second order homotopy Kf between 0 and the composition of the six homotopies attached to the faces of the following cubic diagram: ∪A

A1 ⊗ AL2 LLL LLαL1 ⊗α2 LLL LL & e1 ⊗ A e2 f1 ⊗f2 A

/ A3 f3

e ∪A

DD DD DDα3 DD kα *2 DD ! /A e3

3;  / C3 fe3 DD γ3 eh DD f e DD u kγ DD *2 D  D 3 !  /C e3 eC ∪

fe1 ⊗fe2  ∪C C1 ⊗ CL2 LLL LL(u1 ⊗u2 )1 LL  γ1 ⊗γ2 LLL L&  e1 ⊗ C e2 C

hf

*2

In other words, Kf is a collection of maps   e3 )i−2 Kf = Kfi : (A1 ⊗R A2 )i −→ (C satisfying e C ? (u1 ⊗ u2 )1 , dKf − Kf d = −kγ ? (f1 ⊗ f2 ) − γ3 ? hf + u3 ? ∪A + fe3 ? kα + e hfe ? (α1 ⊗ α2 ) − ∪ i.e. trivializing the boundary of the cube (the homotopy (u1 ⊗ u2 )1 was defined in 1.2.17). (1.3.3.5) A second order homotopy   e3 )i−2 Kg = Kgi : (B1 ⊗R B2 )i −→ (C satisfying an analogous condition, with (A, f, α, u) being replaced by (B, g, β, v): e C ? (v1 ⊗ v2 )1 , dKg − Kg d = −kγ ? (g1 ⊗ g2 ) − γ3 ? hg + v3 ? ∪B + ge3 ? kβ + e he ? (β1 ⊗ β2 ) − ∪ g ∪B

B1 ⊗ BL2 LLL LLβL1 ⊗β2 LL LLL & g1 ⊗g2 e1 ⊗ B e2 B g1 ⊗e g2 e  ∪C C1 ⊗ CL2 LL LLL(v1 ⊗v2 )1 LL  γ1 ⊗γ2 LLL L&  e1 ⊗ C e2 C

/ B3 DD DD DDβ3 g3 kβ DD *2 DD ! /B e3

∪B e

3;  / C3 e g3 DD γ3 eh DD eg DD v kγ DD *2 D  D 3 !  /C e3 e ∪C

hg

*2

(1.3.4) Proposition. (i) Given the data 1.3.3.1–2, then the formula ϕj (aj , bj , cj ) = (αj (aj ), βj (bj ), γj (cj ) + uj (aj ) − vj (bj )) defines a morphism of complexes ej ϕj : Ej −→ E

(j = 1, 2, 3). 34

(ii) Given the data 1.3.3.1–5 and r ∈ R, the formula H ((a1 , b1 , c1 ) ⊗ (a2 , b2 , c2 )) = (kα (a1 ⊗ a2 ), kβ (b1 ⊗ b2 ), −kγ (c1 ⊗ (rf2 (a2 ) + (1 − r)g2 (b2 ))+ e C (ru2 (a2 ) + (1 − r)v2 (b2 ))− +(−1)a1 ((1 − r)f1 (a1 ) + rg1 (b1 )) ⊗ c2 ) + (−1)a1 γ1 (c1 )∪

e C (γ2 (c2 ) + u2 (a2 ) − v2 (b2 )) − Kf (a1 ⊗ a2 ) + Kg (b1 ⊗ b2 )) −(−1)a1 ((1 − r)u1 (a1 ) + rv1 (b1 ))∪ defines a homotopy e e ◦ (ϕ1 ⊗ ϕ2 ) − H :∪ r,h

ϕ3 ◦ ∪r,h ,

i.e. the diagram ∪r,h

E1 ⊗R E2 −−→  ϕ ⊗ϕ y 1 2 ∪ e r,e h e1 ⊗R E e2 −−→ E

E3  ϕ3 y e3 E

is commutative up to homotopy. Proof. The first part is a special case of 1.1.6, while (ii) can be proved by a tedious, but routine calculation. (1.3.5) Transpositions Assume that, in addition to the data (1.3.1–4), we are given the following objects: (1.3.5.1) Morphisms of complexes TA : Aj −→ Aj TB : Bj −→ Bj TC : Cj −→ Cj

(j = 1, 2, 3)

(1.3.5.2) Morphisms of complexes ∪0A : A2 ⊗R A1 −→ A3 ∪0B : B2 ⊗R B1 −→ B3 ∪0C : C2 ⊗R C1 −→ C3

(1.3.5.3) A pair h0 = (h0f , h0g ) of homotopies h0f : ∪0C ◦(f2 ⊗ f1 ) −

f3 ◦ ∪0A

h0g : ∪0C ◦(g2 ⊗ g1 ) −

g3 ◦ ∪0B

(1.3.5.4) Homotopies Uj : fj ◦ TA −

TC ◦ fj

Vj : gj ◦ TB −

TC ◦ gj

(j = 1, 2, 3)

(1.3.5.5) Homotopies tα : ∪0A ◦s12 ◦ (TA ⊗ TA ) −

TA ◦ ∪A

tβ : tγ :

TB ◦ ∪B TC ◦ ∪C

∪0B ∪0C

◦s12 ◦ (TB ⊗ TB ) − ◦s12 ◦ (TC ⊗ TC ) − 35

(1.3.5.6) A second order homotopy Hf trivializing the boundary of the following cube: ∪A

A1 ⊗ AM2 MMM MMTMA ⊗TA MMM MMM & f1 ⊗f2 A1 ⊗ A2

∪0A ◦s12

f1 ⊗f2

 ∪C C1 ⊗ CM2 MMM 1 ⊗U2 )1 MM(U MMM TC ⊗TC MM M M&  C1 ⊗ C2

/ A3 EE EE EETA f3 EE tα )1 EE " / A3

2:  / C3 f3 EE TC 0 E hf ?s12 EE E tγ EE *2 E EU3 "  / C3 , 0 hf

)1

∪C ◦s12

i.e. satisfying dHf − Hf d = −tγ ? (f1 ⊗ f2 ) − TC ? hf + U3 ? ∪A + f3 ? tα + h0f ? (s12 ◦ (TA ⊗ TA )) − (∪0C ◦ s12 ) ? (U1 ⊗ U2 )1 . (1.3.5.7) A second order homotopy Hg trivializing the boundary of an analogous cube in which (A, α, f ) are replaced by (B, β, g). With these data, the formula (a2 , b2 , c2 ) ∪0r,h0 (a1 , b1 , c1 ) = (a2 ∪0A a1 , b2 ∪0B b1 , c2 ∪0C (rf1 (a1 ) + (1 − r)g1 (b1 ))+ +(−1)a2 ((1 − r)f2 (a2 ) + rg2 (b2 )) ∪0C c1 − (h0f (a2 ⊗ a1 ) − h0g (b2 ⊗ b1 ))) (for a fixed r ∈ R) defines a morphism of complexes ∪0r,h0 : E2 ⊗R E1 −→ E3 . The complexes ej = Aj , B ej = Bj , C ej = Cj , A morphisms of complexes fej = fj , gej = gj , αj = TA , βj = TB , γj = TC , e A = ∪0A ◦ s12 , ∪ e B = ∪0B ◦ s12 , ∪ e C = ∪0C ◦ s12 , uj = Uj , vj = Vj , ∪ homotopies

kα = tα , kβ = tβ , kγ = tγ , e hf = h0f ? s12 , e hg = h0g ? s12

and second order homotopies K f = Hf , K g = H g satisfy the conditions (1.3.3.1–5). Applying Proposition 1.3.4 and observing that the following diagram of morphisms of complexes e ∪ r,e h E1 ⊗R E2 −−−−→ E3  s k y 12 E2 ⊗R E1

∪01−r,h0

−−−−→

is commutative, we obtain the following statement. 36

E3

(1.3.6) Proposition. (i) Given the data 1.3.5.1 and 1.3.5.4, then the formula Tj (aj , bj , cj ) = (TA (aj ), TB (bj ), TC (cj ) + Uj (aj ) − Vj (bj )) defines a morphism of complexes Tj : Ej −→ Ej . (ii) Given the data 1.3.5.1–7 and r ∈ R, the diagram ∪r,h

E1 ⊗R E2 −−−−→  s ◦(T ⊗T ) y 12 1 2

E3  T y 3

E2 ⊗R E1

E3

∪01−r,h0

−−−−→

is commutative up to homotopy. (1.3.7) Corollary. Under the assumptions of Proposition 1.3.6(ii), the following diagrams are commutative up to homotopy: adj(∪01−r,h0 )◦T2 • E2 −−−−−−−−−−→ HomR (E1 , E3 )   εE Hom• (T ,id) 1 y 3 y Hom• (adj(∪r,h ),T3 )

• Hom•R (HomR (E2 , E3 ), E3 )

−−−−−−−−−−→

• HomR (E1 , E3 )

adj(∪r,h )

E1  ε ◦T y E3 1

−−−−−−−−−−→ Hom• (adj(∪01−r,h0 )◦T2 ,id)

• HomR (Hom•R (E1 , E3 ), E3 )

−−−−−−−−−−−−−−→

Hom•R (E2 , E3 )  Hom• (id,T ) 3 y Hom•R (E2 , E3 ).

Proof. Apply a homotopy version of Lemma 1.2.14 to the diagram in Proposition 1.3.6(ii). (1.3.8) Bockstein maps Assume that, in addition to the data (1.3.1–4), we are given an R-module ΓR and the following objects: (1.3.8.1) Morphisms of complexes βj,Z : Zj −→ Zj [1] ⊗R ΓR

(j = 1, 2; Z = A, B, C)

(1.3.8.2) Homotopies uj : fj [1] ◦ βj,A −

βj,C ◦ fj

vj : gj [1] ◦ βj,B −

βj,C ◦ gj

(j = 1, 2).

Above, the map fj [1] ⊗ id : Aj [1] ⊗R ΓR −→ Cj [1] ⊗R ΓR is abbreviated as fj [1] (and similarly for gj [1]). (1.3.8.3) Homotopies hZ : ∪Z [1] ◦ (id ⊗ β2,Z ) −

∪Z [1] ◦ (β1,Z ⊗ id)

(Z = A, B, C).

Again, we write ∪Z [1] instead of ∪Z [1] ⊗ id : (Z1 ⊗R (Z2 [1])) ⊗R ΓR −→ Z3 [1] ⊗R ΓR . 37

(1.3.8.4) A second order homotopy Hf trivializing the boundary of the following cubic diagram: β1,A ⊗id

/ A1 [1] ⊗ A2 ⊗ ΓR QQQ QQQ ∪ [1] QQAQ f1 [1]⊗f2 QQQ +3 QQQ hA ( ∪A [1] / A3 [1] ⊗ ΓR

A1 ⊗ A2P PPP PPPid⊗β2,A PPP PPP PP( f1 ⊗f2 A1 ⊗ (A2 [1]) ⊗ ΓR

u1 ⊗f2

f1 ⊗(f2 [1])

 β1,C ⊗id C1 ⊗ C2P PPP PPP f1 ⊗u2 PPP P id⊗β2,C PP P P(  C1 ⊗ (C2 [1]) ⊗ ΓR

2:  / C1 [1] ⊗ C2 ⊗ ΓR f3 [1] QQ∪ QQCQ[1] h [1] f )1 QQQ QQQ hf [1] Q+3 QQ hC Q(  / C3 [1] ⊗ ΓR ,

x

∪C [1]

i.e. such that dHf − Hf d = ∪C [1] ? (u1 ⊗ f2 ) + f3 [1] ? hA + hf [1] ? (id ⊗ β2,A ) − ∪C [1] ? (f1 ⊗ u2 )− −hC ? (f1 ⊗ f2 ) − hf [1] ? (β1,A ⊗ id). Above, we implicitly use the canonical isomorphisms from 1.2.18. (1.3.8.5) A second order homotopy Hg trivializing the boundary of an analogous cube in which (A, f, u) are replaced by (B, g, v). (1.3.9) Proposition. (i) Given the data 1.3.8.1–2, the formula βj,E (aj , bj , cj ) = (βj,A (aj ), βj,B (bj ), −βj,C (cj ) − uj (aj ) + vj (bj )) defines a morphism of complexes βj,E : Ej −→ Ej [1] ⊗R ΓR

(j = 1, 2).

(ii) Given the data 1.3.8.1–5 and r ∈ R, the diagram E1 ⊗R E2  id⊗β E,2 y E1 ⊗R (E2 [1]) ⊗R ΓR

βE,1 ⊗id

−−−−→ ∪r,h [1]

−−−−→

E1 [1] ⊗R E2 ⊗R ΓR  ∪ [1] y r,h E3 [1] ⊗R ΓR

is commutative up to homotopy. Proof. The proof is analogous to that of Proposition 1.3.4. In (ii), we again use the canonical isomorphisms 1.2.18. (1.3.10) Proposition. Given the data 1.3.5.1–7, 1.3.8.1–5 and r ∈ R, the diagram E1 ⊗R E2  s12 y E2 ⊗R E1

βE,1 ⊗id

−−−−→ E1 [1] ⊗R E2 ⊗R ΓR βE,2 ⊗id

−−−−→ E2 [1] ⊗R E1 ⊗R ΓR

∪r,h [1]

−−−−→ T2 [1]⊗T1

−−−−→

E3 [1] ⊗R ΓR

E2 [1] ⊗R E1 ⊗R ΓR

T3 [1]

−−−−→ ∪01−r,h0 [1]

−−−−→

E3 [1] ⊗R ΓR k E3 [1] ⊗R ΓR

is commutative up to homotopy (above, Tj are as in Proposition 1.3.6(i)). Proof. Combine Proposition 1.3.6(ii) and 1.3.9(ii) (using the canonical isomorphisms 1.2.18). 38

(1.3.11) The morphisms βj,X arise naturally in the following context (for simplicity, we suppress the index j from the notation). Assume that ρA

0

−→

A00 −→   00 yf

0

−→

C 00 −→ x  00 g

0

−→

B 00

σ

A −→  f y

0

σ

C −→ x g 

0

σ

B

0

A A0 −→   0 yf

ρC

C C 0 −→ x  0 g

ρB

B0

−→

B −→

−→

is a commutative diagram of morphisms of complexes with exact rows. Assume, in addition, that in each degree i ∈ Z the epimorphism i σX : (X 0 )i −→ X i

(X = A, B, C)

siX : X i −→ (X 0 )i

(X = A, B, C).

admits a section

Writing f −g

E = Cone(A ⊕ B −−→C)[−1] (and similarly for E 0 , E 00 ), then the maps ρiE = (ρiA , ρiB , ρi−1 C ),

i−1 i i i σE = (σA , σB , σC )

define an exact sequence of complexes ρE

σ

E 0 −→ E 00 −→E 0 −→E −→ 0

and siE = (siA , siB , si−1 C ) i is a section of σE (i ∈ Z).

The recipe from 1.1.4 yields morphisms of complexes βX : X −→ X 00 [1]

(X = A, B, C, E)

characterized by ρX [1] ◦ βX = d ◦ sX − sX ◦ d

(X = A, B, C, E)

(1.3.11.1)

and such that ρX

σ

βX

X X 00 −−→X 0 −−→X −−→X 00 [1]

is an exact triangle in the derived category. In order to express βE in terms of βA , βB and βC , we compute d(sE (a, b, c)) − sE (d(a, b, c)) = d(sA (a), sB (b), sC (c)) − (sA (da), sB (db), sC (−dc − f (a) + g(b))) = = ((dsA − sA d)(a), (dsB − sB d)(b), −(dsC − sC d)(c) − (f 0 sA − sC f )(a) + (g 0 sB − sC g)(b)), hence 39

βE (a, b, c) = (βA (a), βB (b), −βC (c) − u(a) + v(b)),

(1.3.11.2)

where u

v

A−−→C 00 ←−−B are morphisms of complexes characterized by ρ C ◦ u = f 0 ◦ s A − sC ◦ f ρC ◦ v = g 0 ◦ sB − sC ◦ g. As −dC 0 ◦ (f 0 ◦ sA − sC ◦ f ) + (f 0 ◦ sA − sC ◦ f ) ◦ dA = (d ◦ sC − sC ◦ d) ◦ f − f 0 ◦ (d ◦ sA − sA ◦ d) (and similarly for (B, g) instead of (A, f )), the morphisms u and v are, in fact, homotopies u : f 00 [1] ◦ βA − 00

v : g [1] ◦ βB − 00

βC ◦ f βC ◦ g.

In the special case when X = X ⊗R ΓR (X = A, B, C) we thus obtain the data (1.3.8.1-2), as well as the formula for βE from Proposition 1.3.9(i).

40

2. Local Duality In this chapter we recall the formalism of Grothendieck’s duality for R-modules ([LC], [RD]). At first reading there is no need to continue beyond 2.10.4 (the subsequent sections are used in the construction of generalized Cassels-Tate pairings in Chapter 10). (2.1) Notation. Throughout Chapters 2-11 (with the exception of Sect. 2.10), R will be a complete noetherian local ring with maximal ideal m and residue field k = R/m. The dimension of R will be denoted by d (it is finite, as R is noetherian and local). The total ring of fractions of R will be denoted by Frac(R). Denote by (R Mod) the category of all R-modules, by (R Mod)f t (resp., (R Mod)cof t ) the category of Rmodules of finite (resp., co-finite) type (i.e. of modules satisfying the ascending (resp., descending) condition for submodules) and (R Mod)f l = (R Mod)f t ∩ (R Mod)cof t the category of R-modules of finite length. (2.2) Dualizing functors. Let I be an R-module. The functor D(−) = HomR (−, I) : (R Mod)op −→ (R Mod) is dualizing if the canonical homomorphism ε : M −→ D(D(M ))

(2.2.1)

is an isomorphism for every M ∈ (R Mod)f l . (2.3) Matlis duality (2.3.1) Matlis Duality. ([LC], Prop. 4.10; [Br-He], Thm. 3.2.13). (i) D is dualizing iff I is an injective hull of k (defined, e.g., in [Br-He], Def. 3.2.3). (ii) Fix such I (it is unique up to a non-unique isomorphism); the functor D is then exact and induces equivalences of categories −→ (R Mod)op f l ←− (R Mod)f l −→

(R Mod)op f t ←− (R Mod)cof t . The map (2.2.1) is an isomorphism for every M in (R Mod)f t or (R Mod)cof t . (2.3.2) From now on, I will be as in 2.3.1(ii). The functor D, being exact, can be derived trivially. For every complex M • of R-modules and n ∈ Z put Dn (M • ) = Hom•R (M • , I[n]) = D(M • )[n] (with the sign conventions of 1.2.1). It follows from 2.3.1(ii) that the canonical map ε = εI[n] : M −→ Dn (Dn (M )) is an isomorphism for every M in Df t (R Mod) or Dcof t (R Mod). (2.3.3) The simplest examples of I are the following: (i) I = R if R = k is a field. (ii) I = K/R if R is a (complete) discrete valuation ring with fraction field K. Pd (iii) I = R[1/x1 . . . xd ]/( i=1 R[1/x1 . . . xbi . . . xd ]) if R = k[[x1 , . . . , xd ]] is a power series ring. (2.3.4) We shall often use the fact that for every projective (resp.. inductive) system (Mn )n∈N of R-modules of finite (resp., co-finite) type such that M = ← lim − Mn (resp., M = lim −→ Mn ) is also of finite (resp., co-finite) type, the canonical map lim D(M ) −→ D(M ) (resp., D(M ) −→ lim n −→ ←− D(Mn )) is an isomorphism. 41

(2.3.5) Lemma. Let f : M −→ N be a homomorphism of R-modules. Then (i) M = 0 ⇐⇒ D(M ) = 0. (ii) The homomorphism ε : M −→ D(D(M )) is injective. (iii) f = 0 ⇐⇒ D(f ) = 0. Proof. (i) If M 6= 0, choose non-zero x ∈ M . By 2.3.1 applied to Rx there exists an injective morphism Rx −→ I; it extends to a morphism f : M −→ I satisfying f (x) 6= 0. This proves (i) and also shows that ε(x) 6= 0, proving (ii). As regards (iii), the morphism f factors as f = gh with g : f (M ) −→ N injective (=⇒ D(g) surjective) and h : M −→ f (M ) surjective (=⇒ D(h) injective). This implies that (i)

D(f ) = 0 ⇐⇒ D(g) = 0 ⇐⇒ D(f (M )) = 0 ⇐⇒ f = 0. (2.3.6) Lemma. If M is an R-module of finite (resp., co-finite) type, then every surjective (resp., injective) R-linear endomorphism f : M −→ M is bijective. Proof. If M is of finite type, see ([Mat], Thm. 2.4). If M is of co-finite type, the previous statement applied to the dual endomorphism D(f ) : D(M ) −→ D(M ) implies that D(f ) is bijective, hence so is f = D(D(f )). (2.4) Cohomology with support at {m} f on X = Spec(R). Its cohomology with support (2.4.1) Every R-module M defines a quasi-coherent sheaf M at the closed point {m} ⊂ X will be denoted by i i f). H{m} (M ) = H{m} (Spec(R), M

An explicit complex representing f) ∈ Db (R Mod) RΓ{m} (X, M can be constructed by using an exact triangle f) −→ RΓ(X, M f) −→ RΓ(X − {m}, M f) −→ RΓ{m} (X, M f)[1] RΓ{m} (X, M

(2.4.1.1)

f) is represented by M in degree zero. To get a complex representing RΓ(X − (2.4.2) First of all, RΓ(X, M f {m}, M ), fix a system of parameters of R, i.e. a d-tuple of elements x1 , . . . , xd ∈ m such that R/(x1 , . . . , xd ) has finite length. Then U = {Ui = Spec(Rxi ) | i = 1, . . . , d } is an open covering of X − {m} such that all ˇ intersections Ui0 ∩ · · · ∩ Uip = Spec(Rxi0 ···xip ) are affine. This implies that the Cech complex Cˇ • (M ) = f) with Cˇ • (M, (xi )) = Cˇ • (U, M Cˇ p (M, (xi )) =

M

Mxi0 ···xip = Cˇ p (R, (xi )) ⊗R M

(0 ≤ p < d)

i0 d − depth(R) (resp., i < 0) by Lemma 2.4.7(ii) (resp., 2.4.7(iii)) and is non-zero for i = 0 and i = d − depth(R) by Lemma 2.4.7(v). Furthermore, dim(H i (ω)) ≤ d − i, by Lemma 2.4.7(iv). ∼ (ii) In particular, R is Cohen-Macaulay (i.e. depth(R) = d) iff ω −→ H 0 (ω) is concentrated in degree zero ∼ d (in which case ω −→ D(H{m} (R))). ∼

(iii) R is Gorenstein (i.e. R is quasi-isomorphic to a bounded complex of injective R-modules) iff ω −→ R. (iv) In order to stress their dependence on R we sometimes denote I, D, D, ω by IR , DR , DR , ωR . (v) The hyper-cohomology spectral sequence E2i,j = ExtiR (M, H j (ω)) =⇒ Exti+j R (M, ω) implies that d Ext0R (M, ω) = HomR (M, H 0 (ω)) = HomR (M, D(H{m} (R))),

for every R-module M . ∼ (vi) If depth(Rp ) = 1 for all p ∈ Spec(R) with ht(p) = 1, then ω −→ H 0 (ω) in D((R Mod)/(pseudo − null)), using the language of 2.8.6 below (this follows from 2.7(i),(ii) applied to the (non-complete - but see 2.10) localizations Rp ). (2.8) Relating D, D and Φ (2.8.1) The functors D : Df t (R Mod)op ←→ Dcof t (R Mod) D : Df t (R Mod)op −→ Df t (R Mod) (which map D± to D∓ ) are related to Φ(−) := RΓ{m} (−)[d] : Df t (R Mod) −→ Dcof t (R Mod) (which maps D± to D± ) as follows. First of all, we have

  L Φ(−) = (−)⊗R Φ(R)[−d] [d].

The natural transformation of functors

η : D =⇒ D ◦ Φ, defined by

  L L D(−) = RHomR (−, ω) −→ RHomR (−)⊗R Φ(R)[−d], ω ⊗R Φ(R)[−d] = t

RHom(id,Tr)

d = RHomR (Φ(−)[−d], Φ(ω)[−d])−→RHom R (Φ(−), Φ(ω))−−−−−−→RHomR (Φ(−), I) = D ◦ Φ(−),

is an isomorphism, by local duality 2.5. The natural transformations εI : id =⇒ D ◦ D,

εω : id =⇒ D ◦ D

are isomorphisms, by Matlis and Grothendieck duality, respectively. As a consequence, the natural transformations 44

D?η

ε ?Φ

I ψ : Φ==⇒D ◦ D ◦ Φ==⇒D ◦ D ψ?D D?εω ξ : Φ ◦ D ==⇒D ◦ D ◦ D ==⇒D

are isomorphisms, too. It follows from the fact that the composition ε ?D

D?ε

I I D==⇒D ◦ D ◦ D==⇒D

is the identity (and from an analogous statement for D) that the following diagrams of natural transformations are commutative: Φ?εω

εI ?D

Φ JJJJ +3 Φ ◦ D ◦ D JJJJJψ JJJJ JJJJ ξ?D J (  D◦D

D JJJJ +3 D ◦ D ◦ D JJJJJη JJJJ JJJJ D?ψ J !)  D◦Φ

εI

id εω

 D ◦D

+3 D ◦ D D?ξ

η?D

 +3 D ◦ Φ ◦ D

(2.8.2) To sum up: the following duality diagram of functors D

Df±t (R Mod)op ←−−−−−−−−→ Df∓t (R Mod)   ←−   −− D −−→   −−−−− Φ Φ −−−−−−− y y − − → − ± ∓ op ← Dcof t (R Mod) Dcof t (R Mod) (and its • • • (2.8.3)

analogue without ±, ∓) is commutative up to various natural isomorphisms of functors: ∼ εI : id =⇒ D ◦ D (Matlis duality). ∼ εω : id =⇒ D ◦ D (Grothendieck duality). ∼ η : D =⇒ D ◦ Φ (local duality). ∼ In particular, ψ(R) induces an isomorphism Φ(R) −→ D(D(R)) = D(ω), hence   L L ∼ ∼ Φ(−) −→ (−)⊗R D(ω)[−d] [d] −→ (−)⊗R D(ω)

(the last arrow is given by s0−d [d], in the notation of 1.2.15). This can also be deduced from the adjunction isomorphism L

∼

adj : RHomR (A, RHomR (B, C)) −→ RHomR (A⊗R B, C) (which is a derived version of 1.2.6; it holds in D+ (R Mod) for all A, B ∈ D− (R Mod), C ∈ D+ (R Mod)) applied to B = D(ω) and C = I: ∼

∼

L

RHomR (−, ω) −→ RHomR (−, D(D(ω))) −→ RHomR ((−)⊗R D(ω), I). ∼

For example, for R = Zp we have I = Qp /Zp , ω −→ Zp and C • (Zp ) = [Zp −→ Qp ] (in degrees 0 and 1); this is quasi-isomorphic to Qp /Zp [−1]. If M is a free Zp -module of finite type, then D(M ) = HomZp (M, Zp ), Φ(M ) = M ⊗Zp Qp /Zp and ∼

D(Φ(M )) −→ HomZp (M ⊗Zp Qp /Zp , Qp /Zp ) = D(M ). (2.8.4) All of the above makes sense on the level of complexes: fixing a system of parameters xi of R and a bounded complex of injective R-modules ω • representing ω, we have Φ(X • ) = (X • ⊗R C • ((xi ), R))[d] • D(X • ) = HomR (X • , I) • • D(X ) = HomR (X • , ω • ).

45

As I is injective, the isomorphism Tr can be represented by a quasi-isomorphism Tr : Φ(ω • ) −→ I, unique up to homotopy. The morphisms η(X • ), ψ(X • ), ξ(X • ) are genuine morphisms of complexes (of course, they are all quasi-isomorphisms). For example, η(X • ) is given by t

d • • D(X • ) = HomR (X • , ω • ) −→ HomR (X • ⊗R C • (R), ω • ⊗R C • (R))−→ • Hom (id,Tr) td • • • • • • • −→Hom R ((X ⊗R C (R))[d], (ω ⊗R C (R))[d])−−−−−−→HomR (Φ(X ), I) = D ◦ Φ(X ).

(2.8.5) For T ∈ Df t (R Mod) put T ∗ = D(T ), A = Φ(T ) = RΓ{m} (T )[d], A∗ = D(T ). Loosely speaking, we can think of these four objects as being related by the diagram D

T ←−−−−−−−−→  ←−  −− D −−→ Φ −−−−−  −−−−−−− y − − → ←− A

T∗   Φ  y A∗

(2.8.6) In the notation of 2.8.5, the hyper-cohomology spectral sequence of D applied to T ∗ is given by E2i,j = ExtiR (H −j (T ∗ ), ω) =⇒ H i+j (D(T ∗ )), i.e. E2i,j = ExtiR (D(H j (A)), ω) =⇒ H i+j (T ).

(2.8.6.1)

It follows from local duality 2.5 and Lemma 2.4.7(iii)-(iv) that E2i,j = 0 for i < 0 and that supp(E2i,j ) has codimension ≥ i in Spec(R). By 2.7(v) we have E20,j = HomR (D(H j (A)), H 0 (ω)). Recall that an R-module M of finite (resp., co-finite) type is pseudo-null (resp., co-pseudo-null) if supp(M ) has codimension ≥ 2 in Spec(R) (resp., if D(M ) is pseudo-null). It follows that, in the quotient category (R Mod)/(pseudo − null), the spectral sequence (2.8.6.1) degenerates to a collection of short exact sequences 0 −→ Ext1R (D(H j−1 (A)), ω) −→ H j (T ) −→ Ext0R (D(H j (A)), ω) −→ 0.

(2.8.6.2)

Recall also that, for each prime ideal p ∈ Spec(R)
with ht(p) ≤ 1, the localization M 7→ Mp defines an exact functor (R Mod)/(pseudo − null) −→ Rp Mod . (2.8.7) Proposition. In the situation of 2.8.5 there are spectral sequences i+d E2i,j = H{m} (H j (T )) =⇒ H i+j (A) 0

i+d E2i,j = H{m} (D(H −j (A))) =⇒ D(H −i−j (T ))

Proof. The first spectral sequence Er is just the hyper-cohomology spectral sequence for Φ. It can be constructed explicitly as follows. Represent T by a complex M • of R-modules; then A is represented by N • = (M • ⊗R C • (R))[d]. Filter N • by the subcomplexes F i N • = (M • ⊗R σ≥i+d (C • (R)))[d]. We have 46

grFi (N • ) = (M • ⊗R C i+d (R))[−i], as C • (R) is a complex of flat R-modules. The corresponding spectral sequence satisfies E1i,j = H j (M • ) ⊗R C i+d (R) = H j (T ) ⊗R C i+d (R), hence

i+d E2i,j = H{m} (H j (T ))

as claimed. The second spectral sequence 0 Er is obtained from (2.8.6.1) by applying D and using local duality 2.5(iii). d (2.8.8) Lemma. For every R-module M of finite type, the R-module Ext0R (M, ω) (resp., H{m} (M )) is 0 torsion-free (resp., divisible). In particular, H (ω) is torsion-free. d Proof. We know from 2.7(v) that Ext0R (M, ω) = HomR (M, D(H{m} (R))). If r ∈ R does not divide zero, then the exact sequence of local cohomology r

d−1 d d d H{m} (R/rR) −→ H{m} (R)−→H{m} (R) −→ H{m} (R/rR) = 0 d (in which the last term vanishes by Lemma 2.4.7(iii)) shows that multiplication by r on D(H{m} (R)), and 0 d hence also on ExtR (M, ω), is injective. It follows that multiplication by r on H{m} (M ) = D(Ext0R (M, ω)) is surjective. The last statement is a consequence of H 0 (ω) = Ext0R (R, ω). (2.8.9) As in 2.8.6, it follows from Lemma 2.4.7(iii)-(iv) that E2i,j , 0 E2i,j in Proposition 2.8.7 are co-pseudonull for i 6= 0, −1. This implies that in the quotient category (R Mod)/(co − pseudo − null) the two spectral sequences degenerate into short exact sequences d−1 d 0 −→ H{m} (H j (T )) −→H j (A) −→ H{m} (H j+1 (T )) −→ 0 d−1 d 0 −→ H{m} (D(H j (A))) −→D(H j (T )) −→ H{m} (D(H j−1 (A))) −→ 0, d (the second one being just D(2.8.6.2)). It follows from Lemma 2.8.8 and Lemma 2.4.7(iv) that H{m} (H j (T )) 1 (resp., ExtR (D(H j−1 (A)), ω)) is the maximal R-divisible (resp., R-torsion) subobject of H j (A) (resp., H j (T )) in (R Mod)/(co − pseudo − null) (resp., (R Mod)/(pseudo − null)). (2.8.10) Much of the previous discussion can be reformulated in terms of an analogue of the duality diagram 2.8.2 for appropriate quotient categories:

Df t (R Mod/(pseudo − null))op    Φ y Dcof t (R Mod/(co − pseudo − null))op

D

←−−−−−−−−→ ←− −− D −−→ −−−−− −−−−−−− − − → − ←

Df t (R Mod/(pseudo − null))    Φ y Dcof t (R Mod/(co − pseudo − null))

(2.8.11) It is sometimes convenient to use another normalization of D and Φ, namely Dd (−) = D(−)[d] = RHomR (−, ω)[d] = RHomR (−, ω[d]) L

Φ−d (−) = Φ(−)[−d] = RΓ{m} (−) = (−)⊗R Φ−d (R). Then the map Tr defines a quasi-isomorphism Tr : Φ−d (ω[d]) −→ I and the diagram (2.8.2) is replaced by D

d Df±t (R Mod)op ←−−−−− −−−→ Df∓t (R Mod) ←−     −− D −−→ −−−−− Φ Φ  −d  −d − − − − − y y − −→ −− − ← ± ∓ op Dcof Dcof t (R Mod) t (R Mod),

47

which is commutative up to natural isomorphisms of functors ∼

ηd : Dd =⇒ D ◦ Φ−d ,

∼

∼

ψd : Φ−d =⇒ D ◦ Dd ,

εω[d] : id =⇒ Dd ◦ Dd ,

∼

∼

ξd : Φ−d ◦ Dd =⇒ D

εI : id =⇒ D ◦ D.

Here ηd is given by   L L Dd (−) = RHomR (−, ω[d]) −→ RHomR (−)⊗R Φ−d (R), ω[d]⊗R Φ−d (R) = RHom(id,Tr)

= RHomR (Φ−d (−), Φ−d (ω[d]))−−−−−−→RHomR (Φ−d (−), I) = D ◦ Φ−d (−), and ψd , ξd are defined as in 2.8.1, with Φ (resp., D) replaced by Φ−d (resp., Dd ). Fixing the same data as in 2.8.4, we can define the functors Φ−d and Dd on the level of complexes; this will be used in the following Lemma. (2.8.12) Lemma. (i) The morphism of complexes ξd (R) : Φ−d (Dd (R)) = Φ−d (ω[d]) −→ I is equal to Tr. (ii) For every complex X • of R-modules with cohomology of finite type, the following diagram of morphisms of complexes is commutative: ev ⊗id

2 (X ⊗R Dd (X)) ⊗R Φ−d (R) −−−−→  o y

Dd (R) ⊗R Φ−d (R) = Φ−d (Dd (R))   yTr

X ⊗R (Dd (X) ⊗R Φ−d (R))

I x ev  2

k X ⊗R Φ−d (Dd (X))

id⊗ξd (X)

−−−−→

X ⊗R D(X)

Proof. This follows from the definitions. (2.9) Relation to Pontrjagin duality In this section we assume that the residue field k = Fpr is a finite field of characteristic p. (2.9.1) Under this assumption, every R-module M of finite type is compact, Hausdorff and totally disconnected in the m-adic topology. The Pontrjagin dual of M is equal to n M D = Homcont (M, R/Z) = Homcont (M, Qp /Zp ) = lim −→ HomZp (M/m M, Qp /Zp ) = n

n n n = lim lim −→ HomR (M/m M, HomZp (R/m , Qp /Zp )) = − → HomR (M, HomZp (R/m , Qp /Zp )) = n

n

n D = HomR (M, − lim → HomZp (R/m , Qp /Zp )) = HomR (M, R ). n

It follows from 2.3.1(i) and Pontrjagin duality that RD = I is an injective hull of k, hence D(M ) = M D for every R-module M of finite type. n (2.9.2) Similarly, if N is an R-module of co-finite type equipped with discrete topology, then N = lim −→ N [m ] n

and the adjunction isomorphism

∼

N D = HomZp (N, Qp /Zp ) −→ HomR (N, HomZp (R, Qp /Zp )) factors through the submodule of the R.H.S. equal to n D HomR (N, − lim → HomZp (R/m , Qp /Zp )) = HomR (N, R ) = D(N ). n

48

Thus the functor D coincides with the Pontrjagin dual on both (R Mod)f t and (R Mod)cof t . (2.10) Non-complete R In this section we assume that R is local and noetherian, but not necessarily complete. As above, d = dim(R). b = lim R/mn the m-adic completion of R, with maximal ideal m b (similarly, put b = mR (2.10.1) Denote by R ← n− c = lim M/mn M for every R-module M ). Recall that R b is faithfully flat over R ([Mat], Thm 8.14(3)). All M ← n− statements in Sect. 2.2-2.4 and 2.4.7 (ii), (iii), (v) are true for R. b m) b has a canonical structure of an (2.10.2) Proposition. (i) An injective hull IR of k = R/m (= R/ b R-module. In fact, IR = IR . b (ii) For every R-module of finite type M we have c c DR (M ) = HomR (M, I) = HomR b(M , I) = DR b(M ) (where I = IR = IR b). (iii) For every M as in (ii), the canonical maps   L ∼ b b c RΓ{m} (M ) −→ RΓ{m} (M ) ⊗R R = RΓ{m} (M )⊗R R −→ RΓ{m b} (M ) ∼

i b are isomorphisms. In particular, each H{m} (M ) has a canonical structure of an R-module. i (iv) For every M as in (ii), each H{m} (M ) is an artinian R-module. (v) For every R-module of finite length N the canonical map ε : N −→ DR (DR (N )) is an isomorphism.

Proof. For (i)-(iii), see [Br-He], Ex. 3.2.14, Lemma 3.5.4(d). For (iv), see [Br-Sh], Thm. 7.1.3. Finally, (v) b. follows from N = N (2.10.3) Dualizing complex. An object ωR of Dfb t (R Mod) is a dualizing complex for R if it can be • represented by a bounded complex ωR of injective R-modules and if it satisfies Grothendieck duality 2.6. If it exists, then ωR is determined (up to isomorphism) up to a shift ωR 7→ ωR [n] ([RD], V.3.1). If R is a quotient of a Gorenstein local ring then ωR exists ([RD], V.10); the converse also holds [Kaw]. (2.10.4) Local Duality. ([RD], Ch. V) Assume that ωR exists. Then (i) The undetermined shift in the normalization of ωR is uniquely determined by the condition ( i H{m} (ωR )

∼

−→

I,

i=d

0,

i 6= d.

∼

d Fix an isomorphism Tr : H{m} (ωR ) −→ I. (ii) For every R-module M of finite type and i ∈ Z, the Yoneda pairing ∼

i d H{m} (M ) × Extd−i R (M, ωR ) −→ H{m} (ωR ) −→ I

induces isomorphisms   ∼ i H{m} (M ) −→ DR Extd−i R (M, ωR )     ∼ \ i i Extd−i (M, ω ) −→ D H (M ) = D H (M ) . R R {m} {m} R b R L

b b b (iii) ωR b = ωR ⊗R R = ωR ⊗R R is a dualizing complex for R. (iv) R is universally catenary. (v) For every prime ideal p ∈ Spec(R), the localization (ωR )p is a dualizing complex for Rp . 49

(vi) If R0 ⊃ R is a local ring, free of finite rank as an R-module, then ωR0 exists and is isomorphic to HomR (R0 , ωR ) = RHomR (R0 , ωR ). (vii) The statements of 2.7(i)-(iii),(vi) hold. (2.10.5) Lemma. Let M be an R-module of finite type. Then 0 (i) If m 6∈ Ass(R), then H{m} (M ) ⊆ Mtors . 0 (ii) If dim(R) = 1, then Mtors ⊆ H{m} (M ). 0 (iii) If dim(R) = depth(R) = 1, then Mtors = H{m} (M ). d (iv) H{m} (M ) is R-divisible. (v) If ωR exists, then H 0 (ωR ) is torsion-free over R. Proof. The statements (i)-(ii) follow from the fact that  Mtors = Ker M −→



M

Mq 

q∈Ass(R)





Y

0 H{m} (M ) = Ker M −→

Mq  .

q∈Spec(R)−{m}

Indeed, in (i) we have Ass(R) ⊆ Spec(R) − {m}, while in (ii) Ass(R) contains all minimal prime ideals q ⊂ R, i.e. all elements of Spec(R) − {m}. The statement (iii) is a conbination of (i) and (ii). As regards (iv) d b b is and (v), Lemma 2.8.8 and Proposition 2.10.2(iii) imply that H{m} (M ) is R-divisible and H 0 (ωR ) ⊗R R b we conclude by the faithful flatness of R b over R (if r ∈ R does not divide zero in R, it torsion-free over R; b is not a zero divisor in R). (2.10.6) Example. If dim(R) = 1 and R is not Cohen-Macaulay, then 0 H{m} (R) 6= 0.

Rtors = 0,

(2.10.7) Lemma. Assume that dim(R) = 1 and fix x ∈ m such that dim(R/xR) = 0. −i (i) Let C • = [R−→Rx ] be the complex C • (R, x) (in degrees 0, 1) from 2.4.3. Then (−i,−i)

(−id,id)

C • ⊗R C • = [R−−−−→Rx ⊕ Rx −−−−→Rx ] and the morphism of complexes u : C • −→ C • ⊗R C • given by idR (resp., (idRx , idRx )) in degree 0 (resp., 1) is a quasi-isomorphism satisfying s12 ◦ u = u. The morphism of complexes v : C • ⊗R C • −→ C • given by idR (resp., by the projection on the first factor) in degree 0 (resp., 1) satisfies vu = id and uv is homotopic to the identity. (ii) If x is not a zero divisor in R, then the canonical map Rx −→ Frac(R) is an isomorphism. Proof. Easy exercise. (2.10.8) Corollary. Assume that dim(R) = 1. Then the morphisms of complexes (X • ⊗R C • ) ⊗R (Y • ⊗R C • )

s

23 −−→

(X • ⊗R Y • ) ⊗R (C • ⊗R C • ) x  id⊗u (X • ⊗R Y • ) ⊗R C •

define a functorial cup product L

∪

L

RΓ{m} (X)⊗R RΓ{m} (Y )−→RΓ{m} (X ⊗R Y ) 50

(X, Y ∈ Df−t (R Mod))

such that the following diagram is commutative: L

−→

∪

RΓ{m} (X ⊗R Y )  (s ) y 12 ∗

L

∪

RΓ{m} (Y ⊗R X).

RΓ{m} (X)⊗R RΓ{m} (Y )  s y 12

RΓ{m} (Y )⊗R RΓ{m} (X) −→

L

L

Proof. Combine Lemma 1.2.4 and Lemma 2.10.7. (2.10.9) Assume that dim(R) = depth(R) = 1. Then the filtration σ≥i C • induces, as in 2.8.7, a filtration on the complex X • ⊗R C • representing RΓ{m} (X • ) (for every complex X • of R-modules with cohomology of finite type). The corresponding spectral sequence Er from Proposition 2.8.7 for T = X = X • ∈ Df t (R Mod) degenerates (E2 = E∞ ) into short exact sequences 1 i 0 0 −→ H{m} (H i−1 (X)) −→ H{m} (X) −→ H{m} (H i (X)) −→ 0

(2.10.9.1)

All terms in the above exact sequence are R-torsion (by Lemma 2.10.5(iii)) and the first term is R-divisible (by Lemma 2.10.5(iv)). This implies that the cup product on cohomology L

j i+j i H{m} (X) ⊗R H{m} (Y ) −→ H{m} (X ⊗R Y )

(X, Y ∈ Df−t (R Mod))

induced by the cup product ∪ from 2.10.8 factors through L

i+j 0 0 ∪ij : H{m} (H i (X)) ⊗R H{m} (H j (Y )) −→ H{m} (X ⊗R Y ),

i.e. L

i+j ∪ij : H i (X)tors ⊗R H j (Y )tors −→ H{m} (X ⊗R Y ),

(2.10.9.2)

(s12 )∗ (x ∪ij y) = (−1)ij y ∪ji x.

(2.10.9.3)

and satisfies

We can drop the upper-boundedness condition if we deal not with objects of the derived category, but with complexes of R-modules X • , Y • , Z • with cohomology of finite type. We obtain products j i+j i H{m} (X • ) ⊗R H{m} (Y • ) −→ H{m} (X • ⊗R Y • )

factoring through i+j ∪ij : H i (X • )tors ⊗R H j (Y • )tors −→ H{m} (X • ⊗R Y • )

and satisfying (2.10.9.3). If u : X • ⊗R Y • −→ Z • is a morphism of complexes of R-modules, then the induced products i+j u∗ ◦ ∪ij : H i (X • )tors ⊗R H j (Y • )tors −→ H{m} (Z • )

depend only on the homotopy class of u. (2.10.10) Assume that dim(R) = depth(R) = 1 and that ωR exists. In the hyper-cohomology spectral sequence E2i,j = ExtiR (H −j (X), ωR ) =⇒ H i+j (D(X)) we have 51

(X ∈ Df t (R Mod))

E2i,j = ExtiR (H −j (X), H 0 (ωR )) and E2i,j = 0 for i 6= 0, 1, which gives short exact sequences 0 −→ Ext1R (H −j+1 (X), H 0 (ωR )) −→ H j (D(X)) −→ HomR (H −j (X), H 0 (ωR )) −→ 0.

(2.10.10.1)

Applying DR to (2.10.10.1) gives, by local duality 2.10.4(ii), the exact sequences (2.10.9.1) for i = −j + 1. (2.10.11) Lemma. Assume that dim(R) = depth(R) = 1 and that ωR exists. Then (i) JR := H 0 (ωR ) ⊗R Frac(R)/R is isomorphic to IR . (ii) The exact sequence (2.10.10.1) is isomorphic to 0 −→ HomR (H −j+1 (X)tors , IR ) −→ H j (D(X)) −→ HomR (H −j (X), H 0 (ωR )) −→ 0. (iii) H j (D(X))tors is isomorphic to ∼

HomR (H −j+1 (X)tors , IR ) −→ HomR (H −j+1 (X)tors , H 0 (ωR ) ⊗R Frac(R)/R). ∼

Proof. As R is Cohen-Macaulay, we have ωR −→ H 0 (ωR ) in D(R Mod). Fix x ∈ m which is not a zero divisor in R. The R-linear map α : IR −→ JR , induced by Rx −→ Frac(R) and (Tr)−1

1 1 IR −−→H{m} (ωR ) = H{m} (H 0 (ωR )) = Coker(H 0 (ωR ) −→ H 0 (ωR ) ⊗R Rx ),

is an isomorphism, since Rx −→ Frac(R) is (Lemma 2.10.7(ii)). This proves the statement (i). If M is an R-module of finite type, so is N = Ext1R (M, H 0 (ωR )). As codimR (supp(N )) ≥ 1, we have Ext1R (M, H 0 (ωR ) ⊗R Frac(R)) = N ⊗R Frac(R) = 0, by Lemma 2.10.5(iii). It follows that [H 0 (ωR ) ⊗R Frac(R) −→ IR ] is an injective resolution of ωR , which gives an isomorphism ∼

HomR (P, IR ) −→ Ext1R (P, H 0 (ωR )) for every torsion R-module P . Taking P = Mtors , the long exact sequence of Ext’s attached to x

0 −→ M/P −→M/P −→ M/(P + xM ) −→ 0 shows that Ext1R (M/P, H 0 (ωR ))/xExt1R (M/P, H 0 (ωR )) = 0, hence Ext1R (M/P, H 0 (ωR )) = 0 by Nakayama’s Lemma. It follows that ∼

Ext1R (M, H 0 (ωR )) = Ext1R (Mtors , H 0 (ωR )) −→ HomR (Mtors , IR ). Taking M = H −j+1 (X) concludes the proof of (ii). Finally, (iii) follows from (ii) and the fact that H 0 (ωR ) is torsion-free (Lemma 2.10.5(v)). (2.10.12) Proposition. Assume that dim(R) = depth(R) = 1 and that ωR exists. Given complexes of R-modules X • , Y • with cohomology of finite type and a morphism of complexes • u : X • ⊗R Y • −→ ωR [n] • (where ωR is a bounded complex of injective R-modules representing ωR ), denote by

∼

∼

1−n 1 • • ∪i,1−n−i : H i (X • )tors ⊗R H 1−n−i (Y • )tors −→ H{m} (ωR [n]) = H{m} (ωR ) −→ IR −→ H 0 (ωR ) ⊗R Frac(R)/R

52

the cup products from 2.10.8 and (2.10.9.2) induced by u and Tr. Complete the map • • adj(u) : X • −→ HomR (Y • , ωR [n]) = Dn (Y • )

to an exact triangle in Df t (R Mod) adj(u)

X • −−→Dn (Y • ) −→ Err −→ X • [1]. Then the morphism adj(∪i,1−n−i ) : H i (X • )tors −→ HomR (H 1−n−i (Y • )tors , IR ) = D(H 1−n−i (Y • )tors ) in (R Mod)f l has the following properties: (i) Ker(adj(∪i,1−n−i )) is isomorphic to a subquotient of H i−1 (Err). (ii) If H i−1 (Err) is R-torsion, then Coker(adj(∪i,1−n−i )) (resp., Ker(adj(∪i,1−n−i ))) is isomorphic to a submodule (resp., a quotient) of H i (Err) (resp., of H i−1 (Err)). In particular, if H i−1 (Err) = H i (Err) = 0, then adj(∪i,1−n−i ) is an isomorphism. Proof. This follows from the Snake Lemma applied to the following diagram, in which the first square is commutative up to a sign and the map f is the isomorphism from Lemma 2.10.11(ii): H i−1 (Err) / H i (X • )/H i (X • )tors

/0

 / Ext0 (H −n−i (Y • ), ωR ) R

/0

adj(∪i,1−n−i )



D H 1−n−i (Y • )tors

0

 / H i (X • )

/ H i (X • )tors

0






f

/ Ext1 (H 1−n−i (Y • ), ωR ) R

adj(u)∗

 / H i (Dn (Y • ))  H i (Err)

(2.10.13) Torsion submodules. (2.10.13.1) The R-torsion submodule of an R-module M is defined as MR−tors = Mtors = Ker (M −→ M ⊗R Frac(R)) = {m ∈ M | (∃r - 0 in R) rm = 0}. Note that (i) If depth(R) = 0, then Frac(R) = R and MR−tors = 0 for all M . (ii) The set of r ∈ R dividing zero is equal to the union of the associated primes of R; thus   M Mtors = Ker M −→ Mp  . p∈Ass(R)

(iii) In particular, (R/m)tors = 0 ⇐⇒ m ∈ Ass(R) ⇐⇒ depth(R) = 0. (iv) Let S ⊂ R be a multiplicative subset of R. If r ∈ R does not divide zero in R, the same is true for its image under the canonical morphism R −→ RS , which then induces a homomorphism of RS -algebras 53

Frac(R)S = Frac(R) ⊗R RS −→ Frac(RS ). Similarly, for every R-module M there is a canonical (injective) homomorphism (MR−tors )S −→ (MS )RS −tors . (v) The maps in (iv) need not be isomorphisms, even if S = R − p for p ∈ Spec(R). For example, if dim(R) = 2, depth(R) = 0, dim(Rp ) = depth(Rp ) = 1, then Frac(R)p = Rp 6= Frac(Rp ). (2.10.13.2) Recall Serre’s conditions (Rn ) Rp is regular for all p ∈ Spec(R) with ht(p) ≤ n. (Sn ) depth(Rp ) ≥ min(ht(p), n) for all p ∈ Spec(R). The following implications hold ([EGA IV.2], Sect. 5.7-5.8):

R is Cohen − Macaulay ⇐⇒ R satisfies (Sn ) for all n ≥ 0 =⇒ R satisfies (S1 ) ⇐⇒ ⇐⇒ R has no embedded primes ⇐= R satisfies (R0 ) and (S1 ) ⇐⇒ R is reduced ⇐= R is a domain. (2.10.13.3) Lemma. If R has no embedded primes, then (i) There is a canonical isomorphism Y ∼ Frac(R) −→ Rp . ht(p)=0

(ii) The canonical map Frac(R)q −→ Frac(Rq ) is an isomorphism, for each q ∈ Spec(R). (iii) If ∆ is a finite abelian group, then the canonical map Frac(R)[∆] −→ Frac(R[∆]) is an isomorphism. Proof. (i) Combine [EGA I], 7.1.8-9 (cf. [Bou], IV.2.5, Prop. 10(iii), in the case when R is reduced). (ii) Fix p ∈ Spec(R) with ht(p) = 0. If p ⊂ q, then (Rp )q (= Rp ⊗R Rq ) = Rp = (Rq )p . If p 6⊂ q, then there is x ∈ p, x 6∈ q. As x ⊗ 1 = 1 ⊗ x is simultaneously nilpotent and invertible in Rp ⊗R Rq , we have (Rp )q = 0. Applying (i) to both R and Rq we get isomorphisms ∼

Frac(R)q −→

Y

(Rp )q =

ht(p)=0

Y

∼

Rq −→ Frac(Rq ).

ht(p)=0 p⊂q

(iii) The map in question is injective for arbitrary R. In order to prove surjectivity, we must show that, for every α ∈ R[∆] which is not a zero divisor, the cokernel of the injective multiplication map multα : R[∆] −→ R[∆],

multα (x) = αx

satisfies ?

Coker(multα ) ⊗R Frac(R) = 0. This follows from (i) and the fact that `Rp (Coker(multα )p ) = `Rp (Ker(multα )p ) = 0 for all p ∈ Spec(R) with ht(p) = 0. 54

(2.10.13.4) Corollary. If R has no embedded primes, then the canonical map (MR−tors )q −→ (Mq )Rq −tors is an isomorphism, for each R-module M and q ∈ Spec(R). Proof. Localize the exact sequence 0 −→ MR−tors −→ M −→ M ⊗R Frac(R) at q and apply Lemma 2.10.13.3(ii). (2.10.14) Assume that ωR exists, but impose no other conditions on R. In this case the statement of Lemma 2.10.7(i) holds if we replace C • by the following complex in degrees 0, 1: •

−i

C = [R−−→Frac(R)] •

(note that C • = C if dim(R) = depth(R) = 1, by Lemma 2.10.7(ii)). Let X • and Y • be complexes of R-modules. The exact sequence •

0 −→ H i−1 (X • ) ⊗R (Frac(R)/R) −→ H i (X • ⊗R C ) −→ H i (X • )tors −→ 0, together with the corresponding sequence for Y • and the construction of Corollary 2.10.8, yield cup products •

∪ij : H i (X • )tors ⊗R H j (Y • )tors −→ H i+j ((X • ⊗R Y • ) ⊗R C ) satisfying (s12 )∗ (x ∪ij y) = (−1)ij y ∪ji x.

(2.10.14.1)

• If ωR is a bounded complex of injective R-modules representing ωR and • u : X • ⊗R Y • −→ ωR [n]

a morphism of complexes, then u induces cup products •

• ∪i,1−n−i : H i (X • )tors ⊗R H 1−n−i (Y • )tors −→ H 1 (ωR ⊗R C ),

with the target sitting in an exact sequence •

• 0 −→ H 0 (ωR ) ⊗R (Frac(R)/R) −→ H 1 (ωR ⊗R C ) −→ H 1 (ωR )tors −→ 0.

If R is Cohen-Macaulay (resp., if depth(Rp ) = 1 for all p ∈ Spec(R) with ht(p) = 1), then H 1 (ωR ) is zero (resp., pseudo-null) and we obtain cup products ∪i,1−n−i : H i (X • )tors ⊗R H 1−n−i (Y • )tors −→ H 0 (ωR ) ⊗R (Frac(R)/R) in (R Mod) (resp., in (R Mod)/(pseudo − null)). If R has no embedded primes, then it follows from Lemma 2.10.13.3(ii) and Corollary 2.10.13.4 that the localization of ∪i,1−n−i at each p ∈ Spec(R) with ht(p) = 1 coincides with the cup product from Proposition 2.10.12, applied to Rp . (2.10.15) Assume that ωR exists and depth(Rp ) = 1 for all p ∈ Spec(R) with ht(p) = 1. In the category (R Mod)/(pseudo − null), the hyper-cohomology spectral sequence E2i,j = ExtiR (H −j (X), ωR ) =⇒ H i+j (D(X)) satisfies E2i,j = ExtiR (H −j (X), H 0 (ωR )) 55

(X ∈ Df t (R Mod))

(by 2.7(vi)) and E2i,j = 0 for i 6= 0, 1. This yields an analogue of (2.10.10.1) 0 −→ Ext1R (H −j+1 (X), H 0 (ωR )) −→ H j (D(X)) −→ HomR (H −j (X), H 0 (ωR )) −→ 0 and an isomorphism ∼

Ext1R (H −j+1 (X), H 0 (ωR )) −→ H j (D(X))R−tors in (R Mod)/(pseudo − null) (using Lemma 2.10.5(v)). (2.10.16) Proposition. Assume that ωR exists and R has no embedded primes. Then, for each X ∈ Df t (R Mod) and j ∈ Z, there is an isomorphism in (R Mod)/(pseudo − null) ∼

∼

HomR (H −j+1 (X)tors , H 0 (ωR ) ⊗R (Frac(R)/R)) −→ Ext1R (H −j+1 (X), H 0 (ωR )) −→ H j (D(X))tors , the localization of which at each p ∈ Spec(R) with ht(p) = 1 coincides with the isomorphism from Lemma 2.10.11(iii), applied to Rp . Proof. The construction in 2.10.15 yields an isomorphism in (R Mod)/(pseudo − null) ∼

H j (D(X))tors −→ Ext1R (M, H 0 (ωR )), where M = H −j+1 (X). For each p ∈ Spec(R) with ht(p) = 1, the canonical map Ext1R (M, H 0 (ωR ))p = Ext1Rp (Mp , H 0 (ωR )p ) −→ Ext1Rp ((Mp )Rp −tors , H 0 (ωR )p ) = Ext1R (MR−tors , H 0 (ωR ))p is an isomorphism (using Corollary 2.10.13.4 and the proof of Lemma 2.10.11(ii)). This yields a canonical isomorphism in (R Mod)/(pseudo − null) ∼

Ext1R (M, H 0 (ωR )) −→ Ext1R (MR−tors , H 0 (ωR )). The boundary map defined by the exact sequence 0 −→ H 0 (ωR ) −→ H 0 (ωR ) ⊗R Frac(R) −→ H 0 (ωR ) ⊗R Frac(R)/R −→ 0 gives an isomorphism in (R Mod) ∼

δ : HomR (MR−tors , H 0 (ωR ) ⊗R Frac(R)/R) −→ Ext1R (MR−tors , H 0 (ωR )). Then the composite isomorphism in (R Mod)/(pseudo − null) ∼

∼

−δ −1

H j (D(X))tors −→ Ext1R (M, H 0 (ωR )) −→ Ext1R (MR−tors , H 0 (ωR ))−−→ −→ HomR (MR−tors , H 0 (ωR ) ⊗R Frac(R)/R)

has the required properties under localization at each p ∈ Spec(R) with ht(p) = 1 (the minus sign comes from the fact that the map in 2.10.11 was defined using an injective resolution, hence differs from that defined in terms of δ by a sign). (2.10.17) Proposition. Assume that R is Cohen-Macaulay (resp., R has no embedded primes) and that ωR exists. Given complexes of R-modules X • , Y • with cohomology of finite type and a morphism of complexes • u : X • ⊗R Y • −→ ωR [n]

as in 2.10.14, let ∪i,1−n−i : H i (X • )tors ⊗R H 1−n−i (Y • )tors −→ H 0 (ωR ) ⊗R (Frac(R)/R) 56

be the cup products in (R Mod) (resp., in (R Mod)/(pseudo − null)) defined in 2.10.14. Complete the map • • adj(u) : X • −→ HomR (Y • , ωR [n]) = Dn (Y • )

to an exact triangle in Df t ((R Mod)/(pseudo − null)) adj(u)

X • −−→Dn (Y • ) −→ Err −→ X • [1]. Then the morphism adj(∪i,1−n−i ) : H i (X • )tors −→ HomR (H 1−n−i (Y • )tors , H 0 (ωR ) ⊗R (Frac(R)/R)) in (R Mod)/(pseudo − null) has the following properties: (i) Ker(adj(∪i,1−n−i )) is isomorphic to a subquotient of H i−1 (Err). (ii) If H i−1 (Err) is R-torsion, then Coker(adj(∪i,1−n−i )) (resp., Ker(adj(∪i,1−n−i ))) is isomorphic to a subobject (resp., a quotient) of H i (Err) (resp., of H i−1 (Err)). In particular, if H i−1 (Err) = H i (Err) = 0 in (R Mod)/(pseudo − null), then adj(∪i,1−n−i ) is an isomorphism in (R Mod)/(pseudo − null). Proof. The proof of Proposition 2.10.12 applies, with Proposition 2.10.16 replacing Lemma 2.10.11. (2.10.18) Proposition. Let X and Y be R-modules of finite type with support of codimension ≥ 1 in Spec(R). Assume that, for each prime ideal p ∈ Spec(R) with ht(p) = 1, there exists a monomorphism (resp., epimorphism, resp., isomorphism) of Rp -modules gp : Xp −→ Yp . Then there exists a monomorphism (resp., epimorphism, resp., isomorphism) g : X −→ Y in (R Mod)/(pseudo − null). Proof. For each prime ideal p in the finite set A(X) = {p ∈ supp(X) | ht(p) = 1}, denote by fp : X −→ Xp the canonical map. The kernel of M

f = (fp ) : X −→

Xp

p∈A(X)

is pseudo-null, and so is the cokernel of the canonical map M Im(f ) −→ Im(fp ). p∈A(X)

It is enough, therefore, to consider only the case when A(X) = A(Y ) = {p} and gp

X ⊂ Xp −→Yp ⊃ Y. There exists r ∈ R − p such that r · gp (X) ⊂ Y ; restricting the map r · gp to X defines the desired morphism of R-modules g : X −→ Y . (2.10.19) Corollary. If R satisfies (R1 ) and X is an R-module of finite type with codimR (supp(X)) ≥ 1, then X is isomorphic in (R Mod)/(pseudo − null) to ∼

X −→

M M

(R/pi )n(p,i) ,

ht(p)=1 i≥1

where n(p, i) ≥ 0 and only finitely many n(p, i) are non-zero. Proof. For each prime ideal p ∈ A(X) (using the same notation as in the proof of Proposition 2.10.18), the localization Rp is a discrete valuation ring, hence 57

∼

Xp −→

M

(Rp /pi Rp )n(p,i)

i≥1

(where the sum is finite). The claim follows from Proposition 2.10.18 applied to X and M M Y = (R/pi )n(p,i) . p∈A(X) i≥1

(2.10.20) Lemma. Assume that dim(R) = depth(R) = 1 and that ωR exists. Let M, N be R-modules of finite type and h : M ⊗R N −→ H 0 (ωR ) a bilinear form. Denote the corresponding adjoint maps by α := adj(h) : M −→ HomR (N, H 0 (ωR )) β := adj(h ◦ s12 ) : N −→ HomR (M, H 0 (ωR )). If, for each minimal prime ideal q ⊂ R, the localization αq is an isomorphism, then: (i) For each minimal prime ideal q ⊂ R, βq is an isomorphism. (ii) Ker(α) = Mtors , Ker(β) = Ntors . ∼ (iii) DR (Coker(α)) −→ Coker(β). (iv) `R (Coker(α)) = `R (Coker(β)). Proof. (i) We have H 0 (ωR )q = IRq , hence hq : Mq ⊗Rq Nq −→ IRq , αq : Mq −→ DRq (Nq ); thus DRq (αq )

ε

βq : Nq −→DRq (DRq (Nq ))−−−−→DRq (Mq ) is an isomorphism by Proposition 2.10.2(v). As regards (ii), we have Mtors ⊆ Ker(α), since H 0 (ωR ) is torsion-free. On the other hand, M Ker(α) ⊆ Ker(M −→ Mq ) = Mtors , q

since all αq are isomorphisms. (iii) According to (ii), the map α factors through M/Mtors −→ HomR (N, H 0 (ωR )) (and similarly for β). Fixing an injective resolution • i : H 0 (ωR ) −→ ωR ,

for each Z = M, N the canonical map HomR (Z/Ztors , H 0 (ωR )) −→ HomR (Z, H 0 (ωR )) resp., • • HomR (Z/Ztors , H 0 (ωR )) −→ HomR (Z/Ztors , ωR ) = D(Z/Ztors )

is an isomorphism (resp., a quasi-isomorphism, by Lemma 2.10.11). This implies that we have an exact triangle in Df t (R Mod) α0

M/Mtors −→D(N/Ntors ) −→ Coker(α). Applying D, we obtain another exact triangle D(α0 )

D(Coker(α)) −→ D(D(N/Ntors ))−−→D(M/Mtors ). 58

As D(α0 ) ◦ ε = β 0 and ε : N/Ntors −→ D(D(N/Ntors )) is an isomorphism, we obtain (again by Lemma 2.10.11) isomorphisms in Df t (R Mod) ∼

∼

Coker(β) −→ D(Coker(α))[1] −→ D(Coker(α)), ∼

hence an isomorphism Coker(β) −→ D(Coker(α)) in (R Mod). Finally, (iv) follows from (iii). (2.10.21) In the situation of Lemma 2.10.20, we use the following notation: `R (det(h)) := `R (Coker(α)) = `R (Coker(β)). (2.11) Semi-local R (2.11.1) Everything in Sect. 2.1-2.9 has a straightforward generalization to the case when R is an equidimensional semi-local noetherian ring, complete with respect to its radical m. In this case R has finitely many maximal ideals m1 , . . . , mr and is isomorphic to ∼

R −→ Rm1 × · · · × Rmr . Similarly, every R-module M decomposes canonically as ∼

M −→ Mm1 ⊕ · · · ⊕ Mmr , and the theory in 2.1-2.9 applies separately to each Rmi -module Mmi . (2.11.2) If R is of the form considered in 2.11.1, so is every finite R-algebra R0 (e.g. R0 = R[∆] for a finite abelian group ∆).

59

3. Continuous Cohomology In this chapter we develop a formalism of continuous cohomology for a certain class of R[G]-modules. Our approach is purely algebraic; the fundamental objects are the “admissible” R[G]-modules, even though the cohomology can be defined even for “ind-admissible” modules (filtered inductive limits of admissible modules). Section 3.6 can be ignored; it is unrelated to the rest of the article. (3.1) Properties of R-modules of finite type We shall repeatedly use the following standard facts about R-modules of finite type ([Bou], Ch. III). Let f : M −→ N be an R-linear map between R-modules of finite type; equip both M and N with the m-adic topology. Then (3.1.1) (3.1.2) (3.1.3) (3.1.4) from N . (3.1.5)

M and N are Hausdorff and linearly compact. f is continuous. Im(f ) is closed in N . f is strict, i.e. the quotient topology on Im(f ) = M/Ker(f ) coincides with the topology induced If f is surjective, then it admits a continuous (not necessarily R-linear) section.

(3.2) Admissible R[G]-modules Let G be a group acting R-linearly on an R-module M . The action can be described either as a map λM : G × M −→ M,

λM (g, m) = g(m),

or by the induced map ρM : R[G] −→ EndR (M ),

X X ρM ( ri gi )(m) = ri gi (m).

Throughout Chapter 3, G will be a (Hausdorff) topological group. (3.2.1) Definition. An R[G]-module M is admissible iff (i) The image of ρM is an R-module of finite type; and ρM i (ii) The map G−→R[G]−→Im(ρM ) is continuous (if Im(ρM ) is equipped with m-adic topology). A morphism between admissible R[G]-modules M, N is an R[G]-linear map f : M −→ N . Admissible R[G]-modules form a full subcategory (ad R[G] Mod) of (R[G] Mod). (3.2.2) Lemma. Let M be an admissible R[G]-module. Then (i) If N ⊂ M is an R-submodule of finite type, then R[G] · N is of finite type over R. (ii) M is the union of its R[G]-submodules that are of finite type over R. (iii) If N ⊂ M is an R[G]-submodule, then both N and M/N are admissible. (iv) If f : H −→ G is a (continuous) homomorphism of topological groups, then f ∗ M (= M viewed as an H-module) is an admissible R[H]-module. (v) If H C G is a closed normal subgroup of G, then M H is an admissible G/H-module. Proof. (i) R[G] · N is contained in the image of ev

Im(ρM ) ⊗R N −→ EndR (M ) ⊗R M −→M. (ii) This follows from (i). (iii) The commutative diagrams ρM

R[G]  ρN y

−→

EndR (N )

,→

α

EndR (M )   0 yα HomR (N, M )

R[G]  ρM/N y EndR (M/N ) 60

ρM

−→ β

,→

EndR (M )   0 yβ HomR (M, M/N )

show that both Im(ρN ) and Im(ρM/N ) are of finite type over R. The map Im(ρM ) ,→ α0 (Im(ρM )) = α(Im(ρN )) induced by α0 is m-adically continuous by (3.1.2), hence G −→ Im(ρN ) is continuous by (3.1.4). The same argument works for G −→ Im(ρM/N ). (iv) This follows from definitions. (v) The commutative diagram R[G]  ρM y

can

−→

R[G/H]

ρM H

EndR (M H )  β y

−→

α

EndR (M )

HomR (M H , M )

,→

shows that Im(ρM H ) = β −1 (α(Im(ρM ))) is of finite type over R. The composite map can

ρ

Hi

β0

M G−→G/H −−−−→Im(ρ M H )−→α(Im(ρM ))

is continuous and the maps can, β 0 (the latter is induced by β) are strict, hence G/H −→ Im(ρM H ) is continuous. (3.2.3) Corollary. (ad R[G] Mod) is an abelian category (satisfying (AB1), (AB2)). (R[G] Mod) preserves finite limits and finite colimits.

Its embedding into

(3.2.4) Lemma. Let T (resp., A) be an R[G]-module of finite (resp., co-finite) type over R. Equip T (resp., A) with m-adic (resp., discrete) topology. Then T (resp., A) is an admissible R[G]-module iff the map λT : G × T −→ T (resp., λA : G × A −→ A) is continuous. Proof. Let M = T or A. If ρM i : G −→ Im(ρM ) is continuous, so is ρM i×idM

ev

λM : G × M −−−−−−→Im(ρM ) × M −→M. Conversely, assume that λT is continuous. By (3.1.4) it is enough to check that ρT i : G −→ EndR (T ) is continuous. By a version of the Artin-Rees Lemma ([Bou], III.3 Prop. 2) there is n0 such that
Ker EndR (T ) −→ HomR (T, T /mn+n0 T ) ⊂ mn EndR (T ) (∀n ≥ 0). By continuity of λT , for each n ≥ 0 there is a neighbourhood of unity Un ⊂ G such that ρM (Un ) acts trivially on T /mn+n0 T ; thus ρT (Un ) ⊂ 1 + mn EndR (T ) and ρT i is continuous. If λA is continuous, then there is, for each n ≥ 0, a neighbourhood of unity Vn ⊂ G stabilizing pointwise A[mn ]. Put T = D(A) with G-action given by (g(t))(a) = t(g −1 (a)). Then, for g ∈ Vn , (g − 1)T ⊆ A[mn ]⊥ = mn T . This means that λT is continuous, hence T is admissible. Admissibility of A follows from the next Proposition. (3.2.5) Proposition. If M, N are admissible R[G]-modules, then both P = M ⊗R N and Q = HomR (M, N ) are admissible. Proof. The modules α

Im(ρP ) ⊂ Im (Im(ρM ) ⊗R Im(ρN ) −→ EndR (M ) ⊗R EndR (N )−→EndR (P )) β

Im(ρQ ) ⊂ Im (Im(ρM ) ⊗R Im(ρN ) −→ EndR (M ) ⊗R EndR (N )−→EndR (Q)) are both of finite type over R (the maps α, β are given by (α(f ⊗ g))(x ⊗ y) = f (x) ⊗ g(y); (β(f ⊗ g))(q) = g ◦ q ◦ f ). The diagonal action (ρM i,ρN i)

G−−−−−−→Im(ρM ) × Im(ρN ) −→ Im(ρM ) ⊗R Im(ρN ) is continuous, hence both ρP i and ρQ i are continuous by (3.1.4). 61

(3.2.6) Proposition. Let M • be a bounded above complex of admissible R[G]-modules with all cohomology groups H i (M • ) of finite type over R. Then there is a subcomplex N • ,→ M • (of admissible R[G]-modules) such that (i) Each N i is of finite type over R. (ii) The inclusion N • ,→ M • is a quasi-isomorphism. Proof. If M i = 0 for all i then take N • = M • . If M j 6= 0 but M i = 0 for i > j, choose an R-submodule of finite type X j ⊂ M j that surjects onto H j (M • ). Then N j = R[G] · X j ⊂ M j is of finite type over R by Lemma 3.2.2(i). Put Y j−1 = d−1 (N j ) ⊃ Z j−1 = Ker(d : M j−1 −→ M j ) and choose an R-submodule of finite type X j−1 ⊂ Y j−1 such that dX j−1 = dY j−1 and that X j−1 ∩ Z j−1 surjects onto H j−1 (M • ). Again N j−1 = R[G] · X j−1 ⊂ Z j−1 is of finite type over R. We put Y j−2 = d−1 (N j−1 ) and continue this process. (3.2.7) Corollary. Let A• be a bounded below (resp., bounded) complex of admissible R[G]-modules with all cohomology groups H i (A• ) of co-finite type over R. Then there is a bounded below (resp., bounded) complex B • of admissible R[G]-modules of co-finite type over R and a map of complexes A• −→ B • which is a quasi-isomorphism. Proof. Applying Proposition 3.2.6 to M • = D(A• ) we get a subcomplex incl : N • ,→ D(A• ). For B • := D(N • ) the canonical map D(incl)

can

A• −→D(D(A• ))−−−−→D(N • ) = B • is a composition of two quasi-isomorphisms (using (2.3.2) and (3.2.6)). If A• is bounded, so is B • . ad (3.2.8) Proposition. Denote by (ad R[G] Mod)R−f t (resp., (R[G] Mod)R−cof t ) the category of admissible R[G]modules of finite (resp., co-finite type) over R. Then the embeddings ad ad (ad R[G] Mod)R−f t −→ (R[G] Mod) ←− (R[G] Mod)R−cof t

induce equivalences of categories ≈

(∗ = −, b)

≈

(∗ = +, b).

∗ ad D∗ ((ad R[G] Mod)R−f t )−→DR−f t (R[G] Mod) ∗ ad D∗ ((ad R[G] Mod)R−cof t )−→DR−cof t (R[G] Mod)

Proof. Essential surjectivity follows from Proposition 3.2.6 and Corollary 3.2.7. Full-faithfulness is a general nonsense ([Ve 2], III.2.4.1). (3.3) Ind-admissible R[G]-modules (3.3.1) Definition. Let M be an R[G]-module. Denote by S(M ) the set of R[G]-submodules Mα ⊂ M satisfying (a) Mα is of finite type over R; (b) The action λMα : G × Mα −→ Mα is continuous (with respect to the m-adic topology on Mα ). (3.3.2) Lemma. (i) If Mα ∈ S(M ), then N ∈ S(M ) for every R[G]-submodule N ⊂ Mα . (ii) If f : M −→ N is a homomorphism of R[G]-modules and Mα ∈ S(M ), then f (Mα ) ∈ S(N ). (iii) If Mα , Mβ ∈ S(M ), then Mα + Mβ ∈ S(M ). Proof. All one needs to do is to check the condition (b) of the definition. In (i) (resp., (ii)) the continuity of λN (resp., λf (Mα ) ) follows from the continuity of λMα and the fact that the inclusion N ,→ Mα (resp., the surjection f : Mα −→ f (Mα )) is a strict map. The statement (iii) follows from (ii), as Mα + Mβ is the image of Mα ⊕ Mβ ∈ S(M ⊕ M ) under the sum map Σ : M ⊕ M −→ M . 62

(3.3.3) Corollary. Let f : M −→ N be a homomorphism of R[G]-modules. Then [

j(M ) :=

Mα

Mα ∈S(M)

is an R[G]-submodule of M , j(j(M )) = j(M ) and f (j(M )) ⊆ j(N ). (3.3.4) Definition. An R[G]-module M is ind-admissible if M = j(M ). (3.3.5) Proposition. (i) Ind-admissible R[G]-modules form a full (abelian) subcategory (ind−ad R[G] Mod) of (R[G] Mod), which is stable under subobjects, quotients, colimits and tensor products. (ii) The embedding functor i : (ind−ad R[G] Mod) −→ (R[G] Mod) is exact and is left adjoint to j : (R[G] Mod) −→ (ind−ad R[G] Mod).

(iii) The functor j is left exact and preserves injectives; the category (ind−ad R[G] Mod) has enough injectives. (iv) Every admissible R[G]-module is ind-admissible. (v) An ind-admissible R[G]-module M is admissible iff Im(ρM ) is an R-module of finite type. (vi) An R[G]-module M of finite (resp., co-finite) type over R is ind-admissible iff it is admissible. (vii) For M, N ∈ (ind−ad R[G] Mod), the canonical maps ∼

HomR[G] (M, N ) −→

∼

lim ←−

Mα ∈S(M)

HomR[G] (Mα , N ) ←−

lim ←−

lim −→

Mα ∈S(M) Nβ ∈S(N )

HomR[G] (Mα , Nβ )

are both isomorphisms. ad (viii) The categories of ind-objects Ind((ad R[G] Mod)R−f t ) and Ind(R[G] Mod) are canonically equivalent to (ind−ad R[G] Mod).

S Proof. (i) If M S = j(M ) is ind-admissible and N is an R[G]-submodule of M , then both N = (N ∩ Mα ) and M/N = (Mα /(N ∩ Mα )) are ind-admissible (Mα ∈ S(M L )). This proves stability by subquotients. Every colimit lim M (β) is a quotient of the direct sum M = M (β). If each M (β) is ind-admissible, so S −→ is M = (M (β ) ⊕ · · · ⊕ M (β ) ) (M (β ) ∈ S(M (β ))). Finally, if M = j(M ) and N = j(N ), then 1 α n α i α i 1 n i S M ⊗R N = Im(Mα ⊗R Nβ −→ M ⊗R N ) = j(M ⊗R N ). (ii) The functors i, j form an adjoint pair almost by definition; i commutes with finite limits by (i). (iii) As i is exact, its right adjoint j preserves injectives (and is left exact by adjointness). For every ind-admissible R[G]-module M there is a monomorphism i(M ) −→ J with J injective in (R[G] Mod); then M −→ j(J) is a monomorphism with j(J) injective in (ind−ad R[G] Mod). (iv) Use (i), Lemma 3.2.2(ii) and Lemma 3.2.4. (v) If M is an ind-admissible R[G]-module, then the canonical map u : Im(ρM ) −→

lim ←−

Mα ∈S(M)

Im(ρMα )

is an isomorphism of R-modules. If, in addition, Im(ρM ) is an R-module of finite type, then u is a homeomorphism with respect to m-adic topologies on Im(ρM ) and Im(ρMα ); thus G −→ Im(ρM ) is continuous and M is admissible. (vi) This follows from (v). (vii) The first arrow is an isomorphism by definition of colimits. As regards the second arrow, note that lim −→

Nβ ∈S(N )

HomR[G] (Mα , Nβ ) −→ HomR[G] (Mα , N )

is an isomorphism, since the image of any R[G]-linear map Mα −→ N is of finite type over R, hence is contained in some Nβ . (viii) For a category C, an object of Ind(C) is a functor F : J −→ C, where J is a small filtered category. Morphisms in Ind(C) are given by 63

0 0 HomInd(C) (F, F 0 ) = lim ←− lim −→ HomC (F (j), F (j )). J

J0

In the special case of C = (ad F (j) in (R[G] Mod) defines functors R[G] Mod), attaching to F the colimit lim −→ J S

T

ind−ad ad Ind((ad R[G] Mod)R−f t )−→Ind(R[G] Mod)−→(R[G] Mod).

It follows from (iv) (resp., (vii)) that S (resp., T ◦ S) is an equivalence of categories. ind−ad ind−ad (3.3.6) Lemma. (i) If M ∈ (ad R[G] Mod)R−f t and N ∈ (R[G] Mod), then HomR (M, N ) ∈ (R[G] Mod). H (ii) If M ∈ (ind−ad ∈ (ind−ad R[G] Mod) and H C G is a closed normal subgroup of G, then M R[G/H] Mod).

S Proof. (i) Write N = Nβ with Nβ ∈ S(N ). The R[G]-modules HomR (M, Nβ ) are all admissible (hence indadmissible) by Proposition 3.2.5; it follows from Proposition 3.3.5(i) that HomR (M, N ) = − lim → HomR (M, Nβ ) β

is ind-admissible as well. (ii) By Proposition 3.3.5(i), M H is ind-admissible as an R[G]-module. The claim follows from the fact that G −→ G/H is a quotient map (i.e. G/H has the quotient topology). b = lim G/U be the pro-finite completion of G with the pro-finite topology (U (3.3.7) Proposition. Let G ←− runs through all normal subgroups of G of finite index). If k is finite, then the action ρM : G −→ AutR (M ) of G on every admissible (resp., ind-admissible) R[G]-module M factors canonically through the natural map b this makes M into an admissible (resp., ind-admissible) R[G]-module. b G −→ G; S Proof. If M = Mα (Mα ∈ S(M )) is ind-admissible, then each group AutR (Mα ) is finite and the map ρM : G −→ lim ←− AutR (Mα ) α

(⊂ AutR (M ))

is continuous with respect to the pro-finite topology on the target, hence factors canonically through a b −→ lim AutR (Mα ). continuous homomorphism G ←− b is continuous, then it induces equivalences of categories (3.3.8) Corollary. If the natural map G −→ G ≈

(ad b Mod)−→(ad R[G] Mod), R[G]

≈

(ind−ad Mod)−→(ind−ad R[G] Mod). b] R[G

(3.3.9) Proposition. Let M • be a bounded above complex of ind-admissible R[G]-modules with all cohomology groups H i (M • ) of finite type over R. Then there is a subcomplex N • ,→ M • (of admissible R[G]-modules) such that (i) Each N i is of finite type over R. (ii) The inclusion N • ,→ M • is a quasi-isomorphism. Proof. The proof of Proposition 3.2.6 applies word by word. (3.3.10) Proposition. The embeddings ind−ad ind−ad ad (ad R[G] Mod)R−f t = (R[G] Mod)R−f t −→ (R[G] Mod) −→ (R[G] Mod)

induce equivalences of categories ≈

≈

− − ind−ad − ind−ad ad D− ((ad Mod)R−f t )−→DR−f R[G] Mod)R−f t ) = D ((R[G] t (R[G] Mod)−→DR−f t (R[G] Mod)

Proof. As in 3.2.8. (3.4) Continuous cochains (3.4.1) Let G be a topological group and M an ind-admissible R[G]-module. 64

(3.4.1.1) Definition. (Non-homogeneous) continuous cochains of degree i ≥ 0 on G with values in M are defined as i i Ccont (G, M ) = lim −→ Ccont (G, Mα ), Mα ∈S(M)

i Ccont (G, Mα )

where In other words,

is the R-module of continuous maps Gi −→ Mα (Mα is equipped with m-adic topology). i i n Ccont (G, Mα ) = ← lim − Ccont (G, Mα /m Mα ). n

(3.4.1.2) The standard differential

(δc)(g1 , . . . , gi+1 ) = g1 c(g2 , . . . , gi+1 ) +

i X

(−1)j c(g1 , . . . , gj gj+1 , . . . , gi+1 ) + (−1)i−1 c(g1 , . . . , gi )

j=1 i+1 i maps Ccont (G, Mα ) to Ccont (G, Mα ) (by Lemma 3.2.4), hence δi

M i+1 i · · · −→ Ccont (G, M )−→C cont (G, M ) −→ · · · • becomes a complex Ccont (G, M ) of R-modules. • (3.4.1.3) Let M • be a complex of ind-admissible R[G]-modules. We define Ccont (G, M • ) to be the simple j i complex attached to Ccont (G, M ): its component of degree n is equal to M j n Ccont (G, M • ) = Ccont (G, M i )

i+j=n

(if M • is bounded below, then the sum is finite and vanishes for n 0 and that H C G is a closed normal subgroup of G, with pro-finite order prime to p. Then the inflation map • • inf : Ccont (G/H, M H ) −→ Ccont (G, M )

is a quasi-isomorphism. Proof. The inflation map is induced by the pair of morphisms G −→ G/H, M H ,→ M (using Lemma 3.3.6(ii)). According to 3.5.1.2 we can assume that M = T is of finite type over R. Corollary 4.1.3 further 75

reduces to the case of M = T /mn T of finite length over R. In this case M is a p-primary torsion discrete G-module and the statement follows from the degeneration of the Hochschild-Serre spectral sequence E2i,j = H i (G/H, H j (H, M )) =⇒ H i+j (G, M )

(4.1.4.1)

(E2i,j = 0 for j 6= 0). (4.2) Finiteness conditions (4.2.1) Consider the following finiteness conditions on G: (F) `R (H i (G, M )) < ∞ for every discrete R[G]-module M of finite length over R and every i ≥ 0. (F’) dimk H i (U, k) < ∞ for every open normal subgroup U C G and every i ≥ 0. By Shapiro’s Lemma, (F) for G implies (F) for every open subgroup of G; in particular (F) implies (F’). (4.2.2) Lemma. We have implications ∼

i i n (F 0 ) ⇐⇒ (F ) =⇒ Hcont (G, T ) −→ lim ←− H (G, T /m T ) n

(∀i ≥ 0).

Proof. Assume (F’) holds. Given M as in (F), there is an open normal subgroup U C G acting trivially on M . Then (F) follows from the Hochschild-Serre spectral sequence (4.1.4.1), as G/U is finite and `R (H j (U, M )) < ∞ by (F’) and d´evissage. The converse is true by Shapiro’s Lemma, as observed in 4.2.1. (1) i−1 n Assuming that (F) holds, the ← lim − -term in Corollary 4.1.3 vanishes, as n 7→ H (G, T /m T ) is a MittagLeffler system. i i (4.2.3) Proposition. If G satisfies (F), then Hcont (G, T ) (resp., Hcont (G, A)) is of finite (resp., co-finite) type over R for every i ≥ 0.

Proof. Induction on d = dim(R). There is nothing to prove for d = 0. If d ≥ 1, choose x ∈ m such that ∼ i dim(R/xR) = d − 1. The R-module M = Hcont (G, T ) −→ lim ←− Mn is the projective limit of a surjective n

projective system of R-modules of finite length Mn = Im(M −→ H i (G, T /mn T )) satisfying mn Mn = 0. Denote by jn : M  Mn the canonical projection. The exact cohomology sequences of x

0 −→ T [x] −→T −→xT −→ 0 0 −→ xT −→T −→ T /xT −→ 0 (valid by Proposition 3.4.2) together with the induction hypothesis show that M/xM is of finite type over R/xR. Fix an epimorphism (R/xR)a  M/xM and lift it to a homomorphism of R-modules f : Ra −→ M . Put N = Coker(f ), Nn = Coker(jn ◦ f ), Kn = Ker(jn ◦ f ). The projective system Nn /xNn consists of R-modules of finite length, has surjective transition maps and its projective limit lim (Nn /xNn ) = (lim Nn )/x(lim Nn ) = 0 ← ← ← n− n− n− vanishes, being a quotient of N/xN = 0. It follows that, for all n, Nn /xNn = 0, hence Nn = 0 by Nakayama’s Lemma. The projective systems of exact sequences 0 −→ mn Ra −→ Kn −→ Kn /mn Ra −→ 0,

0 −→ mn Ra −→ Ra −→ (R/mn )a −→ 0

imply that (1) (1) n a (1) a lim Kn = lim m R =← lim R = 0, ← ← n− n− n−

and the exact sequence 0 −→ Kn −→ Ra −→ Mn −→ 0 yields 76

(1) Ra −→ ← lim Mn (= M ) −→ lim Kn = 0, ← n− n− ∼

i n proving that M is of finite type over R. Dually, P = H i (G, A) −→ lim −→ Pn , where Pn = Im(H (G, A[m ]) −→ n

P ) is an injective inductive system, and P [x] is of co-finite type over R/xR. Fixing a monomorphism P [x] ,→ (I[x])b and extending it to a homomorphism of R-modules g : P −→ I b , the vanishing of lim (Pn ∩ Ker(g))[m] ⊂ lim (Pn ∩ Ker(g))[x] ⊂ Ker(g)[x] = 0 −→ −→ n n implies that all maps Pn ,→ P −→ I b are injective; thus g : P −→ I b is also injective, as required. (4.2.4) Lemma. If G satisfies (F), then the canonical maps   u • • n n lim D (C (G, T /m T )) −→D lim C (G, T /m T ) cont cont −→ ← n n−   v n n • • D lim −→ Ccont (G, A[m ]) −→ lim ←− D (Ccont (G, A[m ])) n

n

are quasi-isomorphisms. Proof. The induced map on cohomology H −i (u) is equal to the composition  
u1
u2 i n i n i lim D H (G, T /m T ) −→D lim H (G, T /m T ) −→D H (G, T ) cont −→ ← n n− The map u1 (resp., u2 ) is an isomorphism by a combination of (2.3.4) and Proposition 4.2.3 (resp., by Lemma 4.2.2). Similarly, the composition of  
H −i (v) −i
v2 n • D H (G, A) −−→ H lim D (Ccont (G, A[m ])) −→ lim D H i (G, A[mn ]) ← − ← − n n i

is an isomorphism by (2.3.4) and v2 is an isomorphism by the argument used in the proof of the second implication in Lemma 4.2.2. + ind−ad (4.2.5) Proposition. If G satisfies (F), then the functor M 7→ RΓcont (G, M ) maps DR−f t (R[G] Mod) + ind−ad + + (resp., DR−cof t (R[G] Mod)) to Df t (R Mod) (resp., Dcof t (R Mod)).

Proof. For M = T or A this is the statement of Proposition 4.2.3. The general case follows from the hyper-cohomology spectral sequence (3.5.4.2). (4.2.6) Lemma. If char(k) = p > 0 and cdp (G) = e < ∞, then i (i) Hcont (G, M ) = 0 for every i > e and every ind-admissible R[G]-module M . (ii) If M • is a bounded below complex of ind-admissible R[G]-modules with H i (M • ) = 0 for i > c, then j Hcont (G, M • ) = 0 for every j > c + e. Proof. (i) By (3.5.1.2) we can assume that M = T is of finite type over R. It follows from Corollary 4.1.3 i that Hcont (G, M ) = 0 for i > e + 1. For i = e + 1 we have e+1 (1) e n Hcont (G, M ) = lim ←− H (G, T /m T ) = 0, n

since n 7→ H e (G, T /mn T ) is a surjective projective system. (ii) This follows from (i) and the spectral sequence (3.5.4.2). 77

(4.2.7) Corollary. Under the assumptions of 4.2.6, (i) The functor + RΓcont (G, −) : D+ (ind−ad R[G] Mod) −→ D (R Mod) b maps Db (ind−ad R[G] Mod) to D (R Mod).

ind−ad ind−ad ∗ ∗ (ii) If, in addition, G satisfies (F), then RΓcont (G, −) maps DR−f t (R[G] Mod) (resp., DR−cof t (R[G] Mod)) ∗ ∗ to Df t (R Mod) (resp., Dcof t (R Mod)) for ∗ = +, b.

Proof. Combine Proposition 4.2.5 and Lemma 4.2.6. (4.2.8) Perfect complexes. Let A be a noetherian ring. Recall ([SGA 6], Exp. I, Cor 5.8.1) that a complex M • of A-modules is perfect (i.e. there exists a quasi-isomorphism P • −→ M • , where P • is a bounded complex of projective A-modules of finite type) iff the following conditions are satisfied: (a) (∀i ∈ Z) H i (M • ) is of finite type over A. (b) H i (M • ) = 0 for all but finitely many i ∈ Z. (c) The complex M • has finite Tor-dimension, i.e. • (∃c ∈ Z)(∀N ∈ (A Mod))(∀i > c) TorA i (M , N ) = 0.

Perfect complexes over A form a full subcategory Dparf (A Mod) of Db (A Mod). A theorem of Serre and Auslander-Buchsbaum ([Br-He], Thm. 2.2.7) implies that, for our ring R, R is regular ⇐⇒ Dparf (R Mod) = Dfb t (R Mod). [a,b]

One says that M • ∈ Dparf (A Mod) has perfect amplitude contained in [a, b] (notation: M • ∈ Dparf (A Mod)) if the complex P • above can be chosen in such a way that P i = 0 for every i < a and i > b. If this is the case, then [a,b−1]

M • ∈ Dparf

(A Mod) ⇐⇒ H b (M • ) = 0;

more generally, [a,b]

[c,d]

[max(a,c),min(b,d)]

Dparf (A Mod) ∩ Dparf (A Mod) = Dparf

(A Mod)

[a,b]

[−b,−a]

([SGA 6], Exp. I, Lemma 4.13). The functor RHomA (−, A) maps Dparf (A Mod) into Dparf

(A Mod).

(4.2.9) Proposition. Assume that G satisfies (F), char(k) = p > 0 and cdp (G) = e < ∞. Let S be a multiplicative subset of R and RS the corresponding localization. If M • is a bounded complex of [a,b] ind-admissible R[G]-modules such that M • ⊗R RS ∈ Dparf (RS Mod) (if we disregard the G-action), then [a,b+e]

RΓcont (G, M • ) ⊗R RS ∈ Dparf

(RS Mod).

Proof. It follows from Corollary 4.2.7 that RΓcont (G, M • ) is an object of Dfb t (R Mod); it remains to verify that RΓcont (G, M • ) ⊗R RS has finite Tor-dimension over RS . As explained to us by O. Gabber, this follows by a standard “way-out” argument from the fact that RΓcont (G, −) commutes with filtered direct limits: let N • be a bounded complex of RS -modules; consider the canonical map • • λN • : (Ccont (G, M • ) ⊗R RS ) ⊗RS N • −→ Ccont (G, (M • ⊗R RS ) ⊗RS N • ).

As RS is flat over R, it follows from Proposition 3.4.4 that λN • is an isomorphism of complexes whenever N • is a complex of flat RS -modules. Given an arbitrary RS -module N , choose its resolution F • by free RS -modules: · · · −→ F −2 −→ F −1 −→ F 0 −→ N −→ 0. Fix k >> 0 and consider the truncated complex Fk• := (σ≥−k F • ) : F −k −→ · · · −→ F 0 ; 78

it satisfies • S (∀j < k − b) H −j ((M • ⊗R RS ) ⊗RS Fk• ) = TorR j (M ⊗R RS , N ) • • • S (∀` < k − b) H −` ((Ccont (G, M • ) ⊗R RS ) ⊗RS Fk• ) = TorR ` (Ccont (G, M ) ⊗R RS , N ).

The cohomology of the bounded complex (of ind-admissible R[G]-modules) B • = (M • ⊗R RS ) ⊗RS Fk• satisfies (∀j > b − k)



 H j (B • ) 6= 0 =⇒ a ≤ j ≤ b .

It follows from Lemma 4.2.6 and the hyper-cohomology spectral sequence i+j i E2i,j = Hcont (G, H j (B • )) =⇒ Hcont (G, B • )

that E2i,j 6= 0 =⇒ [0 ≤ i ≤ e and (a ≤ j ≤ b or j ≤ b − k)] . Using the fact that λFk• is an isomorphism, we get h i −` • • • S (∀` < k − b − e) TorR (C (G, M ) ⊗ R , N ) = H (G, B ) = 6 0 =⇒ −b − e ≤ ` ≤ −a . R S cont cont ` • This finishes the proof that Ccont (G, M • ) ⊗R RS has perfect amplitude contained in [a, b + e], since k can be chosen arbitrarily large.

(4.2.10) Proposition. Let S ⊂ R be a multiplicative subset. If G satisfies (F), then RΓcont (G, −) maps + DR (ind−ad Mod) to Df+t (RS Mod). If, in addition, char(k) = p > 0 and cdp (G) < ∞, then RΓcont (G, −) S −f t RS [G] b maps DR (ind−ad Mod) to Dfb t (RS Mod). S −f t RS [G]

Proof. Combine Lemma 3.7.3, Proposition 3.7.4(ii), Proposition 4.2.5 and Corollary 4.2.7. (4.3) The duality diagram: T, A, T ∗ , A∗ + ind−ad + ind−ad (4.3.1) Let T ∈ DR−f t (R[G] Mod); put A = Φ(T ) ∈ DR−cof t (R[G] Mod). Proposition 3.5.8 then implies that the canonical map

Φ(RΓcont (G, T )) −→ RΓcont (G, A)

(4.3.1.1)

is an isomorphism in D+ (R Mod). If we assume, in addition, that G satisfies (F), then RΓcont (G, T ) (resp., RΓcont (G, A)) lies in Df+t (R Mod) + (resp., Dcof t (R Mod)). Combining the spectral sequence (2.8.6.1) with the isomorphism (4.3.1.1) we get a spectral sequence j i+j E2i,j = ExtiR (D(Hcont (G, A)), ω) =⇒ Hcont (G, T ).

(4.3.1.2)

(4.3.2) The construction from 3.5.9 defines functors op D(−) = RHomR (−, ω) : DR−f t (ad −→ DR−f t (ad R[G] Mod) R[G] Mod) op D(−) = RHomR (−, I) : DR−f t (ad ←→ DR−cof t (ad R[G] Mod) R[G] Mod) ∓ ad b ad b ad which map D± (ad R[G] Mod) to D (R[G] Mod) (hence D (R[G] Mod) to D (R[G] Mod)). Together with Φ these functors define a duality diagram

79

D

op DR−f t (ad ←−−−−−−−−→ DR−f t (ad R[G] Mod) R[G] Mod) ←−     −− D −−→ −−−−−   Φ Φ − − −− −−− y y − − → − ad op ← DR−cof t (R[G] Mod) DR−cof t (ad R[G] Mod),

commutative up to functorial isomorphisms defined by the same formulas as in 2.8.1. This diagram makes sense for an arbitrary topological group G, not necessarily pro-finite. + ind−ad (4.3.3) Proposition. Let T ∈ DR−f t (R[G] Mod); put A = Φ(T ). Assume that G satisfies (F). Then there are spectral sequences j i+j i+d E2i,j = H{m} (Hcont (G, T )) =⇒ Hcont (G, A) 0

−j −i−j i+d E2i,j = H{m} (D(Hcont (G, A))) =⇒ D(Hcont (G, T )).

Proof. Apply Proposition 2.8.7 to RΓcont (G, T ), RΓcont (G, A) instead of T, A (which is legitimate by 4.3.1). Of course, the spectral sequence 0 Er is just D(4.3.1.2). (4.3.4) In the special case when R = Zp and T is free over Zp , the spectral sequence E2i,j degenerates (assuming (F)) into a short exact sequence j j j+1 0 −→ Hcont (G, T ) ⊗Zp Qp /Zp −→ Hcont (G, A) −→ Hcont (G, T )tors −→ 0,

(4.3.4.1)

which coincides - up to a sign - with a piece of the cohomology sequence of 0 −→ T −→ V −→ A −→ 0, where V = T ⊗Zp Qp . (4.3.5) Applying 2.8.6 and 2.8.9 to RΓcont (G, T ), RΓcont (G, A) instead of T, A (again assuming (F)) we obtain exact sequences in (R Mod) /(pseudo − null) j−1 j j 0 −→ Ext1R (D(Hcont (G, A)), ω) −→ Hcont (G, T ) −→ Ext0R (D(Hcont (G, A)), ω) −→ 0

resp., in (R Mod) /(co − pseudo − null) j j j+1 d−1 d 0 −→ H{m} (Hcont (G, T )) −→Hcont (G, A) −→ H{m} (Hcont (G, T )) −→ 0 j j j−1 d−1 d 0 −→ H{m} (D(Hcont (G, A))) −→D(Hcont (G, T )) −→ H{m} (D(Hcont (G, A))) −→ 0, j j−1 d generalizing (4.3.4.1). Again, H{m} (Hcont (G, T )) (resp., Ext1R (D(Hcont (G, A)), ω)) is the maximal R-divisible j j (resp., R-torsion) subobject of Hcont (G, A) (resp., Hcont (G, T )) in (R Mod)/(co − pseudo − null) (resp., (R Mod)/(pseudo − null)).

(4.4) Comparing R+ Γder and RΓcont (4.4.1) Proposition. If cdp (G) ≤ 1, then θG (M ) : R+ Γder (G, M ) −→ RΓcont (G, M ) is an isomorphism for every M ∈ (ind−ad R[G] Mod). i Proof. The functors Hcont (G, −) are effaceable for i = 1 (resp., i > 1) by Proposition 3.6.2(iv) (resp., because they are zero, by Lemma 4.2.6(i)). The claim follows from Proposition 3.6.2(iii) by induction.

(4.4.2) Definition. Denote by (ind−ad ) the full subcategory of (ind−ad R[G] Mod){m} (resp., (R[G] Mod) R[G] Mod) S {m} n (resp., of (R[G] Mod)) consisting of objects M satisfying M = n≥1 M [m ]. (4.4.3) For a given M ∈ (ind−ad R[G] Mod){m} , every Mα ∈ S(M ) is an R-module of finite length, hence it is discrete in the m-adic topology. This implies that the normal subgroup U = Ker(G −→ Aut(Mα )) C G, • which acts trivially on Mα , is open in G, hence M is a discrete G-module and Ccont (G, M ) is the usual 80

complex of locally constant cochains. Using the language of [Bru], (ind−ad R[G] Mod){m} is the category of discrete modules over the pseudo-compact R-algebra R[[G]] = lim ←− R[G/U ]. U

As in Proposition 3.3.5(ii) there is an adjoint pair of functors i0 , j 0 , where ind−ad i0 : (ind−ad R[G] Mod){m} −→ (R[G] Mod)

S is the (exact) embedding functor and j 0 (M ) = n≥1 M [mn ]. As in Proposition 3.3.5, j 0 preserves injectives ind−ad 0 and (ind−ad R[G] Mod){m} has enough injectives of the form j (J), where J is injective in (R[G] Mod). The following statement is not an abstract nonsense. ind−ad (4.4.4) Proposition. The embedding functor i0 : (ind−ad R[G] Mod){m} −→ (R[G] Mod) preserves injectives. ind−ad Proof. Let J be injective in (ind−ad R[G] Mod){m} . We must show that for every diagram in (R[G] Mod) with exact row

0

−→

u

X −→  f y

Y

J there is a morphism g : Y −→ J such that f = gu. A standard argument using Zorn’s Lemma reduces the problem to the case when X and Y are of finite type over R. In this case f factors through X/mnX for suitable n. By Artin-Rees Lemma, there is k ≥ 0 such that Ker(un+k : X/mn+k X −→ Y /mn+k Y ) maps to zero in X/mn X. This implies that the projection X/mn+k −→ X/mnX factors through Im(un+k ). In the diagram 0

−→

X   y

−→

u

Y   y

0

−→

Im(un+k )   y

−→

Y /mn+k Y

X/mnX   y J n+k both Im(un+k ) and Y /mn+k Y are objects of (ind−ad Y −→ J R[G] Mod){m} ; it follows that there is h : Y /m h n+k extending Im(un+k ) −→ J. The composition g : Y −→ Y /m Y −→J satisfies gu = f as required.

(4.4.5) Lemma. If G is a finite group, then (i) (ind−ad R[G] Mod) = (R[G] Mod). i • (ii) Ccont (G, M ) = C • (G, M ), Hcont (G, M ) = H i (G, M ) (M ∈ (R[G] Mod), i ≥ 0). i (iii) (∀i ≥ 1) Hcont (G, −) is effaceable in (ind−ad Mod) = ( Mod). R[G] R[G] i (iv) (∀i ≥ 1) Hcont (G, −) is effaceable in (ind−ad R[G] Mod){m} = (R[G] Mod){m} .

81

i (v) (∀i ≥ 1) Hcont (G, −) vanishes on injective objects of (R[G] Mod) resp., (R[G] Mod){m} .

Proof. The statements (i),(ii) follow from the definitions. As regards (iii) and (iv), every M ∈ (R[G] Mod) embeds to the induced module HomR (R[G], M ) which has trivial cohomology. Finally, (v) follows from (iii) and (iv). (4.4.6) Proposition. Let G be a pro-finite group. Then i i (i) If J is injective in (ind−ad R[G] Mod){m} , then (∀i ≥ 1) Hcont (G, J) = Hder (G, J) = 0. (ii) The map θG (M ) : R+ Γder (G, M ) −→ RΓcont (G, M ) is an isomorphism for every M ∈ (ind−ad R[G] Mod){m} . Proof. (i) Let U C G be an open normal subgroup. The functor Γder (G, G/U, −) : (ind−ad R[G] Mod){m} −→ (ind−ad R[G/U ] Mod){m} preserves injectives; thus

i i U Hcont (G, J) = − lim → Hcont (G/U, J ) = 0 U

(∀i ≥ 1)

i by Lemma 4.4.5(v). The equality Hder (G, J) = 0 (for i ≥ 1) follows from Proposition 4.4.4. + • (ii) Let J be an injective resolution of M in (ind−ad R[G] Mod){m} . Then R Γder (G, M ) is represented by • the complex (J • )G (by Proposition 4.4.4) and the canonical morphism (J • )G −→ Ccont (G, J • ) is a quasiisomorphism by (i) and the spectral sequence (3.5.3.1).

(4.4.7) Lemma. Let H C G be a closed normal subgroup of G. Then ind−ad (i) The functor ResG,H : (ind−ad R[G] Mod){m} −→ (R[H] Mod){m} preserves injectives.

j j (ii) For every M ∈ (ind−ad R[G] Mod){m} and j ≥ 0 the canonical map R Γder (G, G/H, M ) −→ Hder (H, M ) is an isomorphism; it induces an isomorphism between the spectral sequence (3.6.3.1) and the Hochschild-Serre spectral sequence (4.1.4.1). ind−ad Proof. (i) Given an injective object J of (ind−ad R[G] Mod){m} , a monomorphism u : X −→ Y in (R[H] Mod){m} and a morphism f : X −→ ResG,H (J), we must show that there is g : Y −→ ResG,H (J) such that gu = f . As in the proof of Proposition 4.4.4 one can assume that both X and Y are of finite type over R. In this case there is a normal open subgroup U C G such that H ∩ U acts trivially on X and Y . We know that J U ind−ad U is injective in (ind−ad R[G/U ] Mod){m} , hence ResG,H (J ) is injective in (R[H/(H∩U )] Mod){m} . This implies that the composite map f

f 0 : X = X H∩U −→ResG,H (J H∩U ) ,→ ResG,H (J U ) extends to a map g 0 : Y = Y H∩U −→ ResG,H (J U ) such that g 0 u = f 0 ; this defines the required map g0

g : Y −→ResG,H (J U ) ,→ ResG,H (J). (ii) If J • is an injective resolution of M in (ind−ad R[G] Mod){m} , then it follows from (i) that the morphism in Proposition 3.6.4(i) is represented by the identity map id : (J • )H −→ (J • )H . (4.4.8) Question. Let H C G be a closed normal subgroup of G such that both K = H and K = G/H satisfy the condition (*) K satisfies (F) and θK (M ) is an isomorphism for every M ∈ (ind−ad R[K] Mod). Does it follow that G also satisfies (*)? (4.4.9) A positive answer to 4.4.8 in the simplest non-trivial case, when both H and G/H are topologically cyclic, would considerably simplify our treatment of unramified local conditions in Chapter 7. (4.5) Bar resolution 82

(4.5.1) Proposition. Every M ∈ (ind−ad R[G] Mod) has a canonical structure of an R[[G]]-module, where R[[G]] = lim ←− R[G/U ]. U

S

Mα (Mα ∈ S(M )) and Mα = lim Mα /mn Mα , the statement follows from the fact ← n− that each Mα /mn Mα is a module over R/mn R[G/Uα,n ], for a suitable open normal subgroup Uα,n C G. (4.5.2) The completed tensor products Proof. Writing M =

b i = R[[G]]⊗ bR · · · ⊗ b R R[[G]] = lim (R[G/U ] ⊗R · · · ⊗R R[G/U ]) = R[[G]]⊗ ←− U

i = lim ←− R[G/U × · · · × G/U ] = R[[G × · · · × G]] = R[[G ]] U

form a pro-finite bar resolution b i −→ · · · −→ R[[G]] b : · · · −→ R[[G]]⊗ R[[G]]•⊗ of R by projective pseudo-compact R[[G]]-modules. For each M ∈ (ind−ad R[G] Mod){m} , the complex •,naive b HomR[[G]],cont (R[[G]]⊗ • , M)

(where the subscript “cont” refers to homomorphisms continuous with respect to the pseudo-compact topo• logy on R[[Gi ]] and the discrete topology on M ) is canonically isomorphic to Ccont (G, M ). (4.5.3) Conjugation. Let G be a discrete group. For each σ ∈ G, the formula λnσ : g0 [g1 | · · · |gn ] 7→ g0 σ[σ −1 g1 σ| · · · |σ −1 gn σ] defines an isomorphism λσ : Z[G]•⊗ −→ Z[G]⊗ • between the bar resolution of Z and itself. These isomorphisms lift the identity id : Z −→ Z and satisfy λστ = λτ λσ (σ, τ ∈ G). For every complex of G-modules M = M • , the induced map Hom• (λσ , id) on ,naive C • (G, M ) = Hom•Z[G] (Z[G]•⊗ , M ) is equal to the conjugation action Ad(σ). Both λσ and id lift the identity on Z. As the bar resolution is projective over Z[G], there is a homotopy hσ : id − λσ , which induces homotopies hσ (M ) : id − Ad(σ) on C • (G, M ), functorial in M . If we choose another homotopy h0σ : id − λσ , projectivity of the resolution implies that there is a 2-homotopy Hσ : hσ − h0σ , which in turn induces 2-homotopies Hσ (M ) : hσ (M ) − h0σ (M ), functorial in M . If σ, τ ∈ G, then the same argument shows that there is a 2-homotopy Hσ,τ : hστ − λτ ? hσ + hτ , inducing 2-homotopies Hσ,τ (M ) : hστ (M ) −

hσ (M ) ? Ad(τ ) + hτ (M ),

functorial in M . (4.5.4) As in 3.4.5.5, one can apply the above construction in the “universal” case, when G is a free group on countably many generators σ, τ, g0 , g1 , . . .. One obtains homotopies hσ (M ) and 2-homotopies Hσ,τ (M ) functorial in both M and G. (4.5.5) In fact, the formula (3.6.1.4) gives a choice of hσ hσ : [g1 | · · · |gn ] 7→

n+1 X

(−1)j−1 [g1 | · · · |gj−1 |σ|σ −1 gj σ| · · · |σ −1 gn σ],

j=1

which defines such a bi-functorial homotopy hσ (M ) : id − 83

Ad(σ). Similarly, the formula

X

Hσ,τ : [g1 | · · · |gn ] 7→

(−1)k+l−1 [g1 | · · · |gk−1 |τ |τ −1 gk τ | · · · |τ −1 gl−2 τ |

1≤k i0 ) F i H q = mi−i0 F i0 H q . • • • Proof. (i) The spectral sequence Er comes from the complex T C =i C•cont (G, T ) equipped with the filtration • • i • i • 0 • F C = Ccont (G, m T ) (i ≥ 0). As C = F C and i≥0 F C = 0, convergence of Er follows from ([McCl], Thm.3.2). • (ii) The graded grm (R)-module grF• (H q ) is isomorphic to M M i,q−i i,q−i E∞ = Er1 (q) = Hrq1 (q) ,

i≥0

i≥0

hence of finite type over grm (R). We conclude by ([Bou], Prop. III.3.3), which applies thanks to 4.6.3.3. (4.6.5) Hilbert-Samuel functions and multiplicities. We recall some standard facts from ([Mat], Sect. 13,14; [Br-He], Ch. 4). L • If N = i≥0 Ni is a graded grm (R)-module of finite type, put X P (N, t) := `R (Ni )ti ∈ Z[[t]]. •

i≥0

If M is an R-module of finite type equipped with a good filtration F • M (i.e. such that (∀i) F i+1 M and (∃ i0 ) (∀i > i0 ) F i M = mi−i0 F i0 M ) satisfying M = F 0 M , put X f (M, F • , t) := `R (M/F i+1 M )ti ∈ Z[[t]].

m F iM ⊆

i≥0 i

i

In particular, if F M = m M (i ≥ 0) is the m-adic filtration, put X f (M, t) := f (M, m• , t) = `R (M/mi+1 M )ti . i≥0

These generating functions have the following properties: (4.6.5.1) (1 − t)f (M, F • , t) = P (grF• (M ), t). (4.6.5.2) (Hilbert’s Theorem) If N 6= 0, then P (N, t)(1 − t)dim(N ) ∈ Z[t] and P (N, t)(1 − t)dim(N ) |t=1 > 0. The multiplicity of M , defined as • eR (M ) := (1 − t)d+1 f (M, t)|t=1 = (1 − t)d P (grm (M ), t)|t=1 ∈ Z,

(where d = dim(R)) satisfies (4.6.5.3) eR (M ) = 0, eR (M ) > 0,

if dim(M ) < d if dim(M ) = d

• (by 4.6.5.2, as dim(M ) = dim(grm (M ))).

85

(4.6.5.4) If d ≥ 1 and F • is any good filtration on M , then eR (M ) = (1 − t)d+1 f (M, F • , t)|t=1 . (4.6.5.5) ([Mat], Thm. 14.7) eR (M ) =

X

eR/p (R/p) `R/p (Mp ).

ht(p)=0

(4.6.5.6) In particular, if R is a domain, then eR (M ) = eR (R) rkR (M ). (4.6.6) From now on, assume that T • is bounded, char(k) = p > 0 and cdp (G) < ∞. This implies, by Lemma 4.2.6, that (∃ c1 ) E1i,j = 0 whenever i + j > c1 , hence Hrq = 0

for q 6∈ [c0 , c1 ].

It follows from Lemma 4.6.2(ii) that i,j (∃ r2 ) (∀i, j) Eri,j = E∞ . 2 • Each Hrq and Aqr := Ker(dr : Hrq −→ Hrq+1 ) is a graded grm (R)-module of finite type; for each r ≥ 1 put X X Fr (t) := (−1)q P (Hrq , t) = (−1)i+j `R (Eri,j ) ti

q

Gr (t) :=

X

i,j

(−1)

q

P (Aqr , t)

=

X

q

i (−1)i+j `R (Ai,j r )t .

i,j

According to 4.6.5.2, we have (1 − t)d Fr (t), (1 − t)d Gr (t) ∈ Z[t]. (4.6.7) Proposition. (i) (∀r ≥ 1) Fr+1 (t) = (1 − tr )Gr (t) + tr Fr (t). (ii) If d ≥ 1, then X q (1 − t)d Fr (t)|t=1 = (−1)q eR (Hcont (G, T • )) , q

for every r ≥ 1. Proof. (i) This follows from the exact sequence d

i,j r i,j 0 −→ Ai−r,j+r−1 −→ Eri−r,j+r−1 −−→A r r −→ Er+1 −→ 0.

(ii) By (i), the integer (1 − t)d Fr (t)|t=1 does not depend on r ≥ 1. For r ≥ r2 we have Er = E∞ , hence X X i,j i Fr (t) = (−1)i+j `R (E∞ )t = (−1)q (1 − t)f (H q , F • , t), q

i,j

q where H q = Hcont (G, T • ). We conclude by 4.6.5.4 (which applies, by Lemma 4.6.4(ii)).

(4.6.8) Lemma. Assume that, as before, G satisfies (F), char(k) = p > 0 and cdp (G) < ∞. Assume, in addition, that there is c ∈ Q such that X (?c ) (−1)q dimk H q (G, M ) = c · dimk (M ) q

holds, for every discrete k[G]-module M with dimk (M ) < ∞. Then, for every bounded complex M • of discrete R[G]-modules of finite length over R, we have X X (−1)q `R (H q (G, M • )) = c (−1)q `R (M q ). q

q

Proof. Easy d´evissage. 86

(4.6.9) Theorem. Assume that G satisfies (F), char(k) = p > 0, cdp (G) < ∞ and (?c ). If T • is a bounded complex in (ad R[G] Mod)R−f t , then X

q (−1)q eR (Hcont (G, T • )) = c

q

X (−1)q eR (T q ). q

Proof. If d = 0, then eR (−) = `R (−) and the statement reduces to that of Lemma 4.6.8. If d ≥ 1, then Proposition 4.6.7(ii) gives X q (−1)q eR (Hcont (G, T • )) = (1 − t)d F1 (t)|t=1 . q

However, Lemma 4.6.8 implies that F1 (t) =

X

X X
• (−1)q `R H q (G, mi T • /mi+1 T •) ti = c (−1)q `R (mi T q /mi+1 T q ) ti = c (−1)q P (grm (T q ), t),

i,q

q

i,q

hence (1 − t)d F1 (t)|t=1 = c

X

(−1)q eR (T q ).

q

(4.6.10) Corollary. If R is a domain, then X

q (−1)q rkR (Hcont (G, T • )) = c

q

X (−1)q rkR (T q ). q

87

5. Duality Theorems for Galois Cohomology Revisited In this chapter we reformulate - and slightly generalize - Tate’s (and Poitou’s) local and global duality theorems for Galois cohomology. As observed in 0.3, duality with respect to the functor D follows automatically from the classical results for finite modules (cf. 5.2.10); the full duality follows by applying the general result 3.5.8. Throughout Chapter 5 we assume that k = Fpr is a finite field of characteristic p. (5.1) Classical duality results for Galois cohomology Let K be a global field of characteristic char(K) 6= p and S a finite set of primes of K containing all primes above p and all archimedean primes of K (if K is a number field). Denote by Sf the set of non-archimedean primes in S. In Sections 5.1–5.6, we assume that the following condition is satisfied (the general case is treated in 5.7): (P ) If p = 2, then K has no real prime. Fix a separable closure K sep of K. Let KS be the maximal subextension of K sep /K unramified outside S; denote GK,S := Gal(KS /K). For each prime v ∈ S fix a separable closure Kvsep of Kv and an embedding K sep ,→ Kvsep extending the embedding K ,→ Kv . This defines a continuous homomorphism ρv : Gv = αv π sep Gal(Kvsep /Kv )−→G /K)−→GK,S , hence, for each M ∈ (ind−ad K = Gal(K R[GK,S ] Mod), a ‘restriction’ map • • resv : Ccont (GK,S , M ) −→ Ccont (Gv , M )

Denote by Mv := ρ∗v (M ) ∈ (ind−ad R[Gv ] Mod) the R-module M , equipped with the Gv -action induced by ρv . For v ∈ Sf , our assumptions imply the following ([Se 2], [N-S-W]): (5.1.1) Gv and GK,S satisfy the finiteness condition (F’). (5.1.2) cdp (Gv ) = cdp (GK,S ) = 2. (5.1.3) For every n ≥ 0, local class field theory defines an isomorphism ∼

invv : H 2 (Gv , Z/pn Z(1)) = Br(Kv )[pn ] −→ Z/pn Z (where Z/pn Z(1) = µpn ). (5.1.4) Local duality (Tate). For every finite discrete Z/pn Z[Gv ]-module M , the cup product ∪

∼

H i (Gv , M ) × H 2−i (Gv , Hom(M, Z/pn Z(1)))−→H 2 (Gv , Z/pn Z(1)) −→ Z/pn Z is a perfect pairing of finite Z/pn Z-modules (i = 0, 1, 2). P (5.1.5) Reciprocity law. The sum of the local invariants invSf = v∈Sf invv defines a short exact sequence 0 −→ H 2 (GK,S , Z/pn Z(1)) −→

M

invSf

H 2 (Gv , Z/pn Z(1))−−→Z/pn Z −→ 0.

v∈Sf

(5.1.6) Global duality (Poitou-Tate). exact sequence of finite Z/pn Z-modules 0 −→ H 0 (GK,S , M ) −→

For every finite discrete Z/pn Z[GK,S ]-module M there is an M

H 0 (Gv , M ) −→ H 2 (GK,S , M ∗ (1))∗ −→

v∈Sf 1

−→ H (GK,S , M ) −→

M

H 1 (Gv , M ) −→ H 1 (GK,S , M ∗ (1))∗ −→

v∈Sf 2

−→ H (GK,S , M ) −→

M

H 2 (Gv , M ) −→ H 0 (GK,S , M ∗ (1))∗ −→ 0,

v∈Sf

in which (−)∗ = Hom(−, Z/pn Z). The maps H i (GK,S , M ) −→ H i (Gv , M ) are induced by resv ; the maps H i (Gv , M ) −→ H 2−i (GK,S , M ∗ (1))∗ by resv and the pairing (5.1.4); the remaining two maps will be defined in 5.4.3 below. 88

(5.2) Duality for Gv (5.2.1) It follows from (5.1.2-3) that for every R-module A with trivial action of Gv we have ∼

2 invv : Hcont (Gv , A(1)) −→ A i Hcont (Gv , A(1)) = 0

(i > 2).

Here A(1) = A ⊗Zp Zp (1) = A ⊗R R(1), which is admissible by Proposition 3.2.5. This implies that the canonical map of complexes i

v • A[−2]−→τ ≥2 Ccont (Gv , A(1))

(5.2.1.1)

(in which iv is induced by the inverse of invv ) is a quasi-isomorphism. For A = I there is a morphism of complexes r

v • I[−2]←−τ ≥2 Ccont (Gv , I(1))

(5.2.1.2)

which is a homotopy inverse of (5.2.1.1). More generally, if A• is a complex of R-modules with trivial action of Gv , then there are canonical morphisms of complexes • • • II Ccont (Gv , A• (1)) = Tot(i 7→ Ccont (Gv , Ai (1))) −→ τ≥2 Ccont (Gv , A• (1)) =

i

v • • = Tot(i 7→ τ≥2 Ccont (Gv , Ai (1)))←−A [−2]

defining a canonical map in D(R Mod) RΓcont (Gv , A• (1)) −→ A• [−2] (since iv , induced by the inverse of invv , is a quasi-isomorphism). If A• is a bounded below complex of injective R-modules, then iv has a homotopy inverse • II rv = rv,A• : τ≥2 Ccont (Gv , A• (1)) −→ A• [−2],

(5.2.1.3)

unique up to homotopy. (5.2.2) Fix v ∈ Sf , a bounded complex J = J • of injective R-modules and rv = rv,J as in (5.2.1.3) for A• = J. If X • is a bounded complex of admissible R[Gv ]-modules, so is • DJ (X • ) = HomR (X • , J).

The evaluation map ev2 : X • ⊗R DJ (X • )(1) −→ J(1) from 1.2.7 and the cup product defined in 3.4.5.2 induce a morphism of complexes rv,J

∪

• • • • II Ccont (Gv , X • ) ⊗R Ccont (Gv , DJ (X • )(1))−→Ccont (Gv , J(1)) −→ τ≥2 Ccont (Gv , J(1))−→J[−2],

hence by adjunction (see 1.2.6) a morphism of complexes • • • • αJ,X • : Ccont (Gv , X • ) −→ HomR (Ccont (Gv , DJ (X • )(1)), J[−2]) = DJ[−2] (Ccont (Gv , DJ (X • )(1))).

This construction gives a well-defined map in Db (R Mod) (independent on the choice of rv,J ) αJ,X : RΓcont (Gv , X) −→ DJ[−2] (RΓcont (Gv , DJ (X)(1))) for every X ∈ Db (ad R[Gv ] Mod) (where DJ (X) = RHomR (X, J) in the sense of 3.5.9). 89

In the same way, the evaluation map ev1 : DJ (X • )(1) ⊗R X • −→ J(1) gives rise to a morphism of complexes • • α0J,X • : Ccont (Gv , DJ (X • )(1)) −→ DJ[−2] (Ccont (Gv , X • ))

resp., to a morphism α0J,X : RΓcont (Gv , DJ (X)(1)) −→ DJ[−2] (RΓcont (Gv , X)) in Db (R Mod). b The above cup products define, for each X ∈ Db (ad R[Gv ] Mod), morphisms in D (R Mod) L

RΓcont (Gv , X)⊗R RΓcont (Gv , DJ (X)(1)) −→ J[−2] L

RΓcont (Gv , DJ (X)(1))⊗R RΓcont (Gv , X) −→ J[−2], which induce pairings j i Hcont (Gv , X) ⊗R Hcont (Gv , DJ (X)(1)) −→ H i+j−2 (J • ) j i Hcont (Gv , DJ (X)(1)) ⊗R Hcont (Gv , X) −→ H i+j−2 (J • )

(5.2.2.1)

on cohomology. (5.2.3) We shall be interested only in the following two choices of J: (A) J = I[n] for some n ∈ Z (hence DJ = Dn in the notation of 2.3.2) and all cohomology groups of X are of finite (resp., co-finite) type over R. (B) J = ω • [n] for some n ∈ Z (hence DJ = Dn in the notation of 2.8.11) and all cohomology groups of X are of finite type over R. In both cases the canonical map ε = εJ : X −→ DJ (DJ (X)) is a quasi-isomorphism, by Matlis duality 2.3.2 and Grothendieck duality 2.6, respectively. (5.2.4) Proposition. Assume that either b ad b ad (i) J = I[n] and X ∈ DR−f t (R[Gv ] Mod) or X ∈ DR−cof t (R[Gv ] Mod), or b ad (ii) J = ω • [n] and X ∈ DR−f t (R[Gv ] Mod). Then both maps αJ,X , α0J,X are isomorphisms in Db (R Mod). Proof. We consider only αJ,X ; the statement for α0J,X can be proved along the same lines (alternatively, one can use the compatibility result 5.2.7 relating the two maps). (i) If A −→ B −→ C −→ A[1] is an exact triangle in Db (ad R[Gv ] Mod), then (αJ,A , αJ,B , αJ,C ) define a map between exact triangles RΓcont (Gv , A) −→ RΓcont (Gv , B) −→ RΓcont (Gv , C) and DJ[−2] (RΓcont (Gv , D(A)(1))) −→ DJ[−2] (RΓcont (Gv , D(B)(1))) −→ DJ[−2] (RΓcont (Gv , D(C)(1))) . This means that αJ,B is an isomorphism, provided αJ,A and αJ,C are. Applying this observation to truncations 90

τ≤i−1 X −→ τ≤i X −→ H i (X)[−i] −→ (τ≤i−1 X)[1] we reduce to the case when X is a single module in degree zero. Lemma 5.2.5 below further reduces to ad the case J = I. For X = T ∈ (ad R[Gv ] Mod)R−f t (resp., X = A ∈ (R[Gv ] Mod)R−cof t ) there is a commutative diagram • Ccont (Gv , T /mn T ) / lim ← n−

u1

• Ccont (Gv , T )

αT

 • D−2 (Ccont (Gv , D(T )(1))) 

u4

u2

  • n D−2 lim C (G , D(T /m T )(1)) v cont −→ n

 • D−2 (Ccont (Gv , D(T /mn T )(1))) / lim ← − n

u3

in Dfb t (R Mod) (resp., • lim Ccont (Gv , A[mn ]) −→ n

u5

• / Ccont (Gv , A)

u6

 • n lim D (C (G , (D(A)/m D(A))(1))) −2 v cont −→ n

αA

u7



  n • D−2 lim C (G , (D(A)/m D(A))(1)) v cont ← n−

u8

 / D−2 (C • (Gv , D(A)(1))) cont

b in Dcof t (R Mod)). The maps

u4 = lim αI,T /mn T , ← n−

u6 = lim αI,A[mn ] −→ n

are quasi-isomorphisms by (5.1.4) (and by the “universal coefficient theorem” for projective limits of MittagLeffler systems of complexes, used in the proof of Corollary 4.1.3), u1 , u2 , u5 , u8 by Lemma 4.1.2 and u3 , u7 by Lemma 4.2.4. This implies that both αI,T and αI,A are quasi-isomorphisms, as claimed. (ii) Lemma 5.2.5 below implies that we can assume that J = ω • [d] (hence DJ = Dd ). Represent X by a bounded complex X • of admissible R[Gv ]-modules. The functoriality of the morphisms • II iv : A• [−2] −→ τ≥2 Ccont (Gv , A• (1))

from 5.2.1 implies that, for any choices of rv,J for J = I and J = ω • [d], the following diagram is commutative up to homotopy: • Ccont (Gv , ω • [d](1))

(Φ−d )∗

• / Ccont (Gv , Φ−d (ω • [d])(1))

rv,ω• [d]

 (ω • [d])[−2]

Tr∗

• / Ccont (Gv , I(1))

rv,I

Tr◦Φ−d [−2]

This implies, by Lemma 2.8.12, that the complexes 91

 / I[−2]

• A = Ccont (Gv , X • ),

U = I[−2],

• B = Ccont (Gv , D(X • )(1)),

U 0 = (ω • [d])[−2],

• B 0 = Ccont (Gv , Dd (X • )(1)),

C = C • ((xi ), R) = Φ−d (R)

and the maps f, f 0 (resp., b, u) induced by ∪ and rv,J (resp., by ξd : Φ−d ◦ Dd −→ D) satisfy the assumptions of Lemma 1.2.13, which in turn implies that the following diagram in Dfb t (R Mod) is commutative: RΓcont (Gv , X)  αω[d],X y

αI,X

−→

D(ω[d])[−2] (RΓcont (Gv , Dd (X)(1)))  η [−2] y d D−2 ◦ Φ−d (RΓcont (Gv , Dd (X)(1)))

D−2 (RΓcont (Gv , D(X)(1)))  D ((ξ (X)) ) ∗ y −2 d D−2 (RΓcont (Gv , Φ−d ◦ Dd (X)(1)))  D (f ) y −2

==

D−2 ◦ Φ−d (RΓcont (Gv , Dd (X)(1)))

The maps ηd [−2], ξd (X) are isomorphisms by 2.8.11, f : Φ−d (RΓcont (Gv , Dd (X)(1))) −→ Φ−d (RΓcont (Gv , Dd (X)(1))) is an isomorphism by Proposition 3.5.8 and αI,X is an isomorphism by (i); hence αω[d],X is also an isomorphism. It remains to prove the following Lemma. (5.2.5) Lemma. For every J, X • and n ∈ Z (and a fixed choice of rv,J ) the map αJ[n],X • is equal to the composite map αJ,X •

t

n • • • • Ccont (Gv , X • )−−→DJ[−2] (Ccont (Gv , DJ (X • )(1))) −→D J[n−2] (Ccont (Gv , DJ (X )(1))[n]) =
• = DJ[n−2] Ccont (Gv , DJ[n] (X • )(1)) . • • Proof. Apply Lemma 1.2.16 to A• = Ccont (Gv , X • ), B • = Ccont (Gv , DJ (X • )(1)), C • = J[−2] and u the map from 5.2.2 (the composition of rv,J with a truncated cup product).

b ad ∗ b ad (5.2.6) Theorem. If T, T ∗ ∈ DR−f t (R[Gv ] Mod), A, A ∈ DR−cof t (R[Gv ] Mod) are related by the duality diagram D T ←−−−−−−−−→ T ∗  ←−   −− D −−→  Φ Φ −−−−−  −−−−−−−  y y − − → ←− A A∗ so are D RΓcont (Gv , T ) ←−−−−−−−−→ RΓcont (Gv , T ∗ (1))[2] ←−     −− D −−→ −−−−−   Φ Φ −−−−−−− y y − − → ←− RΓcont (Gv , A) RΓcont (Gv , A∗ (1))[2] b (in D(co)f t (R Mod)) and there is a spectral sequence 2−j j i+j E2i,j = ExtiR (Hcont (Gv , T ∗ (1)), ω) = ExtiR (D(Hcont (Gv , A)), ω) =⇒ Hcont (Gv , T ). b ad ∗ Proof. We first explain the assumptions: start with T ∈ DR−f = t (R[Gv ] Mod) and put A = Φ(T ), T ∗ ∗ ∗ D(A), A = D(T ). Then the canonical maps Φ(T ) −→ A , Φ(RΓcont (Gv , T )) −→ RΓcont (Gv , A) and Φ(RΓcont (Gv , T ∗ (1))) −→ RΓcont (Gv , A∗ (1)) are isomorphisms by 4.3.1; this takes care of the vertical arrows. The diagonal and horizontal arrows are given by the isomorphisms of Proposition 5.2.4. The diagram is commutative up to the canonical isomorphisms from 2.8.1; the spectral sequence follows by applying 2.8.6 to the diagram.

92

(5.2.7) Proposition. Let f : X • −→ Y • be a morphism of bounded complexes of admissible R[Gv ]modules. Fix J and rv,J as in 5.2.1-2. Then (i) The composite map α0J,D

(εJ )∗

(X)(1)

J • • • • Ccont (Gv , X • )−−→Ccont (Gv , DJ (DJ (X • )))−−−−−−→D J[−2] (Ccont (Gv , DJ (X )(1)))

resp., αJ,D

DJ[−2] ((εJ )∗ )

(X)(1)

J • • • • • Ccont (Gv , DJ (X • )(1))−−−−−−→D J[−2] (Ccont (Gv , DJ (DJ (X )))) −−−−−−→ DJ[−2] (Ccont (Gv , X ))

is equal to αJ,X • (resp., to α0J,X • ). (ii) The following diagrams are commutative: αJ,X •

• Ccont (Gv , X • ) −−−−→  f y∗ • Ccont (Gv , Y • )

αJ,Y •

−−−−→

• DJ[−2] (Ccont (Gv , DJ (X • )(1)))  D y J[−2] ((DJ (f )(1))∗ ) • DJ[−2] (Ccont (Gv , DJ (Y • )(1)))

α0J,X •

• Ccont (Gv , DJ (X • )(1)) x (D (f )(1)) ∗  J

−−−−→

• Ccont (Gv , DJ (Y • )(1))

−−−−→

α0J,Y •

• DJ[−2] (Ccont (Gv , X • )) x D  J[−2] (f∗ ) • DJ[−2] (Ccont (Gv , Y • ))

(iii) The composite map DJ[−2] (α0J,X • )

εJ[−2]

• • • Ccont (Gv , X • )−−→DJ[−2] (DJ[−2] (Ccont (Gv , X • ))) −−−−−−→ DJ[−2] (Ccont (Gv , DJ (X • )(1)))

resp., εJ[−2]

DJ[−2] (αJ,X • )

• • • Ccont (Gv , DJ (X • )(1))−−→DJ[−2] (DJ[−2] (Ccont (Gv , DJ (X • )(1)))) −−−−−−→ DJ[−2] (Ccont (Gv , X • ))

is homotopic to αJ,X • (resp., to α0J,X • ). Proof. (i) The first statement follows from Lemma 1.2.12 and the following commutative diagram (in which • C • (X • ) = Ccont (Gv , X • )): ∪

C • (X • ) ⊗R C • (DJ (X • )(1))  (ε ) ⊗id y J ∗

−→

C • (DJ (DJ (X • ))) ⊗R C • (DJ (X • )(1))

∪

−→

(ev 2 )∗

C • (X • ⊗R DJ (X • )(1))  (ε ⊗id) ∗ y J

−→

C • (J(1)) k

C • (DJ (DJ (X • )) ⊗R DJ (X • )(1))

(ev 1 )∗

−→

C • (J(1))

(the second square is commutative by Lemma 1.2.9). Exchanging the factors in each tensor product we obtain the second statement. (ii) This follows from Lemma 1.2.11. (iii) The diagram C • (X • ) ⊗R C • (DJ (X • )(1))  s ◦(T ⊗T ) y 12

−→

∪

C • (X • ⊗R DJ (X • )(1))  T ◦(s ) 12 ∗ y

∪

C • (DJ (X • )(1) ⊗R X • )

C • (DJ (X • )(1)) ⊗R C • (X • ) −→

(ev2 )∗

−→

(ev1 )∗

−→

C • (J(1))   yT C • (J(1))

is commutative by (1.2.7.1) and 3.4.5.4. We apply Lemma 1.2.14 and the fact that T is homotopic to the identity. 93

(5.2.8) Corollary. If f : X −→ Y is a morphism in Db (ad R[Gv ] Mod), then αJ,X = α0J,DJ (X)(1) ◦ (εJ )∗ = DJ[−2] (α0J,X ) ◦ εJ[−2] , α0J,X

DJ[−2] ((DJ (f )(1))∗ ) ◦ αJ,X = αJ,Y ◦ f∗

= DJ[−2] ((εJ )∗ ) ◦ αJ,DJ (X)(1) = DJ[−2] (αJ,X ) ◦ εJ[−2] ,

α0J,X ◦ (DJ (f )(1))∗ = DJ[−2] (f∗ ) ◦ α0J,Y

(equalities of morphisms in Db (R Mod)). (5.2.9) Self-dual case. If f : X −→ DJ (X)(1) is a morphism in Db (ad R[Gv ] Mod), denote the morphism DJ (f )(1)

ε

J X −→D J (DJ (X)) = DJ (DJ (X)(1))(1)−−−−→DJ (X)(1)

by g. It follows from (1.2.7.1) and a derived version of Lemma 1.2.10 that the pairings L

id⊗f∗

L

(ev2 )∗

L

id⊗g∗

L

(ev2 )∗

∪f : RΓcont (Gv , X)⊗R RΓcont (Gv , X)−−−−→RΓcont (Gv , X)⊗R RΓcont (Gv , DJ (X)(1))−−→J[−2] ∪g : RΓcont (Gv , X)⊗R RΓcont (Gv , X)−−−−→RΓcont (Gv , X)⊗R RΓcont (Gv , DJ (X)(1))−−→J[−2] are related by ∪g = ∪f ◦ s12 .

(5.2.9.1)

g = ±f =⇒ ∪f ◦ s12 = ± ∪f .

(5.2.9.2)

In particular,

(5.2.10) On the level of cohomology, the diagonal arrows in Theorem 5.2.6 imply that the cup product ∪

∼

2−i i 2 Hcont (Gv , T ) × Hcont (Gv , A∗ (1))−→Hcont (Gv , I(1)) −→ I

induces isomorphisms ∼

2−i i Hcont (Gv , T ) −→ D(Hcont (Gv , A∗ (1))) ∼

j 2−j Hcont (Gv , A∗ (1)) −→ D(Hcont (Gv , T ))

(and similarly for the pair T ∗ (1), A). Note that, as remarked in 0.3, these isomorphisms follow immediately from (5.1.4) by a straightforward limit argument (a rather pompous version of which was given in the proof of Proposition 5.2.4(i)). The only reason for developing the cohomological machinery of Chapters 3-4 was a need to relate cohomology of T and A, as in 4.3.2. (5.2.11) Euler-Poincar´ e characteristic. For every non-archimedean prime v of K, the group Gv satisfies the condition (?c ) from Lemma 4.6.8 with ( −[Kv : Qp ], if Kv is a finite extension of Qp c= 0, otherwise, by Tate’s local Euler-Poincar´e characteristic formula ([Se 2], II.5.7, Thm. 5; [N-S-W], Thm. 7.3.1). As a result, Theorem 4.6.9 applies to Gv and c. (5.3) Cohomology with compact support for GK,S In this section we develop the theory of cohomology groups with compact support. In fact, there are two possible definitions, which differ in their treatment of infinite primes. We use the one appropriate for duality theory (see also the footnote in Sect. 0.7.1). (5.3.1) Cochains with compact support 94

(5.3.1.1) Definition. Let M • be a complex of ind-admissible R[GK,S ]-modules. The complex of continuous cochains with compact support with values in M • is defined as   resSf M • • • Cc,cont (GK,S , M • ) = Cone Ccont (GK,S , M • )−−→ Ccont (Gv , M • ) [−1], v∈Sf

where resSf = (resv )v∈Sf . (5.3.1.2) More precisely, the ‘restriction’ map resv (v ∈ Sf ) is equal to • • resv = ρ∗v : Ccont (GK,S , M • ) −→ Ccont (Gv , ρ∗v (M • )).

If we choose another embedding K sep ,→ Kvsep , then αv : Gv ,→ GK is replaced by α0v = Ad(σv ) ◦ αv (for • some σv ∈ GK ) and the map π(σv ) : ρ∗v (M • ) −→ ρ0∗ v (M ) is an isomorphism of complexes of Gv -modules. In the diagram inf

• resv : Ccont (GK,S , M • ) −−→

k inf

• res0v : Ccont (GK,S , M • ) −−→

α∗

v • Ccont (GK , π ∗ (M • )) −−→  Ad(σ ) v y

α0∗

v • Ccont (GK , π ∗ (M • )) −−→

• Ccont (Gv , ρ∗v (M • ))  π(σ ) y v∗ • • Ccont (Gv , ρ0∗ v (M ))

the first (resp., the second) square is commutative up to homotopy (resp., commutative). Choosing bifunctorial homotopies hσ (M • ) : id − Ad(σ) (e.g. those from 4.5.5), we obtain homotopies • 0 hv = α0∗ v ? hσv (M ) ? inf : resv −

h = (hv )v∈Sf : res0Sf −

π(σv )∗ ◦ resv ,

(π(σv )∗ ) ◦ resSf .

The corresponding morphism of cones Cone(id, ((σv )∗ ), h) : Cone(resSf ) −→ Cone(res0Sf ) is a homotopy equivalence, with homotopy inverse equal to Cone(id, ((σv−1 )∗ ), h0 ) : Cone(res0Sf ) −→ Cone(resSf ), where h0 = (α∗v ? hσv−1 (M • )? inf )v∈Sf . Indeed, this follows from 1.1.7 and the fact that there is a 2-homotopy h0 ? id + ((σv−1 )∗ ) ? h = (resv ? (hσv−1 (M • ) + Ad(σv−1 ) ? hσv (M • )))v ? inf −

0.

i • (5.3.1.3) The cohomology of Cc,cont (GK,S , M • ) will be denoted by Hc,cont (GK,S , M • ). As in 3.5.6, the • • • functor M 7→ Cc,cont (GK,S , M ) preserves homotopy, exact sequences and quasi-isomorphisms for cohomologically bounded below complexes, hence defines an exact functor + RΓc,cont(GK,S , −) : D+ (ind−ad R[GK,S ] Mod) −→ D (R Mod)

such that resSf

RΓc,cont (GK,S , M ) −→ RΓcont (GK,S , M )−−→

M

RΓcont (Gv , M )

(5.3.1.3.1)

v∈Sf

is an exact triangle in D+ (R Mod) for every M ∈ D+ (ind−ad R[GK,S ] Mod). In particular, there is an exact sequence resSf

i i · · · −→ Hc,cont (GK,S , M ) −→ Hcont (GK,S , M )−−→

M v∈Sf

95

i+1 i Hcont (Gv , M ) −→ Hc,cont (GK,S , M ) −→ · · ·

(5.3.1.3.2)

(5.3.2) Lemma. Each of the three functors RΓcont (GK,S , −), RΓcont(Gv , −), RΓc,cont (GK,S , −) maps ind−ad ind−ad ∗ ∗ ∗ ∗ DR−f t (R[GK,S ] Mod) (resp., DR−cof t (R[GK,S ] Mod)) to Df t (R Mod) (resp., Dcof t (R Mod)), for ∗ = +, b. Proof. This is true for RΓcont (GK,S , −) and RΓcont (Gv , −) by Corollary 4.2.7 and (5.1.1-2). The statement for RΓc,cont (GK,S , −) follows from the exact sequence (5.3.1.3.2). (5.3.3) Cup products (5.3.3.1) Let A• , B • be complexes of ind-admissible R[GK,S ]-modules. We shall write elements of M i−1 i i Cc,cont (GK,S , A• ) = Ccont (GK,S , A• ) ⊕ Ccont (Gv , A• ) v∈Sf i−1 i in the form (a, aS ), where a ∈ Ccont (GK,S , A• ), aS = (av )v∈Sf , av ∈ Ccont (Gv , A• ); deg(a, aS ) = i = deg(a) = deg(aS ) + 1. The differential is given by

d(a, aS ) = (da, −resSf (a) − daS ). (5.3.3.2) The first cup product c∪

• • • : Cc,cont (GK,S , A• ) ⊗R Ccont (GK,S , B • ) −→ Cc,cont (GK,S , A• ⊗R B • )

is defined as (a, aS )c ∪b = (a ∪ b, aS ∪ resSf (b)) and satisfies d((a, aS )c ∪b) = (d(a, aS ))c ∪b + (−1)deg(a) (a, aS )c ∪db. (5.3.3.3) The second cup product • • • ∪c : Ccont (GK,S , A• ) ⊗R Cc,cont (GK,S , B • ) −→ Cc,cont (GK,S , A• ⊗R B • )

is defined as a∪c (b, bS ) = (a ∪ b, (−1)deg(a) resSf (a) ∪ bS ) and satisfies d(a∪c (b, bS )) = (da)∪c (b, bS ) + (−1)deg(a) a∪c (d(b, bS )). (5.3.3.4) The involutions T from 3.4.5.4 define morphisms of complexes • T : Cc,cont (GK,S , A• ) −→

(a, aS )

• Cc,cont (GK,S , A• )

7→

(T (a), T (aS ))

which are again involutions homotopic to the identity. As in 3.4.5.4, the following diagram of morphisms of complexes is commutative (and the same is true if the roles of c ∪ and ∪c are interchanged): ∪

c • • Cc,cont (GK,S , A• ) ⊗R Ccont (GK,S , B • ) −→  s ◦(T ⊗T ) y 12

∪

c • • Ccont (GK,S , B • ) ⊗R Cc,cont (GK,S , A• ) −→

• Cc,cont (GK,S , A• ⊗R B • )  T ◦(s ) 12 ∗ y • Cc,cont (GK,S , B • ⊗R A• )

(5.3.3.5) The products c ∪, ∪c (resp., the involutions T ) are compatible with the product ∪ (resp., the • • involution T ) defined in 3.5.4 (via the canonical maps Cc,cont (GK,S , −) −→ Ccont (GK,S , −)). (5.3.4) The statements of Propositions 4.2.9-10 and 4.3.3 also hold for the functor RΓc,cont (GK,S , −). 96

(5.3.5) Euler-Poincar´ e characteristic. If M is a finite discrete Fp [GK,S ]-module, then Tate’s global Euler-Poincar´e characteristic formula ([N-S-W], Thm. 8.6.14) implies that 3 X

(−1)q dimFp Hcq (GK,S , M ) =

q=0

X


dimFp M Gv .

v|∞

A straightforward modification of the arguments in Sect. 4.6 gives the following result. (5.3.6) Theorem. If T • is a bounded complex in (ad R[GK,S ] Mod)R−f t , then X


XX
q (−1)q eR Hc,cont (GK,S , T • ) = (−1)q eR (T q )Gv .

q

q

v|∞

(5.3.7) For R finite and flat over Zp (and not necessarily commutative), Flach [Fl 2] proved a more refined Euler-Poincar´e characteristic formula. (5.4) Duality for GK,S (5.4.1) The sequence (5.3.1.3.2) together with (5.1.5) define an isomorphism ∼

3 Hc,cont (GK,S , Z/pn Z(1)) −→ Z/pn Z.

As in 5.2.1 we get ( i Hc,cont (GK,S , A(1))

∼

−→

A i=3 0

i>3

for every R-module A with trivial action of GK,S , hence a quasi-isomorphism (induced by inv−1 Sf ) i

• A[−3]−→τ≥3 Cc,cont (GK,S , A(1)).

(5.4.1.1)

Let A• be a bounded below complex of R-modules with trivial action of GK,S . For each i ∈ Z the map M 2 2 resSf : Hcont (GK,S , Ai (1)) −→ Hcont (Gv , Ai (1)) v∈Sf

is injective, hence the canonical morphism of complexes  resSf

• • τ≥3 Cc,cont (GK,S , Ai (1)) −→ Cone τ≥2 Ccont (GK,S , Ai (1))−−→

M

 • τ≥2 Ccont (Gv , Ai (1)) [−1]

v∈Sf

is a quasi-isomorphism. This implies that the canonical map
II • • τ≥3 Cc,cont (GK,S , A• (1)) = Tot i 7→ τ≥3 Cc,cont (GK,S , Ai (1)) −→   resSf M II II • • −→Cone τ≥2 Ccont (GK,S , A• (1))−−→ τ≥2 Ccont (Gv , A• (1)) [−1]

(5.4.1.2)

v∈Sf

is also a quasi-isomorphism. As in 5.2.2, fix a bounded complex J = J • of injective R-modules. We claim that there is a homotopy equivalence iS making the following diagram commutative up to homotopy: resSf L j II II • • τ≥2 Ccont (GK,S , J(1)) −−→ v∈Sf τ≥2 Ccont (Gv , J(1)) −→ Cone(resSf ) x x (i ) i  v S L Σ −→ J[−2]. v∈Sf J[−2] 97

Indeed, fix v0 ∈ Sf and put iS = j ◦ iv0 . Then iS is a quasi-isomorphism, hence a homotopy equivalence (as J[−2] is a bounded below complex of injectives). Fix homotopy inverses rv,J of iv (v ∈ Sf ) and also a homotopy inverse rS = rS,J of iS . Then rS ◦ j is homotopic to Σ ◦ (rv,J ). Composing rS [−1] with the map (5.4.1.2) we obtain a quasi-isomorphism • II rJ : τ≥3 Cc,cont (GK,S , J(1)) −→ J[−3],

(5.4.1.3)

unique up to homotopy. (5.4.2) Let X • be a bounded complex of admissible R[GK,S ]-modules. As in 5.2.2, the cup product c ∪, together with the evaluation map ev2 (resp., ev1 ) and the map (5.4.1.3) define morphisms of complexes c βJ,X •

• • • : Cc,cont (GK,S , X • ) −→ Hom•R (Ccont (GK,S , DJ (X • )(1)), Cc,cont (GK,S , J(1))) −→

Hom• (id,rJ )

• • • II −→ HomR (Ccont (GK,S , DJ (X • )(1)), τ≥3 Cc,cont (GK,S , J(1)))−−−−−−→ • • • • −→ HomR (Ccont (GK,S , DJ (X )(1)), J[−3]) = DJ[−3] (Ccont (GK,S , D(X • )(1)))

resp., 0 c βJ,X •

• • • • : Cc,cont (GK,S , DJ (X • )(1)) −→ HomR (Ccont (GK,S , X • ), Cc,cont (GK,S , J(1))) −→

Hom• (id,rJ )

• • II −→ Hom•R (Ccont (GK,S , X • ), τ≥3 Cc,cont (GK,S , J(1)))−−−−−−→ • • • • −→ HomR (Ccont (GK,S , X ), J[−3]) = DJ[−3] (Ccont (GK,S , X • )),

hence canonical maps (independent on the choice of r) c βJ,X

: RΓc,cont (GK,S , X) −→ DJ[−3] (RΓcont (GK,S , DJ (X)(1)))

0 c βJ,X

: RΓc,cont (GK,S , DJ (X)(1)) −→ DJ[−3] (RΓcont (GK,S , X))

resp.,

in Db (R Mod) (for every X ∈ Db (ad R[GK,S ] Mod)). Similarly, using ∪c instead of c ∪, one obtains morphisms of complexes • • βc,J,X • : Ccont (GK,S , X • ) −→ DJ[−3] (Cc,cont (GK,S , DJ (X • )(1)))

0 • • • • βc,J,X • : Ccont (GK,S , DJ (X )(1)) −→ DJ[−3] (Cc,cont (GK,S , X ))

resp., morphisms βc,J,X : RΓcont (GK,S , X) −→ DJ[−3] (RΓc,cont (GK,S , DJ (X)(1))) 0 βc,J,X : RΓcont (GK,S , DJ (X)(1)) −→ DJ[−3] (RΓc,cont (GK,S , X))

in Db (R Mod). As in 5.2.2, ∪c and c ∪ induce, for each X ∈ Db (ad R[GK,S ] Mod), cup products L

RΓc,cont(GK,S , X)⊗R RΓcont (GK,S , DJ (X)(1)) −→ J[−3] L

RΓcont (GK,S , X)⊗R RΓc,cont(GK,S , DJ (X)(1)) −→ J[−3] and pairings on cohomology j i Hc,cont (GK,S , X) ⊗R Hcont (GK,S , DJ (X)(1)) −→ H i+j−3 (J • ) j i Hcont (GK,S , X) ⊗R Hc,cont (GK,S , DJ (X)(1)) −→ H i+j−3 (J • ),

as well as analogous products in which the roles of X and DJ (X)(1) are interchanged. 98

(5.4.2.1)

(5.4.3) Proposition. Assume that either b ad b ad (i) J = I[n] and X ∈ DR−f t (R[GK,S ] Mod) or X ∈ DR−cof t (R[GK,S ] Mod), or b ad (ii) J = ω • [n] and X ∈ DR−f t (R[GK,S ] Mod). 0 0 Then the maps c βJ,X , c βJ,X , βc,J,X , βc,J,X are isomorphisms in Db (R Mod). Proof. As in the proof of Proposition 5.2.4 one reduces to the case when M is a finite discrete Z/pn Z[GK,S ]module. In this case the statement is a variant of the Poitou-Tate global duality theorem (5.1.6) ([Ni], Lemma 6.1; [N-S-W], 8.6.13). This explains the origin of the non-obvious maps in (5.1.6): they are induced by the cup products ∼

∪c : H i (GK,S , M ) × Hc3−i (GK,S , M ∗ (1)) −→ Hc3 (GK,S , Z/pn Z(1)) −→ Z/pn Z c∪

∼

: Hcj (GK,S , M ) × H 3−j (GK,S , M ∗ (1)) −→ Hc3 (GK,S , Z/pn Z(1)) −→ Z/pn Z.

(5.4.4) Proposition. If f : X −→ Y is a morphism in Db (ad R[GK,S ] Mod), then 0 0 βc,J,X = βc,J,D ◦ (εJ )∗ = DJ[−3] (c βJ,X ) ◦ εJ[−3] , J (X)(1) c βJ,X

0 0 = c βJ,D ◦ (εJ )∗ = DJ[−3] (βc,J,X ) ◦ εJ[−3] , J (X)(1)

0 βc,J,X = DJ[−3] ((εJ )∗ ) ◦ βc,J,DJ (X)(1) = DJ[−3] (c βJ,X ) ◦ εJ[−3] , 0 β = DJ[−3] ((εJ )∗ ) ◦ c βJ,DJ (X)(1) = DJ[−3] (βc,J,X ) ◦ εJ[−3] , c J,X

DJ[−3] ((DJ (f )(1))∗ ) ◦ βc,J,X = βc,J,Y ◦ f∗ DJ[−3] ((DJ (f )(1))∗ ) ◦ c βJ,X = c βJ,Y ◦ f∗ 0 0 βc,J,X ◦ (DJ (f )(1))∗ = DJ[−3] (f∗ ) ◦ βc,J,Y 0 c βJ,X

0 ◦ (DJ (f )(1))∗ = DJ[−3] (f∗ ) ◦ c βJ,Y

(equalities of morphisms in Db (R Mod)). Proof. This follows from a variant of Proposition 5.2.7, which is proved in the same way as 5.2.7; the only difference is that the commutative diagram from 3.4.5.4 (with A• = X • , B • = DJ (X • )(1)) has to be replaced by an analogous diagram from 5.3.3.4. b ad ∗ b ad (5.4.5) Theorem. If T, T ∗ ∈ DR−f t (R[GK,S ] Mod), A, A ∈ DR−cof t (R[GK,S ] Mod) are related by the duality diagram D T ←−−−−−−−−→ T ∗  ←−   −− D −−→  Φ Φ −−−−−  −−−−−−−  y y − − → ←− A A∗ so are D RΓcont (GK,S , T ) ←−−−−−−−−→ RΓc,cont (GK,S , T ∗ (1))[3] ←−     −− D −−→   −− −− Φ Φ − − − −− − y y −→ −− − ← RΓcont (GK,S , A) RΓc,cont(GK,S , A∗ (1))[3] b (in D(co)f t (R Mod)) and there are spectral sequences 3−j j i+j E2i,j = ExtiR (Hc,cont (GK,S , T ∗ (1)), ω) = ExtiR (D(Hcont (GK,S , A)), ω) =⇒ Hcont (GK,S , T ) 0

3−j j i+j E2i,j = ExtiR (Hcont (GK,S , T ∗ (1)), ω) = ExtiR (D(Hc,cont (GK,S , A)), ω) =⇒ Hc,cont (GK,S , T ).

Proof. As in 5.2.6, everything follows from Proposition 5.4.3. (5.5) Duality for Poincar´ e groups (5.5.1) Recall ([Ve 1], Sect. 4) that a profinite group G is a Poincar´e group with respect to a prime number p if the following two conditions are satisfied: (5.5.1.1) cdp (G) = n < ∞. 99

(5.5.1.2) The abelian groups Di defined by Di = lim HomZ (H i (U, Z/pm Z), Qp /Zp ) −m → lim −→ U (where U runs through all open subgroups of G and the transition maps are dual to corestrictions) satisfy ( 0, i 6= n ∼ Di −→ Qp /Zp , i = n. There is a natural action of G on each Di ; under the above conditions G acts on Dn by a character χ : G −→ Z∗p . (5.5.2) Examples. (1) G is a compact p-adic Lie group with finite cdp (G). (2) G = Gv in 5.1. In this case n = 2 and χ is equal to the cyclotomic character. (5.5.3) The main duality result for Poincar´e groups ([Ve 1], Prop. 4.4) states that there is a canonical isomorphism ∼

ρ : H n (G, Dn ) −→ Qp /Zp such that for every finite p-primary discrete G-module M and i ∈ Z the cup product ∪

H i (G, M ) × H n−i (G, HomZ (M, Dn ))−→H n (G, Dn ) −→ Qp /Zp induces an isomorphism ∼

H n−i (G, HomZ (M, Dn )) −→ HomZ (H i (G, M ), Qp /Zp ). (5.5.4) The results and proofs in Sect. 5.2 work for a general Poincar´e group G satisfying (F). For example, a generalization of Theorem 5.2.6 yields a digram D

RΓcont (G, T ) ←−−−−−−−−→ RΓcont (G, T ∗ (χ))[n]   ←−   −− D −−→   −−−−− Φ Φ − − − −−− y y − − → ←− RΓcont (G, A) RΓcont (G, A∗ (χ))[n], where M (χ) denotes M with the G-action twisted by the character χ (for every Zp [G]-module M ). A generalization of Proposition 5.2.4 gives canonical isomorphisms ∼

RΓcont (G, X) −→ DJ (RΓcont (G, DJ,G (X))), where DJ,G (X) = DJ (X)(χ)[n]. (5.6) Localization (5.6.1) Let S ⊂ R be a multiplicative subset. Everything in Sections 5.2.1-2, 5.2.7-9, 5.4.1-2 and 5.4.4 remains valid if we replace R by RS . Instead of 5.2.3, we shall be interested in the following case: • (5.6.2) J = ωR [n] for some n ∈ Z (hence DJ = DRS ,n ) and all cohomology groups of X are of finite type S over RS .

100

• (5.6.3) Proposition. Assume that J = ωR [n] for some n ∈ Z. Then: S b ad (i) For every X ∈ DRS −f t (RS [Gv ] Mod), the canonical map

αJ,X : RΓcont (Gv , X) −→ DRS ,n−2 (RΓcont (Gv , DRS ,n (X)(1))) is an isomorphism in Dfb t (RS Mod) (and the same is true for α0J,X ). b (ii) For every X ∈ DR (ad Mod), the canonical map S −f t RS [GK,S ] c βJ,X

: RΓc,cont (GK,S , X) −→ DRS ,n−2 (RΓcont (GK,S , DRS ,n (X)(1)))

0 0 is an isomorphism in Dfb t (RS Mod) (and the same is true for c βJ,X , βc,J,X and βc,J,X ).

Proof. (i) By d´evissage we reduce to the case when X is an admissible RS [Gv ]-module, of finite type over RS . Then X = S −1 Y for some Y ∈ (ad R[Gv ] Mod)R−f t , by Lemma 3.7.3, and αJ,X is the localization S −1 RΓcont (Gv , Y ) −→ S −1 (DR,n−2 (RΓcont (Gv , DR,n (Y )(1)))) of the isomorphism αωR• [n],Y from Proposition 5.2.4. The same argument applies in the case (ii), with 5.4.3 replacing 5.2.4. (5.7) In the absence of (P ) (5.7.1) Assume that the condition (P ) from 5.1 fails, i.e. p = 2 and K is a number field with at least one real prime. In this case the global duality must also take into account the Tate cohomology groups b i (Gv , −) for real primes v (where Gv = Gal(C/R) has order two). The statements in 5.1 then have to be H modified as follows: for every real prime v, (5.7.1.1) Gv and GK,S satisfy the finiteness condition (F’). (5.7.1.2) cd2 (Gv ) = cd2 (GK,S ) = ∞. (5.7.1.3) For every n ≥ 1, local class field theory defines an isomorphism ∼

invv : H 2 (Gv , Z/2n Z(1)) −→ Z/2Z. (5.7.1.4) Local duality. For every finite discrete Z/2n Z[Gv ]-module M , the cup product ∪ ∼ 2 b i (Gv , M ) × H b 2−i (Gv , Hom(M, Z/2n Z(1)))−→H H (Gv , Z/2n Z(1)) −→ Z/2Z

is a perfect pairing of finite Z/2Z-modules (i ∈ Z). P (5.7.1.5) Reciprocity law. The sum of the local invariants invS = v∈S invv defines an exact sequence M invS n 0 −→ H 2 (GK,S , Z/2n Z(1)) −→ H 2 (Gv , Z/2n Z(1))−−→Z/2 Z −→ 0 v∈S

(where invv = 0 for each complex prime v). (5.7.1.6) Global duality (Poitou-Tate). For every finite discrete Z/2n Z[GK,S ]-module M there is an exact sequence of finite Z/2n Z-modules 0 −→ H 0 (GK,S , M ) −→

M

H 0 (Gv , M ) ⊕

v∈Sf 1

−→ H (GK,S , M ) −→

M

−→ H (GK,S , M ) −→

M

v∈Sf

b 0 (Gv , M ) −→ H 2 (GK,S , M ∗ (1))∗ −→ H

Kv =R 1

H (Gv , M ) ⊕

v∈Sf 2

M M

b 1 (Gv , M ) −→ H 1 (GK,S , M ∗ (1))∗ −→ H

Kv =R 2

H (Gv , M ) ⊕

M

Kv =R

in which (−)∗ = Hom(−, Z/2n Z). (5.7.1.7) For each M as in 5.7.1.6 and i > 2, the map 101

b 2 (Gv , M ) −→ H 0 (GK,S , M ∗ (1))∗ −→ 0, H

resS : H i (GK,S , M ) −→

M

M

H i (Gv , M ) ⊕

v∈Sf

b i (Gv , M ) = H

Kv =R

M

b i (Gv , M ) H

Kv =R

is an isomorphism. (5.7.2) In the present situation, the constructions in 5.3 should be modified as follows. For each real prime b • (Gv , M ) of a Gv -module M extends v of K, the usual definition of the complete (Tate) cochain complex C cont naturally to complexes of Gv -modules by M b n (Gv , M • ) = b j (Gv , M i ) C C i+j=n

and using the sign rules from 3.4.1.3. The standard cup products ([C-E], Ch. XII) • • • bcont bcont bcont ∪:C (Gv , M ) ⊗ C (Gv , N ) −→ C (Gv , M ⊗ N )

extend, using the sign rules from 3.4.5.2, to the case of complexes of Gv -modules b • (Gv , M • ) ⊗ C b • (Gv , N • ) −→ C b• (Gv , M • ⊗ N • ). ∪:C cont cont cont

(5.7.2.1)

For every complex M of ind-admissible R[GK,S ]-modules we define   M M • • resS • b• b • (Gv , M • ) [−1],  • C Ccont (Gv , M • ) ⊕ C c,cont (GK,S , M ) = Cone Ccont (GK,S , M )−−→ cont •

v∈Sf

Kv =R

where the map resv for a real prime v is given by • • • bcont Ccont (GK,S , M • ) −→ Ccont (Gv , M • ) ,→ C (Gv , M • ). bi b• The cohomology groups H c,cont (GK,S , −) of Cc,cont (GK,S , −) lie in the exact sequence

res

S i i b c,cont · · · −→ H (GK,S , M • ) −→ Hcont (GK,S , M • ) −−→

M

i Hcont (Gv , M • ) ⊕

v∈Sf

b i+1 (GK,S , M • ) −→ H c,cont

M

b i (Gv , M • ) −→ H

Kv =R

−→ · · ·

The functor

(5.7.2.2)

• bc,cont M • −→ C (GK,S , M • )

gives rise to an exact functor dc,cont (GK,S , −) : Db (ind−ad Mod) −→ D(R Mod). RΓ R[GK,S ] The formulas from 5.3.3.2-3 together with (5.7.2.1) define products • • • bc,cont bc,cont b c : Ccont ∪ (GK,S , M • ) ⊗R C (GK,S , N • ) −→ C (GK,S , M • ⊗R N • )

b c∪

• • • bc,cont bc,cont :C (GK,S , M • ) ⊗R Ccont (GK,S , N • ) −→ C (GK,S , M • ⊗R N • ),

for any pair of complexes of ind-admissible R[GK,S ]-modules M • , N • . (5.7.3) Lemma. (i) If M • = M is concentrated in degree zero, then bi (∀i > 3) H c,cont (GK,S , M ) = 0. (ii) If A is any R-module with trivial action of GK,S , then invS induces an isomorphism ∼ 3 b c,cont H (GK,S , A(1)) −→ A.

Proof. (i) By a standard limit procedure one reduces to the case when M is a finite discrete Z/2n Z[GK,S ]module, in which case the statement follows from 5.7.1.7 and (5.7.2.2). Similarly, it is sufficient to consider only the case A = Z/2n Z in (ii), when we conclude by 5.7.1.5,7 and (5.7.2.2). 102

b c , c∪ b induce (5.7.4) Proposition. If M is a finite discrete Z/2n Z[GK,S ]-module, then the cup products ∪ n perfect pairings of finite Z/2 Z-modules ∼ n n b 3−i (GK,S , Hom(M, Z/2n Z(1))) −→ H b3 H i (GK,S , M ) × H c,cont c,cont (GK,S , Z/2 Z(1)) −→ Z/2 Z ∼ 3−i n n bi b3 H (GK,S , Hom(M, Z/2n Z(1))) −→ H c,cont (GK,S , M ) × H c,cont (GK,S , Z/2 Z(1)) −→ Z/2 Z

(for all i ∈ Z). Proof. This follows from 5.7.1.4,6,7 and the compatibility of the various products involved. • (5.7.5) All constructions from 5.4 work in the present context, provided that Cc,cont (GK,S , −) (resp., • b d RΓc,cont(GK,S , −)) is replaced everywhere by Cc,cont (GK,S , −) (resp., RΓc,cont (GK,S , −)). The final result can be summed up as follows: if D

T ←−−−−−−−−→  ←−  −− D −−→ Φ −−−−−  −−−−−−− y − − → ←− A

T∗   Φ  y A∗

b ad ∗ b ad is a duality diagram with T, T ∗ ∈ DR−f t (R[GK,S ] Mod) and A, A ∈ DR−cof t (R[GK,S ] Mod), then D

RΓcont (GK,S , T ) ←−−−−−−−−→  ←−  −− D −−→ −− −−  Φ −−−−−−− y −− → − ← RΓcont (GK,S , A)

dc,cont (GK,S , T ∗ (1))[3] RΓ    Φ y dc,cont (GK,S , A∗ (1))[3] RΓ

is a duality diagram in D(co)f t (R Mod). b i (Gv , M ) = 0 for each real prime v and each Gv -module M , the canonical map in D(R Mod) (5.7.6) As 2· H dc,cont (GK,S , M • ) RΓc,cont(GK,S , M • ) −→ RΓ

(M • ∈ Db (ind−ad R[GK,S ] Mod))

becomes an isomorphism in Db (R[1/2] Mod), where R[1/2] = R ⊗Z2 Q2 .

103

6. Selmer Complexes Classical Selmer groups consist of elements of H 1 (GK,S , M ) satisfying suitable local conditions in H 1 (Gv , M ) at v ∈ S. We work in the derived category, which means that we have to modify this definition and impose local conditions on the level of complexes rather than cohomology. This is done in Sect. 6.1-2. The main abstract duality result, Theorem 6.3.4, is deduced from our version of the Poitou-Tate duality (Theorem 5.4.5) using the cup products from Sect. 1.3. The symmetry properties of these duality pairings are investigated in Sect. 6.5-6; they require additional data. In Sect. 6.7 we introduce the main example of “elementary” local conditions, following Greenberg [Gre 1-3]. The assumptions of 5.1, including (P ), are in force. (6.1) Definition of Selmer complexes (6.1.1) Let X = X • be a complex of admissible R[GK,S ]-modules. Local conditions for X are given by a collection ∆(X) = (∆v (X))v∈Sf , where each ∆v (X) is a local condition at v ∈ Sf , consisting of a morphism of complexes of R-modules • + i+ v (X) : Uv (X) −→ Ccont (Gv , X).

(6.1.2) The Selmer complex attached to the local conditions ∆(X) is defined as  • ef• (GK,S , X; ∆(X)) = Cone Ccont C (GK,S , X) ⊕

M

resSf −i+ (X) S

Uv+ (X)−−−−−−−−→

v∈Sf

M

 • Ccont (Gv , X) [−1]

v∈Sf

e • (X)), where i+ (X) = (i+ g (sometimes abbreviated as C v (X))v∈Sf . Denote by RΓf (GK,S , X; ∆(X)) (somef S gf (X)) the corresponding object of D(R Mod) and by H e i (GK,S , X; ∆(X)) (sometimes times abbreviated as RΓ f e i (X)) its cohomology. abbreviated as H f

If X and all Uv+ (X) have cohomology of finite type over R, resp., of co-finite type over R, resp., are cohomoe• (GK,S , X; ∆(X)). logically bounded above, resp., are cohomologically bounded below, the same is true for C f (6.1.3) For v ∈ Sf put
−i+ v (X) • Uv− (X) = Cone Uv+ (X)−−−−→C cont (Gv , X) M US± (X) = Uv± (X). v∈Sf

There are exact sequences of complexes (with maps induced by obvious inclusions resp., projections) • 0 −→ Ccont (Gv , X)[−1] −→ Uv− (X)[−1] −→ Uv+ (X) −→ 0 e • (GK,S , X; ∆(X)) −→ U + (X) −→ 0 0 −→ C • (GK,S , X) −→ C

c,cont

0 −→

f

US− (X)[−1]

S

(6.1.3.1)

e • (GK,S , X; ∆(X)) −→ C • (GK,S , X) −→ 0, −→ C f cont

which give rise to the following exact triangles in D(R Mod): Uv+ (X) −→ RΓcont (Gv , X) −→ Uv− (X) −→ Uv+ (X)[1] gf (X) −→ U + (X) US+ (X)[−1] −→ RΓc,cont(GK,S , X) −→ RΓ S US− (X)[−1]

(6.1.3.2)

gf (X) −→ RΓcont (GK,S , X) −→ U − (X). −→ RΓ S

We know from Proposition 5.4.3 that RΓc,cont(GK,S , X) is dual to RΓcont (GK,S , DJ (X)(1))[3] (under suitgf (X) and RΓ gf (DJ (X)(1))[3], we able assumptions). In order to deduce from (6.1.3.2) a duality between RΓ 104

must ensure that US+ (X) is (close to being) isomorphic to the dual of US− (DJ (X)(1))[2]. Taking into account the duality between RΓcont (Gv , X) and RΓcont (Gv , DJ (X)(1))[2], this boils down to a suitable orthogonality relation between Uv+ (X) and Uv+ (DJ (X)(1)), for all v ∈ Sf . This notion of orthogonality is introduced and studied in Sect. 6.2. gf (X) and the dual of RΓ gf (DJ (X)(1))[3]; this is It is essential to have a canonical duality map between RΓ why the additional data 6.2.1.3 enter the picture. The duality map itself is constructed in Sect. 6.3, using the abstract cup products from 1.3. (6.1.4) In the special case when US− (X) −→ τ≥1 US− (X) is a quasi-isomorphism (which is equivalent to M (i+ (X))∗ M S i H i (Uv+ (X))−−−−→ Hcont (Gv , X) v∈Sf

v∈Sf

being bijective for i ≤ 0 and injective for i = 1), the canonical map gf (GK,S , X; ∆(X)) −→ τ≤0 RΓcont (GK,S , X) τ≤0 RΓ is also a quasi-isomorphism, i.e. ∼ i e i (GK,S , X; ∆(X)) −→ H Hcont (GK,S , X) f

(∀i ≤ 0)

and 

M

∼ 1 e 1 (GK,S , X; ∆(X)) −→ H Ker Hcont (GK,S , X) −→ f

 1 1 +  Hcont (Gv , X)/(i+ v (X))∗ (H (Uv (X)))

v∈Sf

coincides with a classical Selmer group given by the local conditions 1 + 1 (i+ v (X))∗ (H (Uv (X))) ⊆ Hcont (Gv , X)

(v ∈ Sf ).

(6.2) Orthogonal local conditions Let J = J • , DJ and rv,J (for each v ∈ Sf ) be as in 5.2.2. (6.2.1) Assume that X1 , X2 are complexes of admissible R[GK,S ]-modules, π : X1 ⊗R X2 −→ J(1) a morphism of complexes of R[GK,S ]-modules and • + i+ v (Xi ) : Uv (Xi ) −→ Ccont (Gv , Xi )

(i = 1, 2; v ∈ Sf )

local conditions for X1 , X2 . (6.2.1.1) Typical examples of π are ev1 : DJ (X2 )(1) ⊗R X2 −→ J(1) ev2 : X1 ⊗R DJ (X1 )(1) −→ J(1). In general, π factors as adj(π)⊗id

ev

1 π : X1 ⊗R X2 −−−−−−→DJ (X2 )(1) ⊗R X2 −−→J(1)

and induces another morphism of complexes 105

s

π

12 π ◦ s12 : X2 ⊗R X1 −→X 1 ⊗R X2 −→J(1),

which factors into adj(π◦s12 )⊗id

ev

1 π ◦ s12 : X2 ⊗R X1 −−−−−−−−→DJ (X1 )(1) ⊗R X1 −−→J(1);

thus π also factors as id⊗adj(π◦s12 )

ev

2 π : X1 ⊗R X2 −−−−−−−−→X1 ⊗R DJ (X1 )(1)−−→J(1).

(6.2.1.2) For each v ∈ Sf , denote by prodv (X1 , X2 , π) the morphism of complexes i+ ⊗i+

∪

v v • • prodv (X1 , X2 , π) : Uv+ (X1 ) ⊗R Uv+ (X2 )−−−−→C cont (Gv , X1 ) ⊗R Ccont (Gv , X2 )−→ II τ≥2

π

∗ II • • • −→ Ccont (Gv , X1 ⊗R X2 )−−→C cont (Gv , J(1))−−→τ≥2 Ccont (Gv , J(1)).

(6.2.1.3) Definition. If there is a null-homotopy hv = hv (X1 , X2 , π) : prodv (X1 , X2 , π) −

0,

we say that ∆v (X1 ) is orthogonal to ∆v (X2 ) with respect to π and hv . Notation: ∆v (X1 ) ⊥π,hv ∆v (X2 ). (6.2.1.4) Definition. We say that ∆(X1 ) is orthogonal to ∆(X2 ) with respect to π and hS = (hv )v∈Sf (notation: ∆(X1 ) ⊥π,hS ∆(X2 )) if ∆v (X1 ) ⊥π,hv ∆v (X2 ) for all v ∈ Sf . (6.2.1.5) If the morphism prodv (X1 , X2 , π) is equal to zero, which happens very often in practice, then ∆v (X1 ) ⊥π,0 ∆v (X2 ). (6.2.2) Local cup products Fix v ∈ Sf and assume that ∆v (X1 ) ⊥π,hv ∆v (X2 ). Recall that, for i = 1, 2, the complex
−i+ v (Xi ) • Uv− (Xi )[−1] = Cone Uv+ (Xi )−−−−→C cont (Gv , Xi ) [−1] has differential d(bi , ci ) = (dbi , i+ v (bi ) − dci ) j

j−1 (for bi ∈ Uv+ (Xi ) , ci ∈ Ccont (Gv , Xi ), j = bi = ci + 1 = (bi , ci )). • • • II II Denote by ∪ = ∪π = τ≥2 ◦ π∗ ◦ ∪ the truncated cup product with values in τ≥2 Ccont (Gv , J(1)) from 6.2.1.2. The formulas •

(b1 , c1 ) ∪−,hv b2 = c1 ∪ i+ v (b2 ) + hv (b1 ⊗ b2 ) •

b1 ∪+,hv (b2 , c2 ) = (−1)b1 i+ v (b1 )∪ c2 + hv (b1 ⊗ b2 ) (cf. Proposition 1.3.2(i)) define morphisms of complexes
rv,J [−1] • II ∪−,π,hv : Uv− (X1 )[−1] ⊗R Uv+ (X2 ) −→ τ≥2 Ccont (Gv , J(1)) [−1]−−−−→J[−3]
rv,J [−1] • II ∪+,π,hv : Uv+ (X1 ) ⊗R (Uv− (X2 )[−1]) −→ τ≥2 Ccont (Gv , J(1)) [−1]−−−−→J[−3], hence, by adjunction, morphisms of complexes 106

u−,π,hv = adj(∪−,π,hv ) : Uv− (X1 )[−1] −→ DJ[−3] (Uv+ (X2 )) u+,π,hv = adj(∪+,π,hv ) : Uv+ (X1 ) −→ DJ[−3] (Uv− (X2 )[−1]). (6.2.3) Error terms. Assuming that ∆v (X1 ) ⊥π,hv ∆v (X2 ), put Errv (∆v (X1 ), ∆v (X2 ), π) = Cone(u+,π,hv ). If ∆(X1 ) ⊥π,hS ∆(X2 ), put Err(∆(X1 ), ∆(X2 ), π) =

M

Errv (∆v (X1 ), ∆v (X2 ), π).

v∈Sf

(6.2.4) Lemma. Fix v ∈ Sf and assume that ∆v (X1 ) ⊥π,hv ∆v (X2 ). The following diagram of morphisms of complexes is commutative and the vertical maps define a morphism of exact triangles in D(R Mod): / U − (X1 )[−1] v

• Ccont (Gv , X1 )[−1]

/ U + (X1 ) v

αJ,X1 [−1]

 • DJ[−3] (Ccont (Gv , DJ (X1 )(1)))

u−,π,hv

u+,π,hv

DJ[−3] ((adj(π◦s12 ))∗ )



• DJ[−3] (Ccont (Gv , X2 ))

DJ[−3] (i+ v (X2 ))

 / DJ[−3] (Uv+ (X2 ))

 / DJ[−3] (Uv− (X2 )[−1])

Proof. This follows from Lemma 1.2.11 and the following equalities: •

(0, c1 ) ∪−,π,hv b2 = c1 ∪π i+ v (b2 ) (b1 , 0) ∪−,π,hv b2 = b1 ∪+,π,hv (b2 , 0). (6.2.5) We shall be interested only in the following two cases: (A) The complexes X1 , X2 are bounded, J = I[n] for some n ∈ Z (hence DJ = Dn ) and - for i = 1 or 2 all cohomology groups of Xi (resp., of X3−i ) are of finite (resp., co-finite) type over R. (B) The complexes X1 , X2 are bounded, J = ω • [n] for some n ∈ Z (hence DJ = Dn ) and all cohomology groups of X1 , X2 are of finite type over R. (6.2.6) Lemma-Definition. Assume that one of the conditions (A) or (B) in 6.2.5 is satisfied. Then the following two conditions are equivalent: adj(π) : X1 −→ DJ (X2 )(1) is a quasi − isomorphism ⇐⇒ adj(π ◦ s12 ) : X2 −→ DJ (X1 )(1) is a Qis. If they are satisfied, we say that π is a perfect duality. Proof. Applying Lemma 1.2.14 to f, g, h = id, λ = π, µ = π ◦ s12 , we see that adj(π ◦ s12 ) is equal to DJ (adj(π))(1)

ε

J X2 −−→D J (DJ (X2 )(1))(1)−−−−−−−−→DJ (X1 )(1)

and adj(π) is equal to ε

DJ (adj(π◦s12 ))(1)

J X1 −−→D J (DJ (X1 )(1))(1)−−−−−−−−−−→DJ (X2 )(1).

The statement follows from the fact that both maps εJ are quasi-isomorphisms, by Matlis duality 2.3.2 (resp., Grothendieck duality 2.6) in the case 6.2.5(A) (resp., 6.2.5(B)). 107

(6.2.7) Lemma-Definition. Assume that one of the conditions (A) or (B) in 6.2.5 is satisfied and π is a perfect duality. If, for a fixed v ∈ Sf , we have ∆v (X1 ) ⊥π,hv ∆v (X2 ), then the following two conditions are equivalent: u+,π,hv is a quasi − isomorphism ⇐⇒ u−,π,hv is a quasi − isomorphism. If they are satisfied, we say that ∆v (X1 ) and ∆v (X2 ) are othogonal complements of each other with respect to π and hv ; notation: ∆v (X1 ) ⊥⊥π,hv ∆v (X2 ). If they are satisfied for all v ∈ Sf , we write ∆(X1 ) ⊥⊥π,hS ∆(X2 ). Proof. Under the assumptions (A) or (B) of 6.2.5 the vertical arrow αJ,X [−1] (resp., DJ[−3] ((adj(π ◦ s12 ))∗ )) in Lemma 6.2.4 is a quasi-isomorphism, by Proposition 5.2.4 (resp., by Definition 6.2.6). (6.2.8) Corollary. Under the assumptions of 6.2.7, ∼

∆v (X1 ) ⊥⊥π,hv ∆v (X2 ) ⇐⇒ Errv (∆v (X1 ), ∆v (X2 ), π) −→ 0 in D(R Mod). (6.3) Global cup products We are going to apply results of Sect. 1.3 to Selmer complexes. (6.3.1) Assume that we are given X1 , X2 , π and ∆(X1 ) ⊥π,hS ∆(X2 ) as in 6.2.1. Our goal is to define a morphism between the exact triangle gf (X1 ) −→ U + (X1 ) RΓc,cont (GK,S , X1 ) −→ RΓ S and the DJ[−3] -dual of the exact triangle gf (X2 ) −→ RΓcont (GK,S , X2 ). US− (X2 )[−1] −→ RΓ Consider the data of the type 1.3.1.1-4 given by the following objects: (1) M • C1 = Ccont (Gv , X1 ) • A1 = Ccont (GK,S , X1 ) • A2 = Ccont (GK,S , X2 )

B1 = US+ (X1 )

• II A3 = τ≥2 Ccont (GK,S , J(1))

B3 = 0

B2 =

v∈Sf

M

C2 =

US+ (X2 )

v∈Sf

M

C3 =

v∈Sf fj

gj

(2) Aj −→Cj ←−Bj given by fj = resSf , gj =

i+ S (Xj )

• Ccont (Gv , X2 )

• II τ≥2 Ccont (Gv , J(1))

(j = 1, 2), g3 = 0. •

(3) Products ∪A , ∪C induced by the truncated cup products ∪π attached to π (cf. 6.2.2) and ∪B = 0. (4) Homotopies h = (hf , hg ) = (0, hS ). For these data, e• (GK,S , Xj ; ∆(Xj )) = C e • (Xj ) Ej = C f f

(j = 1, 2).

As in 5.4.1 we have a quasi-isomorphism rJ [1] : E3 [1] −→ J[−2], unique up to homotopy, which makes the diagram

L

C3  (r ) y v,J

v∈Sf

−→ Σ

J[−2] −→

E3 [1]  r [1] yJ J[−2]

commutative up to homotopy. For every r ∈ R, Proposition 1.3.2 gives cup products r

J ∪π,r,h = ∪r,h : E1 ⊗R E2 −→ E3 −→J[−3],

108

hence, by adjunction, morphisms of complexes   e• (X1 ) −→ DJ[−3] C e• (X2 ) . γπ,r,hS = adj(∪π,r,h ) : C f f The homotopy class of γπ,r,hS is independent of the choices of r ∈ R, (rv,J ) and rS,J from 5.4.1. It may depend on the homotopies (hv ), but it does not change if we replace (hv ) by homotopic homotopies (e hv ) − (hv ) (via a second order homotopy). Denote by   gf (X1 ) −→ DJ[−3] RΓ gf (X2 ) γπ,hS : RΓ the corresponding map in D(R Mod). (6.3.2) If X1 , X2 are bounded and US+ (Xj ) (j = 1, 2) are both cohomologically bounded above, then gf (Xj ) (j = 1, 2) both lie in D− (R Mod) and the cup product ∪π,r,h induces (assuming that ∆(X1 ) ⊥π,h RΓ S ∆(X2 )) a cup product L

gf (X1 )⊗R RΓ gf (X2 ) −→ J[−3] ∪π,hS : RΓ

(6.3.2.1)

e fi (GK,S , X1 ; ∆(X1 )) ⊗R H e j (GK,S , X2 ; ∆(X2 )) −→ H i+j−3 (J • ). H f

(6.3.2.2)

and pairings on cohomology

(6.3.3) Proposition. Fix rv,J and rS,J . If ∆(X1 ) ⊥π,hS ∆(X2 ), then the following diagrams of morphisms of complexes have exact columns and are commutative up to homotopy (the vertical maps are those of 6.1.3 and their DJ[−3] -duals): 0   y US− (X1 )[−1]   y e • (GK,S , X1 ; ∆(X1 )) C f   y • Ccont (GK,S , X1 )   y

0   y u−,π,h

DJ[−3] (US+ (X2 ))   y

γπ,0,h

e • (GK,S , X2 ; ∆(X2 ))) DJ[−3] (C f   y

S −−−−→

S −−−−→

βc

−−−−→

• DJ[−3] (Cc,cont (GK,S , X2 ))   y

0

0

0   y

0   y

• Cc,cont (GK,S , X1 )   y

e • (GK,S , X1 ; ∆(X1 )) C f   y US+ (X1 )   y

cβ

−−−−→

• DJ[−3] (Ccont (GK,S , X2 ))   y

γπ,1,h

e • (GK,S , X2 ; ∆(X2 ))) DJ[−3] (C f   y

u+,π,h

DJ[−3] (US− (X2 )[−1])   y

S −−−−→

S −−−−→

0

0 109

Above, the maps βc resp., c β are equal to DJ[−3] (adj(π◦s12 )∗ )

βc,J,X

1 • • • βc : Ccont (GK,S , X1 )−−−−→D J[−3] (Cc,cont (GK,S , DJ (X1 )(1))) −−−−−−−−−−→DJ[−3] (Cc,cont (GK,S , X2 ))

cβ

DJ[−3] (adj(π◦s12 )∗ )

c βJ,X

1 • • • : Cc,cont (GK,S , X1 )−−−−→D J[−3] (Ccont (GK,S , DJ (X1 )(1))) −−−−−−−−−−→DJ[−3] (Ccont (GK,S , X2 )).

Proof. This follows from Lemma 1.2.11 and the following formulas •

•

(a1 , 0, c1 ) ∪1,h (a2 , 0, 0) = (a1 ∪π a2 , 0, c1 ∪π resSf (a2 )) •

•

(a1 , 0, 0) ∪0,h (a2 , 0, c2 ) = (a1 ∪π a2 , 0, (−1)a1 resSf (a1 )∪π c2 ) •

(0, b1 , c1 ) ∪0,h (0, b2 , 0) = (0, 0, c1 ∪π (i+ S (b2 )) + hS (b1 ⊗ b2 )) •

(0, b1 , 0) ∪1,h (0, b2 , c2 ) = (0, 0, (−1)b1 (i+ S (b1 ))∪π c2 + hS (b1 ⊗ b2 )) (valid in our case, since ∪B = 0 and hf = 0). (6.3.4) Theorem. Assume that one of the conditions (A) or (B) in 6.2.5 is satisfied and π is a perfect duality. If ∆(X1 ) ⊥π,hS ∆(X2 ), then there is an exact triangle in D(R Mod)   γπ,hS gf (X1 )−−−−→D g RΓ J[−3] RΓf (X2 ) −→ Err(∆(X1 ), ∆(X2 ), π). In particular, the map

  gf (X1 ) −→ DJ[−3] RΓ gf (X2 ) γπ,hS : RΓ

is an isomorphism in D(R Mod) if and only if ∆(X1 ) ⊥⊥π,hS ∆(X2 ). Proof. In the second diagram in Proposition 6.3.3, the map c β is a quasi-isomorphism by Proposition 5.4.3. This implies that Cone(γπ,1,hS ) is isomorphic in D(R Mod) to Cone(u+,π,hS ) = Err(∆(X1 ), ∆(X2 ), π), which proves the Theorem (using Corollary 6.2.8). (6.3.5) On the level of cohomology, Theorem 6.3.4 gives exact sequences   e q (X1 ) −→ H q−3 DJ (RΓ gf (X2 )) −→ H q (Err) −→ · · · · · · −→ H q−1 (Err) −→ H f (where Err = Err(∆(X1 ), ∆(X2 ), π)). Under the assumptions of 6.2.5(A), we have J = I[n] and     gf (X2 )) = D H e 3−n−q (X2 ) . H q−3 DJ (RΓ f

Under the assumptions of 6.2.5(B), we have J = ω • [n]. If, in addition, all cohomology groups of US+ (X2 ) are gf (X2 ) ∈ D (R Mod) and there is a spectral sequence of finite type over R, then RΓ ft   e 3−n−j (X2 ), ω) =⇒ H i+j−3 DJ (RΓ gf (X2 )) , E2i,j = ExtiR (H f which degenerates in the category (R Mod)/(pseudo − null) into exact sequences   gf (X2 )) −→ E 0,q −→ 0. 0 −→ E21,q−1 −→ H q−3 DJ (RΓ 2 The term E20,q is torsion-free over R (by Lemma 2.8.8), while codimR (supp(E21,q−1 )) ≥ 1. In particular, there is a monomorphism in (R Mod)/(pseudo − null)    gf (X2 )) e 4−n−q (X2 ), ω), H q−3 DJ (RΓ −→ Ext1R (H f R−tors

which is an isomorphism if R has no embedded primes. (6.3.6) In the situation of 6.2.5(B), there is a straightforward generalization of 6.2.6, 6.3.4 and 6.3.5, if we insist on adj(π) to be a quasi-isomorphism only in D((R Mod)/(pseudo − null)). 110

(6.4) Functoriality of Selmer complexes Let J be as in Sect. 6.2. (6.4.1) Let X1 , X2 , π be as in 6.2.1 and assume that Y1 , Y2 , ρ : Y1 ⊗R Y2 −→ J(1) is another triple of the same kind. Consider the following data: (6.4.1.1) Orthogonal local conditions ∆(X1 ) ⊥π,hS (X) ∆(X2 ),

∆(Y1 ) ⊥π,hS (Y ) ∆(Y2 ).

(6.4.1.2) Morphisms of complexes of R[GK,S ]-modules λj : Xj −→ Yj (j = 1, 2); set λ = λ1 ⊗ λ2 : X1 ⊗R X2 −→ Y1 ⊗R Y2 . (6.4.1.3) Morphisms of complexes of R-modules βj : US+ (Xj ) −→ US+ (Yj ) (6.4.1.4) Homotopies

vj : i+ S (Yj ) ◦ βj −

(j = 1, 2).

(λj )∗ ◦ i+ S (Xj )

(j = 1, 2).

(6.4.1.5) A homotopy k : ρ ◦ λ − π. (6.4.1.6) A second order homotopy •

+ hS (X) + k∗ ? (i+ S (X1 ) ⊗ iS (X2 )) + ∪ρ ? (v1 ⊗ v2 )1 M

i−2 i • II K i : US+ (X1 ) ⊗R US+ (X2 ) −→ τ≥2 Ccont (Gv , J(1)) .

K : hS (Y ) ? (β1 ⊗ β2 ) −

v∈Sf

Given (6.4.1.1-6), we obtain the following data of the type considered in 1.3.3: (Aj , Bj , Cj , fj , gj , ∪∗ , h) as ej , B ej , C ej , fej , gej , ∪ e∗, e in 6.3.1, (A h) the corresponding objects for (Y1 , Y2 , ρ), αj , γj = (λj )∗ ,

β j = βj

(j = 1, 2),

α3 , γ3 = id,

β3 = 0,

uj = 0, vj = vj (j = 1, 2), u3 , v3 = 0, kα , kγ = k∗ , kβ = 0, Kf = 0, Kg = K. Applying Proposition 1.3.4, we obtain the following result. (6.4.2) Proposition. (i) The data 6.4.1.1-4 (it is not necessary to assume orthogonality of local conditions at this point) define morphisms of complexes e • (Xj ) −→ C e• (Yj ), ϕ(λj ) = ϕ(λj , βj , vj ) : C f f given by the formula in Proposition 1.3.4(i). The homotopy class of ϕ(λj , βj , vj ) is unchanged if vj is replaced by vj0 related to vj by a second order homotopy vj0 − vj . (ii) Given the data 6.4.1.1-6, the following diagram of morphisms of complexes is commutative up to homotopy (for every r ∈ R): ∪π,r,hS (X) e • (X1 ) ⊗R C e • (X2 ) −−−−−−→ C J[−3] f f  ϕ(λ )⊗ϕ(λ ) k 2 y 1 e• (Y1 ) ⊗R C e • (Y2 ) C f f

∪ρ,r,h

(Y )

S −−−−−−→

J[−3].

(6.4.3) Corollary. Given the data 6.4.1.1-6, the following diagram is commutative up to homotopy (for every r ∈ R):   γπ,r,hS (X) • e • (X1 ) −−−−−−→ e C D C (X ) 2 J[−3] f f  x ϕ(λ ) D y 1  J[−3] (ϕ(λ2 ))   γρ,r,hS (Y ) e• (Y1 ) −−−−−−→ e• (Y2 ) . C DJ[−3] C f f 111

Proof. Apply Lemma 1.2.11 to the diagram in Proposition 6.4.2(ii). (6.4.4) If we are given the data 6.4.1.1-3 such that hS (X) = hS (Y ) = 0, π = ρ ◦ λ and i+ S (Yj ) ◦ βj = (λj )∗ ◦ i+ (X ) (j = 1, 2) which happens quite often in practice then we can take v = 0 (j = 1, 2), j j S k = 0, K = 0. The formula in Proposition 1.3.4(ii) then gives H = 0, which implies that the diagrams in Proposition 6.4.2(ii) and Corollary 6.4.3 are commutative, not just commutative up to homotopy. (6.4.5) A special case of the functoriality data 6.4.1 occurs if we replace π by a homotopic morphism of complexes π 0 : X1 ⊗R X2 −→ J(1). Taking Yj = Xj ,

∆(Yj ) = ∆(Xj ),

λj = id,

βj = id,

vj = 0 (j = 1, 2),

ρ = π0 ,

k : π0 −

π,

all we need in order to apply Proposition 6.4.2 are new homotopies h0v = hv (X1 , X2 , π 0 ) : prod0v = prodv (X1 , X2 , π 0 ) −

0

and a second order homotopy K : h0S −

+ hS + k∗ ? (i+ S (X1 ) ⊗ iS (X2 )).

For example, if µ : J −→ J is homotopic to the identity via a homotopy ` : µ − can take k = ` ? π,

h0v = µ ? hv ,

id and π 0 = µ ◦ π, then we

K = (Kv = −`hv )v∈Sf ,

since Kv : −(d` + `d) ? hv = h0v − hv −

−` ? (dhv + hv d) = ` ? prodv .

(6.5) Transpositions In order to exchange the roles of X1 and X2 in 6.2.1 we need additional data. (6.5.1) Definition. Given X as in 6.1.1 and a local condition ∆v (X) at v ∈ Sf , a transposition operator for ∆v (X) is a morphism of complexes Tv+ (X) : Uv+ (X) −→ Uv+ (X) such that the diagram i+ (X)

v Uv+ (X) −−→   + yTv (X)

i+ (X)

v Uv+ (X) −−→

• Ccont (Gv , X)   yT • Ccont (Gv , X)

commutes up to homotopy (recall that T denotes the transposition operator defined in 3.4.5.3-4). (6.5.2) Lemma. Assume that we are given π : X1 ⊗R X2 −→ J(1) as in 6.2.1 and local conditions ∆v (Xj ), j = 1, 2 (for some v ∈ Sf ) that both admit transposition operators Tv+ (Xj ). Then the following two conditions are equivalent: ∆v (X1 ) ⊥π,hv ∆v (X2 ) for some hv ⇐⇒ ∆v (X2 ) ⊥π◦s12 ,h0v ∆v (X1 ) for some h0v . Proof. This follows from the fact that the following diagram and its analogue in which the roles of X1 and X2 are interchanged are commutative up to homotopy (by 3.4.5.4): 112

s12 ◦(T + ⊗T + )

Uv+ (X1 ) ⊗R Uv+ (X2 )  + + yiv ⊗iv

v v −−−−−−−−→

• • Ccont (Gv , X1 ) ⊗R Ccont (Gv , X2 )  • y∪π

−−−−−−→

• • Ccont (Gv , X2 ) ⊗R Ccont (Gv , X1 )  • y∪π◦s12

II • τ≥2 Ccont (Gv , J(1))

==

II • τ≥2 Ccont (Gv , J(1))

s12 ◦(T ⊗T )

Uv+ (X2 ) ⊗R Uv+ (X1 )  + + yiv ⊗iv

(6.5.3) Assume that we are given X1 , X2 , π as in 6.2.1. Consider the following additional data: (6.5.3.1) Orthogonal local conditions ∆(X1 ) ⊥π,hS ∆(X2 ),

∆(X2 ) ⊥π◦s12 ,h0S ∆(X1 ).

(6.5.3.2) For each Z = X1 , X2 and v ∈ Sf a transposition operator Tv+ (Z) : Uv+ (Z) −→ Uv+ (Z); put T + (Z) = (Tv+ (Z))v∈Sf . (6.5.3.3) For each Z = X1 , X2 and v ∈ Sf a homotopy + VZ,v : i+ v (Z) ◦ Tv (Z) −

T ◦ i+ v (Z);

put VZ = (VZ,v )v∈Sf (the existence of VZ follows from 6.5.3.2, by definition). (6.5.3.4) For each Z = X1 , X2 and v ∈ Sf , homotopies kZ : id −

satisfying

T

• on Ccont (GK,S , Z)

+ kZ,v : id −

Tv+ (Z)

on Uv+ (Z)

kZ,v : id −

T

• on Ccont (Gv , Z)

resv ? kZ = kZ,v ? resv

and such that there exists a second order homotopy + i+ v (Z) ? kZ,v + VZ,v −

kZ,v ? i+ v (Z).

(6.5.3.5) For each v ∈ Sf , a second order homotopy •

Hv : T ? hv (X1 , X2 , π) + ∪π ? (VX1 ,v ⊗ VX2 ,v )1 − h0v (X2 , X1 , π ◦ s12 ) ? (s12 ◦ (Tv+ (X1 ) ⊗ Tv+ (X2 )))
i
i−2 II • Hvi : Uv+ (X1 ) ⊗R Uv+ (X2 ) −→ τ≥2 Ccont (Gv , J(1)) . (6.5.4) Proposition. (i) Given the data 6.5.3.1-3, the formula TZ (a, b, c) = (T (a), T + (Z)(b), T (c) − VZ (b))

(Z = X1 , X2 )

defines a morphism of complexes ef• (GK,S , Z; ∆(Z)) −→ C ef• (GK,S , Z; ∆(Z)). TZ : C If, in addition, we are given 6.5.3.4, then TZ is homotopic to the identity. (ii) Given the data 6.5.3.1-3 and 6.5.3.5, then the following diagrams commute up to homotopy (for every r ∈ R). ∪π,r,h e• (X1 ) ⊗R C e• (X2 ) −−−−−−−−→ C J[−3] f f  s ◦(T ⊗T ) k y 12 X1 X2 ∪

0

π◦s12 ,1−r,h e• (X2 ) ⊗R C e• (X1 ) −−−−−−−−→ C f f

113

J[−3]

γπ,r,h

e• (X1 ) C f  ε y J[−3] ◦TX1

e• (X2 )) DJ[−3] (C f

S −−−−−−−−−−−−−−→

e • (X1 ))) DJ[−3] (DJ[−3] (C f e • (X2 ) C f  εJ[−3] y

e• (X2 )) DJ[−3] (C f

S −−−−−−−−−−−−−−→

γπ◦s12 ,1−r,h0 ◦TX2

e• (X1 )) DJ[−3] (C f  D y J[−3] (TX1 )

S −−−−−−−−−−−−→

e• (X2 ))) DJ[−3] (DJ[−3] (C f

k

DJ[−3] (γπ◦s12 ,1−r,h0 ◦TX2 )

DJ[−3] (γπ,r,h )

e• (X1 )) DJ[−3] (C f

S −−−−−−−−−−−−→

Proof. (i) This follows from 1.1.7. (ii) We are going to apply Proposition 1.3.6 to the data 6.3.1.1-4 together with

h0f

TA , TC = T , = 0,

h0g

=

h0S ,

•

TB = T + (X1 ) resp., T + (X2 ),

Uj = 0 (j = 1, 2, 3), tα = tβ = tγ = 0, Hf = 0,

∪0A , ∪0C = ∪π , V3 = 0, Vj = VXj Hg = (Hv )v∈Sf .

∪0B = 0 (j = 1, 2),

Proposition 1.3.6 then implies that the first diagram is commutative up to homotopy (as T3 = T − id and ∪01−r,h0 = ∪π◦s12 ,1−r,h0 ). Commutativity up to homotopy of the second and third diagrams then follows from Corollary 1.3.7. (6.5.5) Corollary. Given the data 6.5.3.1-5 and r ∈ R, the diagram ∪π,r,h e• (X1 ) ⊗R C e• (X2 ) −−−−−−−−→ C f f  s y 12 ∪π◦s12 ,1−r,h0

e• (X2 ) ⊗R C e• (X1 ) −−−−−−−−→ C f f

J[−3] k J[−3]

is commutative up to homotopy and the composite maps DJ[−3] (γπ◦s

,1−r,h0

)

εJ[−3] 12 S e• (X1 )−−−−→D e• e• C J[−3] (DJ[−3] (Cf (X1 )))−−−−−−−−−−−−→DJ[−3] (Cf (X2 )) f DJ[−3] (γπ,r,hS ) εJ[−3] e• (X2 )−−−−→D e• e• C J[−3] (DJ[−3] (Cf (X2 )))−−−−−−−−−−−−→DJ[−3] (Cf (X1 )) f

are homotopic to γπ,r,hS and γπ◦s12 ,1−r,h0S , respectively. Under the assumptions as in (6.3.2.1), the corresponding cup products make the diagram ∪π,hS :

∪π◦s12 ,h0S :

L

gf (X1 )⊗R RΓ gf (X2 ) RΓ  s y 12 L

gf (X2 )⊗R RΓ gf (X1 ) RΓ

−→

J[−3] k

−→

J[−3]

commutative in D− (R Mod). (6.6) Self-dual case (6.6.1) Let X and ∆(X) be as in 6.1.1. Assume that we are given a morphism of complexes of R[GK,S ]modules π : X ⊗R X −→ J(1) such that ∆(X) ⊥π,hS ∆(X) for suitable homotopies hS = (hv )v∈Sf , and that π 0 := π ◦ s12 is equal to π 0 = π ◦ s12 = c · π, •

•

This implies that ∪π0 = c · ∪π and 114

c = ±1.

h0S = c · hS .

∆(X) ⊥π0 ,h0S ∆(X),

The formula in Proposition 1.3.2(i) implies that the cup products e • (X) ⊗R C e• (X) −→ J[−3] ∪π,r,h , ∪π0 ,r,h0 : C f f are related by ∪π0 ,r,h0 = c · ∪π,r,h .

(6.6.1.1)

(6.6.2) Proposition. Under the assumptions of 6.6.1, assume, in addition, that Z = X admits transposition operators Tv+ (X) (v ∈ Sf ) and the data 6.5.3.3-5, where (6.5.3.5) consists of second order homotopies •

Hv : T ? hv + ∪π ? (VX,v ⊗ VX,v )1 −

c · hv ? (s12 ◦ (Tv+ (X) ⊗ Tv+ (X)))

(v ∈ Sf ).

Then the following diagram is commutative up to homotopy: ∪π,r,h e• (X) ⊗R C e• (X) −−−−−−−−→ C f f  s y 12 c·∪π,1−r,h

e• (X) ⊗R C e• (X) −−−−−−−−→ C f f

J[−3] k J[−3]

and the morphism D

(γ

)

εJ[−3] π,r,hS J[−3] ef• (X)−−−−→D e• e• C J[−3] (DJ[−3] (Cf (X)))−−−−−−−−−−−−→DJ[−3] (Cf (X))

is homotopic to c · γπ,1−r,hS . Proof. This is a special case of Corollary 6.5.5 for X1 = X2 = X, if we take into account (6.6.1.1). e • (X) is cohomologically bounded above, then the cup product (6.3.2.1) (6.6.3) Corollary. If, in addition, C f L

gf (X)⊗R RΓ gf (X) −→ J[−3] ∪π,hS : RΓ satisfies ∪π,hS = c · (∪π,hS ◦ s12 ) . (6.6.4) Hermitian case. In 6.6.4-7 we assume that R is equipped with an involution ι, i.e. with a ring homomorphism ι : R −→ R satisfying ι ◦ ι = id. For an R-module X, set X ι = X ⊗R,ι R; this is an R-module in which r(x ⊗ r0 ) = x ⊗ rr0 ,

(rx) ⊗ r0 = x ⊗ ι(r)r0

(r, r0 ∈ R, x ∈ X).

For any R-modules X, Y there are canonical isomorphisms of R-modules ∼

(X ι )ι −→ X, ∼

X ι ⊗R Y ι −→ (X ⊗R Y )ι , ∼

ι(= ι ⊗ id) : Rι = R ⊗R,ι R −→ R,

(x ⊗ r) ⊗ r0 7→ ι(r)r0 x, (x ⊗ r) ⊗ (y ⊗ r0 ) 7→ (x ⊗ y) ⊗ rr0 , r ⊗ r0 7→ ι(r)r0 ,

(6.6.4.1)

f ⊗ r 7→ (x ⊗ r0 7→ f (x) ⊗ rr0 ).

∼

HomR (X, Y )ι −→ HomR (X ι , Y ι ),

For f : X −→ Y we denote by f ι : X ι −→ Y ι the image of f ⊗ 1 under the last isomorphism in (6.6.4.1). With this notation, the isomorphism 115

∼

ιι : R −→ (Rι )ι −→ Rι is given by r −→ 1 ⊗ r. The functor X 7→ X ι extends in an obvious way to complexes of R-modules, and the isomorphisms (6.6.4.1) also hold for X • ⊗R Y • and Hom•R (X • , Y • ). Assume that we are given a morphism of complexes ν : J ι −→ J (where J is as in 6.2) such that ∼

ν ι : J −→ (J ι )ι −→ J ι is a homotopy inverse of ν. ι

(6.6.5) Let X, ∆(X) be as in 6.1.1. The local conditions ∆(X) define local conditions ∆(X ι ) = ∆(X) for X ι: ι

ι

∼

ι

• • ι + + ι i+ v (X ) = iv (X) : Uv (X) −→ Ccont (Gv , X) −→ Ccont (Gv , X )

and a canonical isomorphism of complexes ∼ e• e • (X)ι −→ C Cf (X ι ). f

Assume that π : X ⊗R X ι −→ J(1) is a morphism of complexes of R[GK,S ]-modules such that ∆(X) ⊥π,hS ∆(X ι ) for suitable homotopies hS = (hv )v∈Sf , and that π 0 := π ◦ s12 is equal to π 0 = π ◦ s12 = c · (ν ◦ π ι ), •

c = ±1.

•

It follows that ∪π0 = c · (ν ◦ (∪π )ι ) and ∆(X ι ) ⊥π0 ,h0S ∆(X),

h0S = c · (ν ◦ hιS ).

As in (6.6.1.1), the cup products ef• (X) ⊗R C ef• (X)ι −→ J[−3] ∪π,r,h : C

e• (X)ι ⊗R C e• (X) −→ J[−3] ∪π0 ,r,h0 : C f f are related by ∪π0 ,r,h0 = c · (ν ◦ (∪π,r,h )ι ). (6.6.6) Proposition. Under the assumptions of 6.6.5, assume, in addition, that Z = X admits transposition operators Tv+ (X) (v ∈ Sf ) and the data 6.5.3.3-4. Applying the functor M 7→ M ι , we obtain the same data for Z = X ι . Assume, furthermore, that the triple (X, X ι , π) admits the data 6.5.3.5, i.e. second order homotopies •

Hv : T ? hv + ∪π ? (VX,v ⊗ VX ι ,v )1 −

c · (ν ◦ hιv ) ? (s12 ◦ (Tv+ (X) ⊗ Tv+ (X ι )))

Then the following diagram is commutative up to homotopy: e • (X) ⊗R C e • (X)ι C f f  s y 12

∪π,r,h

−−−−−−−−→ ι

c·(∪π,1−r,h ) e • (X)ι ⊗R C e • (X) −−−−−−−−→ C f f

J[−3] x ν 

J[−3]   ι yν

J ι [−3] == J ι [−3].

Proof. This is a special case of Corollary 6.5.5 for X1 = X, X2 = X ι . 116

==

(v ∈ Sf ).

e • (X) is cohomologically bounded above, then the cup product (6.3.2.1) (6.6.7) Corollary. If, in addition, C f gf (X)⊗R RΓ gf (X)ι −→ J[−3] ∪π,hS : RΓ L

satisfies ∪π,hS = c · (ν ◦ (∪π,hS )ι ◦ s12 ) . (6.7) An example of local conditions In this section we consider local conditions analogous to those studied by Greenberg [Gre 1-3]. Let J be as in 6.2, X, Y complexes of admissible R[GK,S ]-modules, and π : X ⊗R Y −→ J(1) a morphism of complexes of R[GK,S ]-modules. (6.7.1) Fix v ∈ Sf . Assume that we are given for Z = X, Y a complex of admissible R[Gv ]-modules Zv+ and a morphism of complexes of R[Gv ]-modules jv+ (Z) : Zv+ −→ Z. These data define local conditions j + (Z)

v • • ∆v (Z) : Uv+ (Z) = Ccont (Gv , Zv+ )−−−−→C cont (Gv , Z).

Put −j + (Z)

v Zv− = Cone(Zv+ −−−−→Z); • then Uv− (Z) = Ccont (Gv , Zv− ).

(6.7.2) Definition. We say that Xv+ ⊥π Yv+ if the morphism j + (X)⊗j + (Y )

π

Xv+ ⊗R Yv+ −−−−−−−−−−→X ⊗R Y −−→J(1) is zero. (6.7.3) Lemma. (i) Xv+ ⊥π Yv+ =⇒ ∆v (X) ⊥π,0 ∆v (Y ). (ii) Xv+ ⊥π Yv+ ⇐⇒ Yv+ ⊥π◦s12 Xv+ . Proof. This follows from the definitions. (6.7.4) The morphism π ◦ (j + (X) ⊗ j + (Y )) in 6.7.2 factors through X ⊗R Yv+ . By adjunction we obtain a morphism of complexes X −→ Hom•R (Yv+ , J(1)) = DJ (Yv+ )(1).

(6.7.4.1)

If Xv+ ⊥π Yv+ , then (6.7.4.1) induces a morphism of complexes Xv− −→ Hom•R (Yv+ , J(1)) = DJ (Yv+ )(1).

(6.7.4.2)

We say that Xv+ ⊥⊥π Yv+ if (6.7.4.2) is a quasi-isomorphism. (6.7.5) We shall be interested only in the following two cases: (A) The complexes X, Y, Xv+ , Yv+ are bounded, J = I[n] for some n ∈ Z and either all cohomology groups of X, Xv+ (resp., of Y, Yv+ ) are of finite (resp., co-finite) type over R, or all cohomology groups of X, Xv+ (resp., of Y, Yv+ ) are of co-finite (resp., finite) type over R. (B) The complexes X, Y, Xv+ , Yv+ are bounded, J = ω • [n] for some n ∈ Z and all cohomology groups of X, Y, Xv+ , Yv+ are of finite type over R. 117

(6.7.6) Proposition. Assume that one of the conditions (A) or (B) of 6.7.5 is satisfied, Xv+ ⊥π Yv+ and π is a perfect duality. Then (i) Xv+ ⊥⊥π Yv+ ⇐⇒ Yv+ ⊥⊥π◦s12 Xv+ . (ii) Xv+ ⊥⊥π Yv+ =⇒ ∆v (X) ⊥⊥π,0 ∆v (Y ). (iii) Yv+ ⊥⊥π◦s12 Xv+ =⇒ ∆v (Y ) ⊥⊥π◦s12 ,0 ∆v (X). (iv) Completing the morphism (6.7.4.2) to an exact triangle Wv −→ Xv− −→ DJ (Yv+ )(1) −→ Wv [1] in b Db (ad R[Gv ] Mod), then there is an isomorphism in D (R Mod) ∼

Errv (∆v (X), ∆v (Y ), π) −→ RΓcont (Gv , Wv ). Proof. The map (6.7.4.2) and the dual of its analogue for π ◦ s12 fit into a morphism of exact triangles in Db (ad R[Gv ] Mod) Xv+   y

−→

Hom•R (Yv− , J(1)) −→

Xv−   y

X   y

−→

Hom•R (Y, J(1))

−→

(6.7.6.1)

Hom•R (Yv+ , J(1)),

in which the middle vertical arrow is an isomorphism, since π is a perfect duality; this proves (i). As regards (iv), applying the functor RΓcont (Gv , −) to (6.7.6.1) we obtain a morphism of exact triangles in Db (R Mod) RΓcont (Gv , Xv+ )  u+,π,0 y

−→

RΓcont (Gv , DJ (Yv− )(1)) −→

RΓcont (Gv , X)   y

−→

RΓcont (Gv , Xv− )   yλ

RΓcont (Gv , DJ (Y )(1))

−→

RΓcont (Gv , DJ (Yv+ )(1)),

in which the middle vertical arrow is again an isomorphism. This gives isomorphisms ∼

∼

Errv (∆v (X), ∆v (Y ), π) = Cone(u+,π,0 ) −→ Cone(λ)[−1] −→ RΓcont (Gv , Wv ) in Db (R Mod), proving (iv) and (ii) (hence also (iii), if we replace π by π ◦ s12 ). (6.7.7) Proposition. Assume that we are given Zv+ −→ Z (Z = X, Y ) satisfying Xv+ ⊥π Yv+ for all v ∈ Sf . Assume that one of the conditions 6.7.5(A) or (B) is satisfied. Then M γπ,0 gf (X)−−−−→D g RΓ RΓcont (Gv , Wv ), J[−3] (RΓf (Y )) −→ v∈Sf b where Wv was defined in Proposition 6.7.6(iv), is an exact triangle in Dfb t (R Mod) (resp., Dcof t (R Mod)). In + + particular, if Xv ⊥⊥π Yv for all v ∈ Sf , then the map

gf (X) −→ DJ[−3] (RΓ gf (Y )) γπ,0 : RΓ b is an isomorphism in Dfb t (R Mod) (resp., Dcof t (R Mod)).

Proof. Apply Theorem 6.3.4. (6.7.8) Transpositions. Assume that we are given Zv+ −→ Z (Z = X, Y ) satisfying Xv+ ⊥π Yv+ for all v ∈ Sf . Then the following objects are the data of the type 6.5.3.1-5: ∆v (X) ⊥π,0 ∆v (Y ),

∆v (Y ) ⊥π◦s12 ,0 ∆v (X),

+ kZ , kZ,v , kZ,v

Tv+ (Z) = T ,

− given by a functorial homotopy id −

VZ,v = Hv = 0, T.

(6.7.9) Assume that X, Y, Xv+ , Yv+ (v ∈ Sf ) satisfy the condition 6.7.5(B) with J = ω • . Write T = X, Tv+ = + Xv+ , T ∗ (1) = Y , T ∗ (1)+ v = Yv and put 118

− A+ v = D(Yv )(1)

A = D(Y )(1), A∗ (1) = D(X)(1),

− A∗ (1)+ v = D(Xv )(1).

If Xv+ ⊥⊥π Yv+ for all v ∈ Sf , then the previous discussion and Theorem 6.3.4 imply that the Selmer complexes of T, A, T ∗ (1), A∗ (1) are related by the duality diagram D gf (T ) ←−−−− gf (T ∗ (1))[3] RΓ −−−−→ RΓ ←−     −− D −−→   −−−−− Φ Φ −−−−−−− y y − − → ←− gf (A) gf (A∗ (1))[3] RΓ RΓ b (in D(co)f t (R Mod)). This diagram gives a spectral sequence

e 3−j (T ∗ (1)), ω) = Exti (D(H e j (A)), ω) =⇒ H e i+j (T ). E2i,j = ExtiR (H R f f f (6.8) Localization (6.8.1) Let S ⊂ R be a multiplicative subset. Everything in Sections 6.1-7 is still valid for RS instead of R; the only difference is that references to 5.2.3 should be replaced by those to 5.6.2. For example, the same proof as in 5.6.3 gives a localized version of the duality Theorem 6.3.4. (6.9) In the absence of (P ) b• (Gv , X) and local conditions (6.9.1) In the situation of 5.7, we must also consider the complexes C cont • bcont Uv+ (X) −→ C (Gv , X)

at all real primes v of K. Everything in 6.1-7 works with obvious modifications, provided we consider only bounded complexes X, Y . The easiest method is to put Uv+ (X) = Uv+ (Y ) = 0 for all real primes v; then the complex M Errv (∆v (X), ∆v (Y ), π) Kv =R

becomes acyclic in D(R[1/2] Mod), where R[1/2] = R ⊗Z2 Q2 .

119

7. Unramified cohomology Let v ∈ Sf , v - p. The aim of this chapter is to define a suitable generalization of unramified cohomology 1 Hur (Gv , M ). In the absence of a good Hochschild-Serre spectral sequence on the level of complexes, we use explicit “small” complexes computing continuous cohomology in this case. Unramified local conditions turn out to be orthogonal with respect to the Pontrjagin duality (Proposition 7.5.5, 7.6.6), but not with respect to the Grothendieck duality; generalized local Tamagawa factors appear at this point (7.6.7-12). Combining unramified local conditions with those from Sect. 6.7, we obtain Selmer complexes attached to the Greenberg local conditions; these are studied in Sect. 7.8. (7.1) Notation (7.1.1) We use the standard notation: Kvur (resp., Kvt ) denotes the maximal unramified (resp., tamely ramified) extension of Kv contained in Kvsep and Iv = Gal(Kvsep /Kvur ) (resp., Ivw = Gal(Kvsep /Kvt )) the inertia (resp., wild inertia) group. Put Gv = Gv /Ivw = Gal(Kvt /Kv ) I v = Iv /Ivw = Gal(Kvt /Kvur ) (= the tame inertia group) ur (7.1.2) For M ∈ (ind−ad R[Gv ] Mod) we define the unramified local conditions ∆v (M ) to be inf

• • Uv+ (M ) = Ccont (Gv /Iv , M Iv )−−→Ccont (Gv , M ).

The inflation maps induce isomorphisms  G v i=0  M  ∼ i Iv res 1 1 1 Hcont (Gv /Iv , M ) −→ Hur (Gv , M ) = Ker(Hcont (Gv , M )−−→Hcont (Iv , M )) i = 1    0 i>1 (this is well-known for discrete modules; the general case follows by taking limits). • • We would like to define ∆ur v (M ) for a (bounded below) complex M of ind-admissible R[Gv ]-modules. The naive definition inf

• • Ccont (Gv /Iv , (M • )Iv )−−→Ccont (Gv , M • )

is not very useful, as it does not factor through the derived category. Note that M Iv is quasi-isomorphic to • • τ≤0 Ccont (Iv , M ); it would be natural (especially from a perverse point of view) to define ∆ur v (M ) as ∼

• • • • • “Ccont (Gv /Iv , τ≤0 Ccont (Iv , M • )) −→ Ccont (Gv /Iv , Ccont (Iv , M • )) −→ Ccont (Gv , M • )”.

(7.1.2.1)

Unfortunately, we have not been able to make sense of the Hochschild-Serre spectral sequence for continuous cohomology even in this very simple case. The problem is, as explained in 3.6.1.4, that in general • τ≤0 Ccont (Iv , M • ) is not a complex of Gv /Iv -modules (let alone of ind-admissible R[Gv /Iv ]-modules). In• stead of interpreting (7.1.2.1) literally, we use explicit “small” complexes quasi-isomorphic to Ccont (G, M • ) for G = Gv , Iv , Gv /Iv . (7.2) Complexes C(M ) (7.2.1) For every M ∈ (ind−ad R[Gv ] Mod), the inflation map • • inf : Ccont (Gv , M t ) −→ Ccont (Gv , M )

w

(M t = M Iv )

• is a quasi-isomorphism, by Lemma 4.1.4. This means that it will be sufficient to consider only Ccont (Gv , M ) ind−ad t for “tame” modules M = M ∈ (R[G ] Mod). v ∼ b Fix a topological generator t = tv of I v −→ Z/Z l (where l 6= p is the characteristic of the residue field k(v) of v) and a lift f = fv ∈ Gv of the geometric Frobenius element Fr(v) ∈ Gv /Iv = Gv /I v . Then

120

Gv = hti o hf i has topological generators t and f , with a unique relation tf = f tL ,

L = |k(v)| = N (v).

The element θ = f (1 + t + · · · + tL−1 ) ∈ Z[Gv ] satisfies θ(t − 1) = f (tL − 1) = (t − 1)f (θ − 1)(t − 1) = (t − 1)(f − 1). For every Gv -module M denote by C(M ) the complex (f −1,t−1)

(1−t,θ−1)

C(M ) = [M −−−−−−→M ⊕ M −−−−−−→M ] in degrees 0, 1, 2 and by C + (M ) (resp., C − (M )) the subcomplex of C(M ) equal to   f −1 C + (M ) = M t=1 −−−−→M t=1 in degrees 0, 1 (resp., the quotient complex equal to Lf −1

θ−1

C − (M ) = [M/(t − 1)M −−−−→M/(t − 1)M ] = [M/(t − 1)M −−−−→M/(t − 1)M ] in degrees 1, 2). The canonical projections define a quasi-isomorphism Qis

C(M )/C + (M )−−→C − (M ), hence an exact triangle C + (M ) −→ C(M ) −→ C − (M ) −→ C + (M )[1]. On the level of cohomology this gives

0

−→

H 0 (C + (M ))

−→

∼

H 0 (C(M ))

H 1 (C + (M ))

−→

H 1 (C(M ))

−→

H 1 (C − (M ))

H 2 (C(M ))

∼

H 2 (C − (M )).

−→

−→

0

We are now going to define functorial quasi-isomorphisms µ

λ

• C(M )−→Ccont (Gv , M )−→C(M )

(M ∈ (ind−ad Mod)) R[G ] v

satisfying λ ◦ µ = id. (7.2.2) Let G = hσi be a topologically cyclic pro-finite group with a fixed topological generator σ. Assume that the order of G is divisible by p∞ ; then cdp (G) = 1. In this case the complex σ−1

[M −−→M ]

(M ∈ (ind−ad R[G] Mod))

• in degrees 0, 1 is canonically quasi-isomorphic to Ccont (G, M ). Indeed, writing M = − lim → Mα (Mα ∈ S(M )) n and Mα = ← lim M /p M , it is sufficient to construct functorial quasi-isomorphisms α α −

σ−1

• λ : Ccont (G, M ) −→ [M −−→M ]

for discrete p-primary torsion G-modules M (such as Mα /pn Mα above). The formulas 121

λ0 = id,

λ1 (c) = c(σ),

λi = 0

(i > 1)

define such a λ. There is another functorial quasi-isomorphism in the opposite direction σ−1

• µ : [M −−→M ] −→ Ccont (G, M ),

given by µ0 = id (µ1 (m))(σa ) = (1 + σ + · · · + σ a−1 )m

(a ∈ N0 )

(with the convention that 1 + σ + · · · + σ a−1 = 0 for a = 0). This formula defines the values of µ1 (m) only at 1, σ, σ 2 , . . ., but µ1 (m) extends uniquely by continuity to a continuous 1-cochain (in fact a 1-cocycle). As λ ◦ µ = id, the maps λ, µ induce mutually inverse isomorphisms in the derived category. All of the above applies, in particular, to Gv /Iv = hf i and I v = hti. (7.2.3) Proposition. The formulas λ0 = id λ1 (c) = (c(f ), c(t)) λ2 (z) = −z(t, f ) + z(f, tL ) + f

L−2 X

ti z(t, tL−1−i )

i=0

λi = 0

(i > 2)

define functorial quasi-isomorphisms • λ : Ccont (Gv , M ) −→ C(M )

(M ∈ (ind−ad Mod)). R[G ] v

Proof. As in 7.2.2 we can assume that M is a discrete p-primary torsion Gv -module. Let us first explain the origin of the map λ. We begin with a morphism of complexes h i δ0 ⊕(2L+1) δ1 • λ0 : Ccont (Gv , M ) −→ M −−→M −−→M ⊕2L given by λ00 = id λ01 (c) = (c(f ), c(f t), . . . , c(f tL ), c(t), . . . , c(tL )) λ02 (z) = (z(f, t), z(f, t2 ), . . . , z(f, tL ), z(t, t), . . . , z(t, tL−1 ), z(t, f )). The differentials are uniquely determined by δ ◦ λ0 = λ0 ◦ δ: δ0 (m) = ((f − 1)m, (f t − 1)m, . . . , (f tL − 1)m, (t − 1)m, . . . , (tL − 1)m) δ1 (x0 , . . . , xL , y1 , . . . , yL ) = (f (y1 ) − x1 + x0 , . . . , f (yL ) − xL + x0 , t(y1 ) − y2 + y1 , . . . , t(yL−1 ) − yL + y1 , t(x0 ) − xL + y1 ). We want to construct λ as a composition λ = λ00 ◦ λ0 for a suitable morphism of complexes h i δ0 ⊕(2L+1) δ1 λ00 : M −−→M −−→M ⊕2L −→ C(M ). It is natural to require λ0 = id, λ1 (c) = (c(f ), c(t)), which implies that λ000 = id,

λ001 (x0 , . . . , xL , y1 , . . . , yL ) = (x0 , y1 ). 122

The condition λ00 ◦ δ = δ ◦ λ00 forces us to define λ002 (z1 , . . . , zL , u1 , . . . , uL−1 , w) = −w + zL + f

L−2 X

ti uL−1−i .

i=0

This leads to the formulas for λ and at the same time shows that λ is a morphism of complexes. ∼

Why is λ a quasi-isomorphism? First of all, (λ0 )∗ : M Gv −→ M t=1,f =1 is an isomorphism for trivial reasons. The Hochschild-Serre spectral sequence E2i,j = H i (Gv /I v , H j (I v , M )) =⇒ H i+j (Gv , M ) simplifies to Gv /I v

0 −→ H 1 (Gv /I v , M I v ) −→ H 1 (Gv , M ) −→ H 1 (I v , M )

−→ 0

and ∼

H 2 (Gv , M ) −→ H 1 (Gv /I v , H 1 (I v , M ))

(7.2.3.1)

(we have dropped the subscript “cont”, as we consider usual cohomology groups of discrete modules). The quasi-isomorphism t−1

• λt : Ccont (I v , M ) −→ [M −−→M ]

gives isomorphisms on cohomology (λt0 )∗ : M I v = M t=1 and ∼

M/(t − 1)M

7→

c(t) (mod (t − 1)M ).

(λt1 )∗ : H 1 (I v , M ) −→ [c]

Under (λt1 )∗ , the action of f on H 1 (I v , M ) corresponds to the action of θ (= the action of Lf ) on M/(t−1)M , since (f ∗ c)(t) = f (c(f −1 tf )) = f (c(tL )) = f (1 + t + · · · + tL−1 )c(t) = θ(c(t)).

(7.2.3.2)

Similarly, we have a quasi-isomorphism h i f −1 • λf : Ccont (Gv /I v , M I v ) −→ M I v −−→M I v . Taken together, (λf1 , λ1 , λt1 ) induce a map of exact sequences 0 −→ H 1 (Gv /I v , M I v )   f y(λ1 )∗ 0 −→

H 1 (C + (M ))

−→

H 1 (Gv , M ) −→  (λ ) y 1∗

−→

H 1 (C(M ))

−→

Gv /I v

H 1 (I v , M )   t y(λ1 )∗

H 1 (C − (M ))

−→

0

−→

0.

This shows that (λ1 )∗ is an isomorphism. In degree 2, let us recall an explicit description of the isomorphism (7.2.3.1): for every continuous 2-cocycle 2 z 0 ∈ Ccont (Gv , M ) there is a cohomologous 2-cocycle z = z 0 + δc vanishing on Gv × I v ; such a 2-cocycle is called normalized. For fixed g ∈ Gv the function 1 (zg : h 7→ z(h, g)) ∈ Ccont (I v , M )

depends only on the coset g = gI v ∈ Gv /I v and is a 1-cocycle. Furthermore, the function 1 (g 7→ [zg ]) ∈ Ccont (Gv /I v , H 1 (I v , M ))

123

is again a 1-cocycle and its class in H 1 (Gv /I v , H 1 (I v , M )) coresponds to [z] = [z 0 ] ∈ H 2 (Gv , M ) under (7.2.3.1). This recipe, the formulas for λt , λf and (7.2.3.2) give an isomorphism ∼

H 2 (Gv , M ) −→ (M/(t − 1)M )/(θ − 1)(M/(t − 1)M ) = M/(t − 1, 1 − θ)M [z]

7→

z(t, f ) (mod (t − 1, 1 − θ)M ),

which coincides with −(λ2 )∗ , since z is normalized. This finishes the proof that λ is a quasi-isomorphism. (7.2.4) Proposition. (i) For every discrete p-primary torsion Gv -module M , the formulas µ0 = id 0

a b

(µ1 (m, m ))(f t ) = (1 + f + · · · + f a−1 )m + f a (1 + t + · · · + tb−1 )m0 (µ2 (m))(f a tb , f c td ) = −f a (1 + t + · · · + tb−1 )(1 + θ + · · · + θc−1 )m (a, b, c, d ∈ N0 ) extend uniquely by continuity to a morphism of complexes • µ : C(M ) −→ Ccont (Gv , M ).

(ii) The morphism µ is a quasi-isomorphism and is functorial in M , hence defines a functorial quasiisomorphism • µ : C(M ) −→ Ccont (Gv , M ) (M ∈ (ind−ad Mod)) R[G ] v

satisfying λ ◦ µ = id. Proof. We leave it to the reader to check that the function µ1 (m, m0 ) (resp., µ2 (m)) defined in (i) on a dense subset of Gv (resp., Gv × Gv ) extends by continuity to all of Gv (resp., Gv × Gv ). In order to verify that µ commutes with differentials it is enough to check this on the above dense subsets, which in turn follows from an explicit calculation based on the following formulas: ta f b = f b taL

b

θb = f b (1 + t + · · · + t(L b

θ (t − 1) = (t − 1)f

b

−1)

)

b

(1 + t + · · · + ta−1 )θb = f b (1 + t + · · · + t(aL

b

−1)

)

(a, b ∈ N0 ).

It follows from the definitions that λ◦ µ = id; thus µ is a quasi-isomorphism. (See 7.4.8 for a more conceptual proof.) (7.2.5) Corollary. The map µ induces isomorphisms ∼

0 (µ0 )∗ : H 0 (C + (M )) −→ Hcont (Gv , M ) ∼

1 (µ1 )∗ : H 1 (C + (M )) −→ Hur (Gv , M )

for every M ∈ (ind−ad Mod). In other words, R[G ] v

µ

• C + (M ) −→ C(M )−→Ccont (Gv , M )

is an alternative way of defining unramified local conditions ∆ur v (M ). (7.2.6) Assume that G = G1 × · · · × Gr , where each Gi = hσi i is as in 7.2.2. Consider the Koszul complex K • = KΛ• (Λ, x) for the sequence x = (σ1 − 1, . . . , σr − 1) over Λ = Zp [[G]]. Then K • [r] is a Λ-free resolution of Zp , which implies ([Bru], Lemma 4.2(i)) that the complex •,naive •,naive HomZ (K • [r], M ) = HomZ (K • [r], M ) p [[G]],cont p [[G]]

124

(7.2.6.1)

• is quasi-isomorphic to Ccont (G, M ), for every discrete p-primary torsion G-module M . By the usual limit argument (and functoriality of (7.2.6.1)), the same property holds for all M ∈ (ind−ad R[G] Mod). For example, for r = 2, the complex (7.2.6.1) is equal to

(σ1 −1,σ2 −1)

(1−σ2 ,σ1 −1)

[M −−−−−−→M ⊕ M −−−−−−→M ] . ∼

(7.2.7) Continuous homology of G −→ Zrp . Self-duality of the Koszul complex (Br-He], 1.6.10) implies that the complex (7.2.6.1) is isomorphic to K • ⊗Λ M = KΛ• (M, x), hence ∼

H i (G, M ) −→ H i (KΛ• (M, x)) = Hr−i (G, M )

(7.2.7.1)

(cf. [Bru], 4.2(ii)). It is natural to generalize (7.2.7.1) and use it as a definition of continuous homology • Hj,cont (G, M ) := H r−j (KR[[G]] (M, x))

(7.2.7.2)

of any R[[G]]-module M (up to isomorphism, this is independent of the choice of the γi ’s). For example, ∼

Hr,cont (G, M ) −→ M G .

H0,cont (G, M ) = MG , If A is a discrete R[[G]]-module, then

∼

Hj,cont (G, D(A)) −→ D(H j (G, A)),

(7.2.7.3)

where D(−) = DR[[G]] (−). (7.3) Explicit resolutions (discrete case) One can reinterpret the map µ in terms of (a pro-finite version of) Fox’s free differential calculus, which we now briefly summarize (see [Bro], Ex. II.5.3, Ex. IV.2.3,4). (7.3.1) Let F = F (S) be the free group on a set S (= generators), and T (= relations) a subset of F . Let N C F be the smallest normal subgroup of F containing T ; put G = F/N . The augmentation ideal JF = Ker(Z[F ] −→ Z) is a free left Z[F ]-module with basis s − 1 (s ∈ S). One defines partial derivatives ∂ : F −→ Z[F ] ∂s

(s ∈ S)

by the formulas f −1=

X  ∂f  s∈S

∂s

(s − 1)

(f ∈ F ).

They satisfy the 1-cocycle identity ∂(f1 f2 ) ∂f2 ∂f1 = f1 + . ∂s ∂s ∂s Denote by a 7→ a the projections F −→ G and Z[F ] −→ Z[G]. The latter projection has kernel JN Z[F ] = Z[F ]JN (where JN is the augmentation ideal of Z[N ]). Tensoring the exact sequence of Z[F ]-modules M ∂1 0 −→ Z[F ]es −−→Z[F ] −→ Z −→ 0 s∈S

(in which ∂1 (es ) = s − 1) with Z[G] (or, equivalently, taking its homology H• (N, −)), we obtain an exact sequence of Z[G]-modules 125

i

0 −→ N ab −→

M

∂

1 Z[G]es −−→Z[G] −→ Z −→ 0,

(7.3.1.1)

s∈S

in which G acts on N ab by conjugation and M  ∂n 

i(n (mod [N, N ])) =

∂s

s∈S

es ,

∂1 (es ) = s − 1

(7.3.1.2)

∼

2 2 (we use the standard isomorphism JN /JN −→ N ab , n − 1 (mod JN ) ←→ n (mod [N, N ])).

(7.3.2) Define a surjective homomorphism of Z[G]-modules η:

M

Z[G]e0t −→ N ab

t∈T

by η(e0t ) = t (mod [N, N ]).

(7.3.2.1)

Then we have cd(G) ≤ 2 ⇐⇒ N ab is a projective Z[G] − module. When is η an isomorphism? A necessary condition is that cd(G) ≤ 2 and T being a minimal set of relations of G. It is unclear when is this condition also sufficient. A classical result of Lyndon [Ly] states that, for T = {t} consisting of one relation, we have η is an isomorphism ⇐⇒ cd(G) ≤ 2 ⇐⇒ t 6= un for any u ∈ F, n ≥ 2. For general T , if we assume that η is an isomorphism, then M

∂

2 Z[G]e0t −−→

t∈T

M

∂

1 Z[G]es −−→Z[G]

(∂2 = i ◦ η)

(7.3.2.2)

s∈S

is a Z[G]-free resolution of Z (the “Lyndon-Fox resolution”). Fix a section σ : G −→ F of the projection F −→ G. Then the formulas α0 : [ ] 7→ 1 X  ∂(σ(g))  α1 : [g] 7→ es ∂s

(7.3.2.3)

s∈S

−1

α2 : [g1 , g2 ] 7→ σ(g1 )σ(g2 )σ(g1 g2 ) αi = 0

(mod [N, N ])

(i > 2)

(extended by Z[G]-linearity) define a morphism of complexes L

g1 ,g2 ,g3 ∈G Z[G]

  y 0

· [g1 |g2 |g3 ] −→

−→

L g1 ,g2 ∈G

Z[G] · [g1 |g2 ] −→  α y 2

N ab

−→

L g1

L

Z[G] · [g1 ]  α y 1

s∈S

Z[G]es

−→

Z[G] · [ ]  α y 0

∂

Z[G]

1 −−→

from the bar resolution Z[G]⊗ • to the resolution (7.3.1.1). If η is an isomorphism, then (η −1 ◦ α2 , α1 , α0 ) give an explicit quasi-isomorphism from Z[G]•⊗ to the Lyndon,naive Fox resolution (7.3.2.2). Applying Hom•Z[G] (−, M ) we obtain functorial quasi-isomorphisms 126

" δ1

µ : M −−→

M

δ2

M −−→

s∈S

M

# M −→ C • (G, M )

t∈T

for all G-modules M . Here δ1 (m) = ((s − 1)m)s∈S δ2 ((ms )s∈S ) =

(m0t )t∈T ,

m0t

 =

 ∂t ms . ∂s

(7.3.2.4)

(7.4) Explicit resolutions (pro-finite case) Consider now a pro-finite version of the constructions in 7.3. (7.4.1) Let F = F (S) be the free group on a finite set S. Denote by Fb = ← lim − F/U , where U runs through all subgroups of finite index in F , the pro-finite completion of F . The pro-finite group algebra n Zp [[Fb]] = lim ←− Zp [F/U ] = lim ←− Z/p Z[F/U ] U

n,U

has augmentation ideal JFb = lim JF/U = lim JF/U ⊗ Z/pn Z, ← − ← − U n,U where JF/U = Ker(Zp [F/U ] −→ Zp ) is the augmentation ideal of Zp [F/U ]. (7.4.2) Lemma. JFb is a free left Zp [[Fb]]-module with basis s − 1 (s ∈ S). Proof. Applying H• (U, −) to the exact sequence 0 −→ JF ⊗ Z/pn Z −→ Z/pn Z[F ] −→ Z/pn Z −→ 0, we obtain an exact sequence 0 −→ U ab ⊗ Z/pn Z −→ (JF /JU JF ) ⊗ Z/pn Z −→ JF/U ⊗ Z/pn Z −→ 0. The projective system [U ab ⊗ Z/pn Z]U,n is ML-zero, hence the map n n lim ← −(JF /JU JF ) ⊗ Z/p Z −→ lim ← −(JF/U ⊗ Z/p Z) = JFb n,U n,U

is an isomorphism. As JF is a free Zp [F ]-module with basis s − 1 (s ∈ S), the same is true for (JF /JU JF ) ⊗ Z/pn Z as a module over Z/pn Z[F ]/JU Z/pn Z[F ] = Z/pn Z[F/U ]. The claim follows by taking the projective limit. (7.4.3) Corollary. The formula f −1=

X  ∂f  s∈S

∂s

(s − 1)

(f ∈ Fb )

defines maps (in fact, 1-cocycles) ∂ b : F −→ Zp [[Fb ]] ∂s 127

(s ∈ S).

(7.4.4) For a given finite subset T ⊂ Fb , let N C Fb be the smallest closed normal subgroup containing T ; put G = Fb /N . The kernel of the projection Zp [[Fb]] −→ Zp [[G]] is equal to JFbZp [[N ]] = Zp [[N ]]JFb. Taking the completed tensor product of the exact sequence of Zp [[Fb]]-modules 0 −→

M

∂1 b Zp [[Fb ]]es −−→Z p [[F ]] −→ Zp −→ 0

s∈S

(∂1 (es ) = s − 1) with Zp [[G]] gives an exact sequence of Zp [[G]]-modules ([Bru], 5.2.2) i

b p −→ 0 −→ N ab ⊗Z

M

∂

1 Zp [[G]]es −−→Z p [[G]] −→ Zp −→ 0,

(7.4.4.1)

s∈S

b p = lim N ab /pn N ab and the in which N ab = N/[N, N ]cl ([N, N ]cl denotes the closure of [N, N ]), N ab ⊗Z ←− maps i, ∂1 are given by the same formulas as in (7.3.1.2). (7.4.5) According to ([Bru], Lemma 4.2(i)), cohomology of discrete Zp [[G]]-modules (= discrete p-primary torsion G-modules) can be computed using arbitrary projective pseudo-compact Zp [[G]]-resolutions of Zp . One such resolution is given by the pro-finite bar resolution b b b Zp [[G]]⊗ • : · · · −→ Zp [[G]]⊗ · · · ⊗Zp [[G]] −→ · · · −→ Zp [[G]]. Fix a continuous section σ : G −→ Fb of the projection Fb −→ G. The formulas (7.3.2.3) define a morphism of resolutions of Zp " # M ∂1 i ⊗ ab b b α : Z [[G]] −→ N ⊗Z −→ Z [[G]]e −−→Z [[G]] . p

p

•

p

s

p

s∈S

The formula (7.3.2.1) defines a surjective homomorphism η:

M

b p, Zp [[G]]e0t −→ N ab ⊗Z

t∈T

hence a morphism from the complex M

∂2 =i◦η

Zp [[G]]e0t −−−−→

t∈T

M

∂

1 Zp [[G]]es −−→Z p [[G]]

(7.4.5.1)

s∈S

to the resolution (7.4.4.1). As in the discrete case we have ([Bru], Thm. 5.2) b p is a projective pseudo − compact Zp [[G]] − module cdp (G) ≤ 2 ⇐⇒ N ab ⊗Z Under what conditions is η an isomorphism? If G is a pro-p-group, then it is shown in ([Bru], Cor. 5.3) that η is an isomorphism ⇐⇒ cdp (G) ≤ 2 and T is a minimal set of relations of G. Whenever η is an isomorphism we obtain (as in 7.3.2) functorial quasi-isomorphisms " # M M δ1 δ2 • µ : M −−→ M −−→ M −→ Ccont (G, M ) s∈S

t∈T

for all discrete p-primary torsion G-modules M (here δi are given by (7.3.2.4)). As in 7.2.2, the maps µ define corresponding quasi-isomorphisms for all M ∈ (ind−ad R[G] Mod). (7.4.6) Let us apply the previous discussion to G = Gv with generators f, t, i.e. take F = F (S) for S = {α, β}, T = {r = αβ L α−1 β −1 } (L ∈ N, p - L). Then Fb −→ G sends α to α = f and β to β = t. 128

b b ∈ Q Zq ⊂ Z. b We define a Every element g ∈ G can be expressed uniquely as g = f a tb with a ∈ Z, q6=l a b a b continuous section σ : G −→ Fb by σ(f t ) = α β . A short calculation shows that ∂r = 1 − αβ L α−1 , ∂α

∂r = α(1 + β + · · · + β L−1 ) − αβ L α−1 β −1 ; ∂β

thus 

∂r ∂α



 = 1 − t,

∂r ∂β

 = θ − 1.

This implies that the complex (7.4.5.1) is given by ∂2 (e0r ) = (1 − t)eα + (θ − 1)eβ ∂1 (eα ) = f − 1, ∂1 (eβ ) = t − 1, hence the complex " δ1

M −−→

M

δ2

M −−→

s∈S

M

# M

t∈T

coincides with (f −1,t−1)

(1−t,θ−1)

C(M ) = [M −−−−−−→M ⊕ M −−−−−−→M ] . b p (7.4.7) Lemma. For the presentation G = Gv = Fb/N considered in 7.4.6 the map η : Zp [[G]] −→ N ab ⊗Z is an isomorphism. b p is a projective pseudo-compact Zp [[G]]-module; thus η has a section and Proof. As cdp (G) = 2, N ab ⊗Z ab b Zp [[G]] = (N ⊗Zp ) ⊕ X. We want to show that X = 0. Let M be an arbitrary discrete p-power torsion G-module. The morphisms of complexes of projective pseudo-compact Zp [[G]]-modules b Zp [[G]]⊗ •

α

−→

b p [N ab ⊗Z x η 

L1 =

L2 = [Zp [[G]]e0r

i

−→ ∂

2 −−→

L s∈S

L s∈S

∂

Zp [[G]]]

∂

Zp [[G]]]

Zp [[G]]es

1 −−→

Zp [[G]]es

1 −−→

induce morphisms of complexes η∗ α∗ •,naive •,naive •,naive • b Ccont (G, M ) = HomZ (Zp [[G]]⊗ • , M )←−−HomZp [[G]],cont (L1 , M )−−→HomZp [[G]],cont (L2 , M ) p [[G]],cont

such that η ∗ = λ ◦ α∗ . As both λ and α∗ are quasi-isomorphisms (by Proposition 7.2.3 and cdp (G) = 2, respectively), so is η ∗ . This means that HomZp [[G]],cont(X, M ) = 0 for all M . Writing X = lim ←− Xu with Xu discrete p-power torsion and taking M = Xu , we get idX ∈ ← lim Hom (X, X ) = 0, hence X = 0. u Z [[G]],cont p − u

(7.4.8) As a corollary of Lemma 7.4.7 we obtain a morphism η −1 α = (η −1 ◦ α2 , α1 , α0 ) from the pro-finite bar resolution of Gv to the Lyndon-Fox resolution (7.4.5.1) and functorial quasi-isomorphisms •,naive • µ = HomZ (η −1 α, id) : C(M ) −→ Ccont (Gv , M ). [[G ]] p

v

Let us verify that µ is indeed given by the fromulas from Proposition 7.2.4. If g1 = f a tb , g2 = f c td with a, b, c, d ∈ N0 , then g1 g2 = f a+c t(bL

c

+d)

,

c

n := σ(g1 )σ(g2 )σ(g1 g2 )−1 = αa β b αc β −bL α−a−c .

It follows immediately from the definitions that 129



   ∂(σ(g1 )) ∂(σ(g2 )) eα + eβ = (1 + f + · · · f a−1 )eα + f a (1 + t + · · · tb−1 )eβ , ∂α ∂β

which gives the formula for µ1 . A slightly tedious calculation shows that 

   ∂n ∂n eα + eβ = f a (1 + t + · · · tb−1 )(1 + θ + · · · θc−1 )((t − 1)eα + (1 − θ)eβ ), ∂α ∂β

verifying the formula for µ2 . In fact, this gives an alternative proof of Proposition 7.2.4. (7.4.9) Lemma. For every ind-admissible R[Gv ]-module M there exists a homotopy bv : µλ − id on • Ccont (Gv , M ), which is functorial in M and for which there is a 2-homotopy bv ? µ − 0, again functorial in M. Proof. It is enough to consider the case when M is a discrete p-primary torsion Gv -module. We have µ = Hom•Z,naive (η −1 α, id), [[G ]] p

λ = Hom•Z,naive (γ, id), [[G ]]

v

p

v

where " ∂2

γ : Zp [[Gv ]]e0r −−→

M

#

b Zp [[Gv ]]es −−→Zp [[Gv ]] −→ Zp [[Gv ]]•⊗ ∂1

s∈S

is given by γ0 (1) = [ ],

γ1 (ef ) = [f ],

γ2 (e0r ) = −[t, f ] + [f, tL ] +

L−2 X

γ1 (et ) = [t]

f ti [t, tL−1−i ],

γi = 0

(i > 2).

i=0

b −→ Z [[G ]]⊗ b The map γ ◦η −1 α : Zp [[Gv ]]•⊗ p v • is a morphism of projective pseudo-compact Zp [[Gv ]]-resolutions lifting the identity on Zp , which means that there is a homotopy cv : γ ◦ η −1 α − id. For any such cv , the homotopy bv = Hom•,naive (cv , id) : µλ −

id

has the desired properties. As η −1 α ◦ γ = id, both 0 and η −1 α ? cv are homotopies η −1 α − η −1 α. Again, projectivity of the pro-finite bar resolution implies that there exists a 2-homotopy H : η −1 α ? cv − 0, which in turn induces the desired 2-homotopy bv ? µ − 0. (7.5) Duality (7.5.1) Cup products. For M, N ∈ (ind−ad Mod) we define R[G ] v

∪ : C(M ) ⊗R C(N ) −→ C(M ⊗R N ) to be the composite morphism of complexes µ⊗µ

∪

λ

• • • C(M ) ⊗R C(N )−−−−→Ccont (Gv , M ) ⊗R Ccont (Gv , N )−→Ccont (Gv , M ⊗R N )−→C(M ⊗R N ).

Explicitly, the components of ∪ ∪ab : C a (M ) ⊗R C b (N ) −→ C a+b (M ⊗R N ) are equal to 130

m ∪00 n = m ⊗ n m ∪01 (n, n0 ) = (m ⊗ n, m ⊗ n0 )

(m, m0 ) ∪10 n = (m ⊗ f (n), m ⊗ t(n)) m ∪02 n = m ⊗ n m ∪20 n = m ⊗ f t(n) (m, m0 ) ∪11 (n, n0 ) = −m0 ⊗ t(n) + m ⊗ θ(n0 ) +

X

f ti (m0 ) ⊗ f tj (n0 ).

0≤i 1

)

( −→ H

j

(Errur v (Φ, T ))

−→

j+d−1 H{m} (H)fv =1 ,

if j ≤ 0

0,

if j > 0

) −→ 0,

1 where we have abbreviated H := Hcont (Iv , T ). (v) The following statements are equivalent: ∼

Errur v (Φ, T ) −→ 0 (∀q = 0, . . . , d − 1)

b in Dcof t (R Mod) ⇐⇒

fv −1

q q 1 1 αq : H{m} (Hcont (Iv , T ))−−→H{m} (Hcont (Iv , T )) is an isomorphism ⇐⇒

(∀q = 0, . . . , d − 1)

Ker(αq ) = 0.

w

Proof. As before, we can assume that T = T Iv . The statement (i) follows from (3.5.4.2) and Lemma 4.2.6, as cdp (Iv ) = 1. As regards (ii), let C • be as in the proof of Proposition 7.6.7. The assumption T = σ≤0 T implies that L(T ) = σ≤1 L(T ) and Φ(T ) = T ⊗R C • = σ≤0 Φ(T ). Tensoring the exact triangle in Dfb t (R Mod) τ≤0 L(T ) −→ L(T ) −→ H[−1] −→ (τ≤0 L(T )) [1] 1

1 (in which H = H (L(T )) = Hcont (Iv , T )) with the complex C • (of flat R-modules) yields an exact triangle b in Dcof t (R Mod)

140

(τ≤0 L(T )) ⊗R C • −→ L(T ) ⊗R C • −→ H ⊗R C • [−1] −→ (τ≤0 L(T )) ⊗R C • [1]. b As C • = σ≤0 C • , applying the truncation τ≤0 we obtain another exact triangle in Dcof t (R Mod)

(τ≤0 L(T )) ⊗R C • −→ τ≤0 (L(T ) ⊗R C • ) −→ (τ≤−1 (H ⊗R C • )) [−1] −→ (τ≤0 L(T )) ⊗R C • [1], which is equal to Φ (τ≤0 L(T )) −→ τ≤0 (L(Φ(T ))) −→ (τ≤−1 Φ(H)) [−1] −→ Φ (τ≤0 L(T )) [1]; we conclude as in the proof of Proposition 7.6.7(i). 1 (iii) If H = Hcont (Iv , T ) is zero or a Cohen-Macaulay R-module of dimension d = dim(R), then the complex
∼ τ≤−1 Φ(H) −→ τ≤d−1 RΓ{m} (H) [d] is acyclic, by Lemma 2.4.7(ii). The statement (iv) is a consequence of (ii), and the first equivalence in (v) follows from (iv). The non-trivial implication ‘⇐=’ in (v) follows from Lemma 2.3.6. (7.6.12) Corollary. Under the assumptions of Proposition 7.6.11, (i) There is an isomorphism in Dfb t (R Mod) (using the notation of (7.6.5.1))
fv−1 −1 ∼ 1 1 D(Errur v (D, T )) −→ Cone τ≥1 D(Hcont (Iv , T )))−−−−→τ≥1 D(Hcont (Iv , T ))) [1]. (ii) The cohomology groups of W := D(Errur v (D, T )) sit in the following exact sequences: ( 0 −→

Extj+1 R (H, ωR )/(fv − 1), if j ≥ 0 0,

)

if j < 0

( j

−→ H (W ) −→

fv =1 Extj+2 , if j ≥ −1 R (H, ωR )

0,

if j < −1

) −→ 0.

(iii) The following statements are equivalent: ∼

Errur v (D, T ) −→ 0

in Dfb t (R Mod) ⇐⇒

fv−1 −1

1 1 (∀q = 1, . . . , d) α0q : ExtqR (Hcont (Iv , T ), ωR )−−−−→ExtqR (Hcont (Iv , T ), ωR )

(∀q = 1, . . . , d)

Coker(α0q )

is an isomorphism ⇐⇒

= 0.

∼

Proof. As W −→ D(Errur v (Φ, T )) by the proof of Proposition 7.6.7(ii), it is enough to apply D to the statements of Proposition 7.6.11 and use local duality 2.5. (7.6.13) It follows immediately from the definitions that the functor w

M • 7→ U + ((M • )Iv ) respects quasi-isomorphisms and homotopies. As a result, it defines a functor b RΓur (Gv , −) : D∗ (ind−ad R[Gv ] Mod) −→ D (R Mod),

∗ = +, b.

(7.7) Transpositions Fix v ∈ Sf , v - p. We are going to construct transposition operators satisfying 6.5.3.1-5 for the local conditions ∆ur v . 141

(7.7.1) Lemma. (i) For every complex of ind-admissible R[Gv ]-modules M • , the map µ

T

λ

• • C(M • )−→Ccont (Gv , M • )−→Ccont (Gv , M • )−→C(M • )

is equal to λ ◦ µ = id. (ii) For every M ∈ (ind−ad Mod) the map R[G ] v

µ+

T

• • C + (M )−→Ccont (Gv , M )−→Ccont (Gv , M )

µ

ν

• is equal to µ+ : C + (M )−→C(M )−→Ccont (Gv , M ).

Proof. (i) It is sufficient to consider the case when M • = M ∈ (ind−ad Mod). Then the statement is trivial R[G ] v

in degree 0. In degree 1, we have to check that −g(µ1 (m, m0 )(g −1 )) = m (resp., = m0 ) for g = f (resp., g = t), which follows from the fact that D(σ(g −1 )) = D(σ(g)−1 ) = −σ(g)−1 = −g −1 for (g, D) = (f, ∂/∂α) (resp., (g, D) = (t, ∂/∂β)). In degree 2 we use the fact that µ2 (m) vanishes on Gv × I v ; thus
λ2 ◦ T ◦ µ2 (m) = λ2 (g, g 0 ) 7→ −gg 0 (µ2 (m)(g2−1 , g1−1 )) = −f tL (µ2 (m)(t−L , f −1 )). As n := σ(t−L )σ(f −1 )σ(t−L f −1 )−1 = β −L α−1 βα satisfies 

   ∂n ∂n eα + eβ = −f −1 t−1 [(1 − t)eα + (θ − 1)eβ ], ∂α ∂β

we have µ2 (m)(t−L , f −1 ) = −f −1 t−1 (m),

λ2 ◦ T ◦ µ2 (m) = −f tL (−f −1 t−1 (m)) = m.

(ii) We only have to check what happens in degree 1. For m ∈ M t=1 the continuous 1-cochain µ1 (m, 0) satisfies µ1 (m, 0)(f a tb ) = (1 + f + · · · + f a−1 )m for a, b ∈ N0 . This implies that µ1 (m, 0) is a 1-cocycle, hence −g(µ1 (m, 0)(g −1 )) = µ1 (m, 0)(g). (7.7.2) Assume that J = J • is as in 7.6.5, i.e. satisfies J • = σ≥0 J • . Let X, Y be bounded complexes of admissible R[Gv ]-modules and π : X ⊗R Y −→ J(1) a morphism of complexes of R[Gv ]-modules. The ur following data 7.7.2.1-5 define transposition operators for the local conditions ∆ur v (X), ∆v (Y ), satisfying 6.5.3.1-5. (7.7.2.1) Put hv = h0v = 0; then 6.5.3.1 holds by 7.6.5. (7.7.2.2) Define Tv+ (Z) = id (Z = X, Y ). (7.7.2.3) By Lemma 7.7.1(i), λT µ = λµ = id; thus the formula VZ,v = inf ? bv ? (T ◦ i+ v (Z)) (in which bv : µλ −

(Z = X, Y )

id is as in 7.4.9) defines a homotopy

VZ,v : i+ v (Z) = inf ◦ µν = inf ◦ µλ ◦ T µν − 142

inf ◦ T µν = T ◦ inf ◦ µν = T ◦ i+ v (Z).

(7.7.2.4) For each Z = X, Y , let kZ and kZ,v be the functorial homotopies id − T induced by a homotopy a from 3.4.5.5; then resv ? kZ = kZ,v ? resv by functoriality. + We define kZ,v = 0. We must show that there is a second order homotopy ?

VZ,v −

kZ,v ? i+ v (Z).

As

kZ,v ? i+ v (Z) = inf ? kZ t ,v ? µν

VZ,v = inf ? bv ? T µν,

• (where kZ t ,v : id − T on Ccont (Gv , Z t ) is also induced by a), it is sufficient to show that there is a second order homotopy ? bv ? T µ − kZ,v ? µ

for tame Z = Z t . By construction, the L.H.S. (resp., the R.H.S.) is equal to Hom•,naive (η −1 α ◦ T ? cv , idZ ) : µ = µλT µ − resp.,

Hom•,naive (η −1 α ? a, idZ ) : µ −

Tµ

T µ,

in the notation of 7.4.9 (resp., 3.4.5.5). It is enough to observe that the homotopies η −1 α ◦ T ? cv , η −1 α ? a : η −1 α −

η −1 α ◦ T

between the morphisms of pseudo-compact Zp [[Gv ]]-resolutions of Zp η

−1

α, η

−1

"

⊗ b

α ◦ T : Zp [[Gv ]]• −→

∂2 Zp [[Gv ]]e0r −−→

M

# ∂1

Zp [[Gv ]]es −−→Zp [[Gv ]]

s∈S

are 2-homotopic, as both morphisms η −1 α, η −1 α ◦ T lift the identity on Zp and the resolutions are projective. (7.7.2.5) We have hv = h0v = 0 by definition. The homotopy •

h := ∪π ? (VX,v ⊗ VY,v )1 : 0 − is between the zero maps

0

0

• II Uv+ (X) ⊗R Uv+ (Y )−→τ≥2 Ccont (Gv , J(1)).

However, the domain (resp., the target) of 0 is concentrated in degrees ≤ 2 (resp., ≥ 2, since J = σ≥0 J), hence h = 0. This means that we can take Hv = 0. We can summarize the previous discussion in the following statement. (7.7.3) Proposition. Let J = σ≥0 J be a bounded complex of injective R-modules, X and Y bounded complexes of admissible R[Gv ]-modules and π : X ⊗R Y −→ J(1) a morphism of complexes. Then the ur unramified local conditions ∆ur v (X), ∆v (Y ) admit transposition operators satisfying 6.5.3.1-5 (with hv = 0 hv = 0). (7.8) Greenberg’s local conditions In this section we develop the theory of Greenberg’s local conditions. These seem to be the only local conditions that can be handled by ‘elementary’ methods. (7.8.1) Fix a subset Σ ⊂ Sf containing all primes above p and put Σ0 = Sf − Σ. We are going to combine the local conditions of the type considered in 6.7 (for v ∈ Σ) with those from 7.6.5 (for v ∈ Σ0 ). The corresponding Selmer complexes are then analogues of Greenberg’s Selmer groups [Gre 2-4]. 143

We consider J = J • of the form J = I or J = ω • = σ≥0 J (cf. 2.5(ii)). Let rv,J (v ∈ Sf ) be as in 5.2.2. For each v ∈ Σ0 fix fv and tv as in 7.2.1. (7.8.2) Let π : X ⊗R Y −→ J(1) be as in 6.2.1; we assume that π is a perfect duality in the sense of 6.2.6 (in particular, the complexes X, Y are bounded). Assume that we are given, for each v ∈ Σ, jv+ (Z) : Zv+ −→ Z

(Z = X, Y )

as in 6.7.1, which satisfy 6.7.5(A) or (B) (in particular, the complexes Xv+ , Yv+ are bounded) and such that Xv+ ⊥⊥π Yv+ for all v ∈ Σ. As we have fixed fv , tv for all v ∈ Σ0 , we can define the following local conditions for Z = X, Y : ( • • Ccont (Gv , Zv+ ) −→ Ccont (Gv , Z), (v ∈ Σ) ∆v (Z) = • • Cur (Gv , Z) −→ Ccont (Gv , Z), (v ∈ Σ0 ) gf (X), RΓ gf (Y ) ∈ Db (R Mod). Our assumptions imply that RΓ (7.8.3) By Proposition 6.7.6, (∀v ∈ Σ)

∆v (X) ⊥⊥π,0 ∆v (Y ),

∆v (Y ) ⊥⊥π◦s12 ,0 ∆v (X).

By 7.6.5 and Proposition 7.6.6, ( ∆v (X) ⊥⊥π,0 ∆v (Y ), ∆v (Y ) ⊥⊥π◦s12 ,0 ∆v (X), 0 (∀v ∈ Σ ) ∆v (X) ⊥π,0 ∆v (Y ), ∆v (Y ) ⊥π◦s12 ,0 ∆v (X),

if J = I if J = ω • .

According to 6.7.8 and Proposition 7.7.3, the local conditions ∆v admit transpositions satisfying (6.5.3.1-5). (7.8.4) The general machinery of 6.3 and 6.5 then defines, for each r ∈ R, cup products ef• (X) ⊗R C ef• (Y ) −→ J[−3] ∪π,r,0 : C ef• (Y ) ⊗R C ef• (X) −→ J[−3] ∪π◦s12 ,r,0 : C such (7.8.4.1) (7.8.4.2) (7.8.4.3)

that The homotopy class of ∪π,r,0 (resp., ∪π◦s12 ,r,0 ) does not depend on r ∈ R. ∪π◦s12 ,r,0 ◦ s12 is homotopic to ∪π,1−r,0 , hence to ∪π,r 0 ,0 , for all r, r0 ∈ R. If J = I, then the adjoint morphism in Db (R Mod)   gf (X) −→ DJ[−3] RΓ gf (Y ) γπ,0 = adj(∪π,r,0 ) : RΓ

is an isomorphism. (7.8.4.4) If J = ω • , then there is an exact triangle in Db (R Mod)   M γX ur gf (X)−−→D g RΓ Errv (∆ur J[−3] RΓf (Y ) −→ v (X), ∆v (Y ), π). v∈Σ0

More generally, if one assumes only that Xv+ ⊥π Yv+ (v ∈ Σ), then one has to add to the third term of the triangle the sum M Errv (∆v (X), ∆v (Y ), π), v∈Σ

given by the formulas from Proposition 6.7.6(iv). ur 0 (7.8.4.5) In Db ((R Mod)/(pseudo − null)), the error terms Errv (∆ur v (X), ∆v (Y ), π) (v ∈ Σ ) are given by the formulas in Proposition 7.6.7(ii) and Corollary 7.6.8 (cf. 7.6.10.6-9). In particular, they vanish after localizing at each prime ideal q ∈ Spec(R) with ht(q) = 0. ur 0 b (7.8.4.6) If X = σ≤0 X, then the error terms Errv (∆ur v (X), ∆v (Y ), π) (v ∈ Σ ) in Df t (R Mod) are given by the formulas in Corollary 7.6.12. 144

(7.8.5) In practice, it is often the case that the canonical maps τ≤0 X −→ X τ≤0 Xv+ −→ Xv+

(v ∈ Σ)

are all quasi-isomorphisms. If true, then it follows from Lemma 4.2.6 that the maps • • τ≤2 Ccont (GK,S , X) −→ Ccont (GK,S , X)

τ≤2 Uv+ (X) −→ Uv+ (X)

(v ∈ Sf ),

and hence e • (GK,S , X; ∆(X)) −→ C e• (GK,S , X; ∆(X)), τ≤3 C f f are also quasi-isomorphisms. gf (X) is (7.8.6) Theorem (Euler-Poincar´ e characteristic). The Euler-Poincar´e characteristic of RΓ equal to X

  XX X
X
e q (X) = (−1)q eR H (−1)q eR (X q )Gv − [Kv : Qp ] (−1)q eR (Xv+ )q . f

q

v|∞

q

q

v|p

Proof. The middle exact triangle in 6.1.3 implies that the L.H.S. is equal to the sum of (1) =

X


q (−1)q eR Hc,cont (GK,S , X)

q

and X

(2)v =

v∈Sf

XX v∈Sf


(−1)q eR H q (Uv+ (X)) .

q

However, (1) =

XX v|∞

(−1)q eR (X q )Gv




q

by Theorem 5.3.6, while (2)v = 0 for v ∈ Σ0 . Finally, for v ∈ Σ, we have (2)v =

X X

q (−1)q eR Hcont (Gv , Xv+ ) = cv · (−1)q eR (Xv+ )q q

q

with ( cv =

−[Kv : Qp ],

v|p

0,

v - p,

by 5.2.11. (7.8.7) Corollary. Assume that:
P −1 q (i) d+ = [K : R] (−1) eR (X q )Gv does not depend on v|∞. v ∞ q P q + q (ii) d+ p = q (−1) eR ((Xv ) ) does not depend on v|p. Then   X e q (X) = [K : Q](d+ − d+ ). (−1)q eR H ∞ p f q

145

(7.8.8) Proposition (Change of S). Let X be an ind-admissible R[GK,S ]-module and S 0 ⊃ S a finite set of primes of K. Then: (i) The canonical morphisms of complexes • • inf : Ccont (GK,S , X) −→ Ccont (GK,S 0 , X),

M

• res : Ccont (GK,S , X) −→

• Ccont (Gv /Iv , X)

v∈S 0 −S

(the second depending on the choice of embeddings K ,→ K v for all v ∈ S 0 − S) give rise to an exact triangle (inf,res)

RΓcont (GK,S , X)−−−−→RΓcont (GK,S 0 , X) ⊕

M

(0,inf )

RΓur (Gv , X)−−−−→

v∈S 0 −S

M

RΓcont (Gv , X).

v∈S 0 −S

• (ii) Assume that we are given local conditions ∆v (X) for all v ∈ Sf . Defining Uv+ (X) = Cur (Gv , X) for all 0 v ∈ S − S and keeping the given local conditions for v ∈ Sf , there is a canonical isomorphism

∼

RΓ(GK,S , X; (∆v (X))v∈Sf ) −→ RΓ(GK,S 0 , X; (∆v (X))v∈Sf0 ). Proof. (i) By a standard limit argument we can assume that X = M is a p-primary torsion discrete GK,S module. As recalled in 9.2.1 below, M defines an ´etale sheaf Met on Spec(OK,S ) and RΓcont (GK,S , M ) is canonically isomorphic to RΓ(Spec(OK,S ), Met ). The statement of (i) then follows from the excision triangles for ´etale cohomology (cf. [Mi], p. 214): M

RΓ{v} (Ovh , Met ) −→ RΓ(Spec(OK,S ), Met ) −→ RΓ(Spec(OK,S 0 ), Met ),

v∈S 0 −S

RΓ{v} (Ovh , Met ) −→ RΓ(Ovh , Met ) −→ RΓ(Spec(Kv ), Met )

(v ∈ S 0 − S)

(where Ovh denotes the henselianization of OK,S at v), if we take into account canonical isomorphisms ∼

RΓ(Ovh , Met ) −→ RΓcont (Gv /Iv , M ),

∼

RΓ(Spec(Kv ), Met ) −→ RΓcont (Gv , M ).

The statement (ii) is a straightforward consequence of (i). (7.8.9) Corollary - Definition. Under the assumptions of Proposition 7.8.8(ii), the cohomology groups e i (GK,S 0 , X; (∆v (X))v∈S 0 ) do not depend – up to a canonical isomorphism – on the choice of S 0 . We shall H f f e i (K, X) = H e i (K, X; ∆(X)). denote them by H f

f

(7.8.10) Localization. The above discussion works whenever R is replaced by RS , under the assumption 6.7.5(B), localized at S . For example, if S = R − p for p ∈ Spec(R) with ht(p) = 1, then everything in 7.6.10 holds if T is a bounded complex of admissible Rp [Gv ]-modules with cohomology of finite type over Rp . Another example is provided by the following Proposition. (7.8.11) Proposition (Euler-Poincar´ e characteristic: self-dual case). Assume that, in the situation of 7.8.6, R is an integral domain with fraction field K = Frac(R) of characteristic char(K) = 0, all complexes X = H 0 (X) and Xv+ = H 0 (Xv+ ) (v ∈ Σ) are concentrated in degree zero and all morphisms Xv+ −→ X (v ∈ Σ) are injective. Put V = X ⊗R K, Vv± = Xv± ⊗R K (where Xv− = X/Xv+, v ∈ Σ). Assume, in addition, that V is a simple K[GK,S ]-module and that there exists a non-degenerate skewsymmetric GK,S -equivariant bilinear form V ⊗K V −→ K(1), which induces isomorphisms of K[Gv ]-modules ∼

Vv± −→ HomK (Vv∓ , K)(1) for all v ∈ Σ. Then, for each homomorphism χ : GK,S −→ R∗ , e 1 (X ⊗ χ) = rkR H e 2 (X ⊗ χ) = rkR H e 1 (X ⊗ χ−1 ) = rkR H e 2 (X ⊗ χ−1 ) rkR H f f f f 146

(with respect to Greenberg’s local conditions given by Xv+ ⊗ χ resp., Xv+ ⊗ χ−1 ). Proof. For α ∈ {χ, χ−1 }, put e q (V ⊗ α) = rkR H e q (X ⊗ α). hqα = dimK H f f Self-duality of V and of the local conditions Vv+ (v ∈ Σ) imply that, by the localized duality theorem (cf. 7.8.4.4-5),   ∼ e q (V ⊗ χ) −→ e 3−q (V ⊗ χ−1 ), K H HomK H =⇒ hqχ = h3−q (7.8.11.1). f f χ−1 . For each v ∈ Σ, Vv+ is a Lagrangian (= maximal isotropic) subspace of V ; it follows that (∀v|p)

dimK (Vv+ ⊗ α) = dimK (V )/2 = rkR (X)/2.

∼

Self-duality V −→ V ∗ (1) implies that, for each real embedding K ,→ R, the corresponding complex conjugation acts on V by a matrix with eigenvalues +1, −1, each with the same multiplicity dimK (V )/2 = rkR (X)/2; thus (∀v|∞)

[Kv : R]−1 dimK H 0 (Gv , V ⊗ α) = dimK (V )/2 = rkR (X)/2.

Applying Corollary 7.8.7, we obtain (using 7.8.5) 3 X

(−1)q hqα = 0

(α = χ, χ−1 )

(7.8.11.2).

q=0

However, h0χ = h0χ−1 (= h3χ ) = 0, because V is an irreducible representation. Combining (7.8.11.1-3), we obtain h1χ = h2χ = h1χ−1 = h2χ−1 , as required.

147

(7.8.11.3)

8. Iwasawa theory ∼

In this chapter we study continuous cohomology in towers K∞ /K, where Gal(K∞ /K) = Γ −→ Zrp × ∆, for a finite abelian group ∆. Our main tool is Shapiro’s Lemma, which allows us to reduce to statements about cohomology over K, but replaces R by the bigger coefficient ring R[[Γ]]. Once we establish a correspondence between the duality diagrams over K and over K∞ (Sect. 8.4) and the compatibility of the Greenberg local conditions with Shapiro’s Lemma (Sect. 8.5-8.8), we can apply the duality formalism over K to the induced modules and obtain – after an analysis of the local Tamagawa factors – Iwasawa-theoretical duality results (Sect. 8.9). In Sect. 8.10 we study an abstract version of “Mazur’s Control Theorem”; in our context it is a consequence of a fundamental base-change property of Selmer complexes (Proposition 8.10.1). Throughout Chapter 8 we assume that the residue field of R is a finite field of characteristic p. (8.1) Shapiro’s Lemma Let us recall basic facts about Shapiro’s Lemma ([Ve 1], 1.1-3; [Bro], III.6.2, Ex. III.8.2). (8.1.1) Let U ⊂ G be an open subgroup of a pro-finite group G. For every discrete U -module X, the induced module IndG U (X) = {f : G −→ X | f locally constant, f (ug) = uf (g) ∀u ∈ U, ∀g ∈ G} is a discrete G-module with respect to the (left) action (g ∗ f )(g 0 ) = f (g 0 g). The inclusion U ,→ G and the map δU : IndG U (X) −→ X f 7→ f (1) define a morphism of pairs (G, IndG U (X)) −→ (U, X) in the sense of 3.4.1.6, hence a morphism of complexes (functorial in X) • • sh : Ccont (G, IndG U (X)) −→ Ccont (U, X).

Shapiro’s Lemma asserts that sh is a quasi-isomorphism. If X, Y are discrete U -modules, then there is a commutative diagram ∪

G • • Ccont (G, IndG U (X)) ⊗Z Ccont (G, IndU (Y )) −→   ysh⊗sh ∪

• • Ccont (U, X) ⊗Z Ccont (U, Y )

−→

• Ccont (G, IndG U (X ⊗Z Y ))   ysh • Ccont (U, X ⊗Z Y ),

in which the upper horizontal arrow is induced by the map G IndG U (X) ⊗Z IndU (Y ) −→

f1 ⊗ f2

7→

IndG U (X ⊗Z Y ) (g 7→ f1 (g) ⊗ f2 (g)).

(8.1.2) Restriction. If V ⊂ U is another open subgroup of G, then there is an inclusion IndG U (X) ⊂ IndG V (X) making the following diagram of morphisms of complexes commutative: • Ccont (G, IndG U (X))  incl y ∗

sh

−→ sh

• Ccont (G, IndG V (X)) −→

148

• Ccont (U, X)  res y • Ccont (V, X).

(8.1.3) If X is a discrete G-module, then there are two other natural discrete G-modules isomorphic to G IndG U (X) (more precisely, to IndU (X0 ), where X0 is equal to X as a set, but viewed as an U -module), namely X XU = Z[G/U ] ⊗Z X = { β ⊗ xβ | β ∈ G/U, xβ ∈ X} UX

∼

= HomZ (Z[G/U ], X) −→ {a : G/U −→ X}

with G-action (ga)(β) = g(a(g −1 β)).

g(β ⊗ x) = gβ ⊗ gx, Denote by δβ : G/U −→ Z Kronecker’s delta-function ( 1 0 δβ (β ) = 0

β = β0 β 6= β 0 .

Then the following formulas define G-equivariant isomorphisms, functorial in X: ∼

IndG U (X) −→ U X,

X

∼

XU −→ U X, ∼

IndG U (X) −→ XU ,

f 7→ (gU 7→ g(f (g −1 ))) X β ⊗ xβ 7→ xβ δβ X f 7→ gU ⊗ g(f (g −1 )). gU ∈G/U

We use the above isomorphisms to pass freely between IndG The map δU from 8.1.1 will U (X), XU and U X. P then be, indeed, identified with Kronecker’s delta-function δU : XU −→ X, δU ( β ⊗ xβ ) = xU . (8.1.4) Corestriction. S Fix a section U \G, α 7→ α of the canonical projection G −→ U \G (i.e. a set of coset representatives G = U αi ). For every discrete G-module X, the formula X (cor(c))(g1 , . . . , gn ) = α−1 c(αg1 (αg1 )−1 , . . . , αg1 . . . gn−1 gn (αg1 . . . gn )−1 ) α∈U \G

defines a morphism of complexes • • cor : Ccont (U, X) −→ Ccont (G, X)

inducing the corestriction maps on cohomology (see [Bro], III.9(D), for a conceptual explanation of this formula). In fact, the homotopy class of cor is independent of any choices: if cor0 corresponds to another section α 7→ α0 , then the formula X

(hn+1 (c))(g1 , . . . , gn ) =

α−1

αg1 . . . gi−1 gi (αg1 . . . gi )

(−1)i c(αg1 (αg1 )−1 , . . . , αg1 . . . gi−1 gi (αg1 . . . gi )−1 ,

i=0

α∈U \G 0 −1

n−1 X

0

, αg1 . . . gi gi+1 (αg1 . . . gi+1 0 )−1 , . . . , αg1 . . . gn−1 0 gn (αg1 . . . gn 0 )−1 )

defines a homotopy h : cor − cor0 . If V ⊂ U is another open subgroup of G, fix coset representatives [ [ G= U αi , U= V βj . i

j

Then the composition of the corresponding corestriction morphisms cor

cor

• • • Ccont (V, X)−→Ccont (U, X)−→Ccont (G, X)

is equal to the corestriction map attached to the coset decomposition 149

G=

[

V β j αi .

i,j

(8.1.5) Lemma. For every discrete G-module X, the composite morphism ∼

sh

cor

• • • • Ccont (G, U X) −→ Ccont (G, IndG U (X))−→Ccont (U, X)−→Ccont (G, X)

(where cor depends on a fixed section of G −→ U \G) is homotopic to the map induced by Tr : U X −→ X,

X

a 7→

a(β).

β∈G/U

Proof. The formula X n−1 X (−1)i c(g1 , . . . , gi , (αg1 . . . gi )−1 , αg1 . . . gi gi+1 (αg1 . . . gi+1 )−1 , . . . ,

(hn (c))(g1 , . . . , gn−1 ) =

α∈U \G i=0

. . . , αg1 . . . gn−2 gn−1 (αg1 . . . gn−1 )−1 )(α−1 ), e.g. h1 (c) =

X

c(α−1 )(α−1 ),

α∈U \G

defines a homotopy h : Tr∗ −

cor ◦ sh.

(8.1.6) Functoriality. Let V ⊂ U be open subgroups of G. The previous discussion gives the following functoriality properties of IndG U (X), U X and XU (for variable U ).

G (8.1.6.1) Restriction. The map res is induced by the inclusion IndG U (X) ,→ IndV (X), which corresponds to the map UX

−→ V X

induced by the canonical projection pr : Z[G/V ] −→ Z[G/U ] resp., to the trace map Tr : XU −→ XV ,

gU ⊗ x 7→

X

guV ⊗ x.

u∈U/V

(8.1.6.2) Corestriction. The homotopy class of cor corresponds to the homotopy class of the map • • Ccont (G, V X) −→ Ccont (G, U X)

induced by Tr : Z[G/U ] −→ Z[G/V ],

X

gU 7→

u∈U/V

resp., to the map • • Ccont (G, XV ) −→ Ccont (G, XU )

induced by 150

guV,

pr : Z[G/V ] −→ Z[G/U ]. (8.1.6.3) Conjugation. If U C G is an open normal subgroup of G, then the formula (Ad(gU )f )(g 0 ) = (gU f )(g 0 ) = g(f (g −1 g 0 )) defines a (left) G/U -action on IndG U (X) commuting with the action of G. This Ad-action of G/U corresponds to the action (gU a)(g 0 U ) = a(g 0 gU ) on

UX

(resp., to the action  gU



X



X

hU ⊗ xhU  =

hU ∈G/U

hg −1 U ⊗ xhU

hU ∈G/U

• on XU ) and corresponds via sh to the homotopy action of G/U on Ccont (U, X), defined in 3.6.1.4. More precisely, for each g ∈ G, the commutative diagram of morphisms of pairs

(G, IndG U (X))

(incl,can)

/ (U, X)

(id,Ad(gU ))

 (G, IndG U (X))

(Ad(g−1 ),g)

(Ad(g−1 ),g)

 (G, IndG U (X))

(incl,can)

 / (U, X)

induces a commutative diagram of morphisms of complexes sh

• Ccont (G, IndG U (X))

• / Ccont (U, X)

Ad(gU )∗

 • Ccont (G, IndG U (X))

Ad(g)

Ad(g)

 • Ccont (G, IndG U (X))

sh

 / C • (U, X). cont

In the left column, the two vertical maps commute with each other. A homotopy hg : id − 4.5.3 induces a homotopy (sh ◦ Ad(gU )∗ ) ? hg : sh ◦ Ad(gU )∗ −

Ad(g) from

sh ◦ Ad(gU )∗ ◦ Ad(g) = sh ◦ Ad(g) ◦ Ad(gU )∗ = Ad(g) ◦ sh.

(8.1.6.4) Products. If, in the situation of 8.1.6.3, G/U is abelian, then XU becomes a Z[G/U ][G]-module, with G (resp., G/U ) acting as in 8.1.3 (resp., in 8.1.6.3). Let (XU )ι be the following Z[G/U ][G]-module: as a G-module, (XU )ι = XU , but gU acts on (XU )ι as g −1 U acted on XU . The map mU : XU ⊗Z[G/U ] (YU )ι −→ Z[G/U ] ⊗Z (X ⊗Z Y ) [g1 U ] ⊗ x ⊗ [g2 U ] ⊗ y 7→ [g2 g1−1 U ] ⊗ x ⊗ y 151

is a morphism of Z[G/U ][G]-modules, provided we let G (resp., Z[G/U ]) act trivially (resp., by multiplication) on the factor Z[G/U ] on the R.H.S. If G/V is also abelian, then the diagram m

Z[G/V ] ⊗Z (X ⊗Z Y )  pr⊗id y

m

Z[G/U ] ⊗Z (X ⊗Z Y )

XV ⊗Z[G/V ] (YV )ι  pr⊗pr y

V −−→

XU ⊗Z[G/U ] (YU )ι

U −−→

is commutative. (8.1.6.5) Lemma. If G/U is abelian, then the diagram sh⊗sh

• • Ccont (G, XU ) ⊗Z Ccont (G, (YU )ι )   y

−−−−→

• • Ccont (U, X) ⊗Z Ccont (U, Y )

∪◦(mU )∗

∪

• −−→ Ccont (U, X ⊗Z Y )  cor y

δ

• • Ccont (G, XU ) ⊗Z[G/U ] Ccont (G, (YU )ι ) −−−−→

U • • Z[G/U ] ⊗Z Ccont (G, X ⊗Z Y ) −−→ Ccont (G, X ⊗Z Y )

is commutative up to homotopy. Proof. This follows from Lemma 8.1.5 and the commutative diagram ∪

• • Ccont (U, X) ⊗Z Ccont (U, Y ) O

/ C • (U, X ⊗Z Y ) cont O

sh⊗sh

sh ∪

• • Ccont (G, XU ) ⊗Z Ccont (G, (YU )ι )

/ C • (G, XU ⊗Z YU ) cont

q∗

/ C • (G, (X ⊗Z Y )U ) cont Tr∗



• • Ccont (G, XU ) ⊗Z[G/U ] Ccont (G, (YU )ι )

∪◦(mU )∗

• / Z[G/U ] ⊗Z Ccont (G, X ⊗Z Y )

δU

 • / Ccont (G, X ⊗Z Y ),

where q : XU ⊗Z YU −→ (X ⊗Z Y )U is the morphism of G-modules given by the formula X X q: β ⊗ xβ ⊗ β 0 ⊗ yβ 0 7→ β ⊗ xβ ⊗ yβ . β,β 0

β

(8.1.7) Semilocal case (8.1.7.1) Assume that α : G −→ G is a continuous homomorphism of pro-finite groups and U C G is an open normal subgroup of G. Then U = α−1 (U ) is an open normal subgroup of G and α : G/U −→ G/U is injective. Fix coset representatives σi ∈ G of G/U =

[

σi α(G/U ) =

i

[

α(G/U)σi−1

i

and set, for each i, α

Ad(σi )

αi : G−→G−−−−→G, Ad(σi )

α

αi : G/U −→G/U −−−−→G/U. 152

For every discrete G-module X, the above coset decomposition yields a decomposition of the G-module α∗ (XU ) into a direct sum M M α∗ (Z[G/U ] ⊗Z X) = α∗ (Z[α(G/U)]σi−1 ⊗Z X) = α∗ (XU )i ; i

i

denote by pri : α∗ (XU ) −→ α∗ (XU )i the projection on the i-th factor. The isomorphism of G-modules ∼

σi ⊗ σi : α∗ (XU )i −→ α∗i (Z[αi (G/U)] ⊗Z X), together with the canonical identification ∼

α∗i (Z[αi (G/U )] ⊗Z X) −→ (α∗i X)U α∗i (gU ) ⊗ x 7→ [gU ] ⊗ x, give a G-isomorphism ∼

ωi0 : α∗ (XU )i −→ (α∗i X)U . Putting all ωi = ωi0 ◦ pri together, we obtain a G-isomorphism M ∼ ω = (ωi ) : α∗ (XU ) −→ (α∗i X)U . i

(8.1.7.2) The commutative diagram of morphisms of pairs (G, XU )  (α,id) y (G, α∗ (XU ))

(Ad(σi−1 ),σi )

−−−−−−→

(G, XU )

(id,ωi )

sh

−−→ sh

(G, (α∗i X)U ) −−→

−−−−−−→

(U, X)  (α ,id) y i (U , α∗i X)

induces a commutative diagram of morphisms of complexes Ad(σi )

C • (G, XU )   ∗ yα

−−−−→

C • (G, (α∗ (XU )))

i −−−−→

ω

C • (G, XU )

−−→

sh

C • (U, X)  α∗ y i

sh

C • (U , α∗i X).

C • (G, (α∗i X)U ) −−→

This implies that the following diagram ω◦α∗

C • (G, XU ) −−−−→   ysh C • (U, X)

(α∗ )

i −−−−→

L

C • (G, (α∗i X)U )   ysh L • ∗ i C (U , αi X) i

is commutative up to the homotopy h = (α∗i ◦ sh ◦ hσi )i : (α∗i ) ◦ sh − quasi-isomorphism (functorial in X)

sh ◦ ω ◦ α∗ , which induces a

Cone(sh, sh ◦ ω, h) : Cone(α∗ ) −→ Cone((α∗i )).

(8.1.7.2.1)

Above, hσ denotes fixed bi-functorial homotopies hσ (M ) : id − Ad(σ), e.g. those from 4.5.5. As any two choices of hσ (M ) are 2-homotopic, it follows from 1.1.7 that the homotopy class of the quasi-isomorphism (8.1.7.2.1) does not depend on the choice of hσ . (8.1.7.3) Conjugation. (8.1.7.3.1) Fix g ∈ G. Then we have, for each i, g −1 σi = u−1 i σg(i) α(g i ),

ui ∈ U, g i ∈ G. 153

∼

The G-isomorphisms σi : α∗ X −→ α∗i X give rise to morphisms of complexes −1 (σg(i) )∗

Ad(g−1 )

(σi )∗

i • F (g)i : C • (U , α∗g(i) X)−−−−→C • (U , α∗ X)−−−−→C (U , α∗ X)−−→C • (U , α∗i X)

and F (g) = (F (g)i )i :

M

C • (U , α∗i X) −→

i

M

C • (U , α∗i X),

i

functorial in X. (8.1.7.3.2) The faces of the cubic diagram ω◦α∗

C • (G, XU ) FF FF FF FF sh FF FF FF FF F" Ad(gU )∗ C • (U, X)

/ L C • (G, (α∗ X) ) i i MMM U MMM MMM MMsh MMM ω◦Ad(gU)∗ ◦ω −1 MMM ,4 MM h MM& (α∗ ) L i • ∗ / i C (U , αi X)

Ad(g)

`

.6

 / L C • (G, (α∗ X) ) 19 F (g) i i MMM U MMMsh MMM ,4 k2 MMM M MMM m ,4 MMM h MM&  (α∗ L i) • / C (U , α∗i X) i

 ω◦α∗ C • (G, XU ) FF FF FF FF -5 FF k1 F FF sh FF FF "  C • (U, X) commute up to the following homotopies: `=0 h = ((α∗i ◦ sh) ? hσi )i m = −((F (g)i ◦ α∗g(i) ) ? hui )i

(as in 8.1.7.2) (as α∗i ◦ Ad(g) = F (g)i ◦ α∗g(i) ◦ Ad(ui ))

k1 = (sh ◦ Ad(gU )∗ ) ? hg

(as in 8.1.6.3)

k2 = −((F (g)i ◦ sh) ? hgi )i

(as sh ◦ ωi ◦ Ad(gU )∗ = F (g)i ◦ sh ◦ ωg(i) ◦ Ad(g i )).

(8.1.7.3.3) Lemma. The boundary of the cube in 8.1.7.3.2 is trivialized by a 2-homotopy (α∗i ) ? k1 + m ? sh + F (g) ? h − k2 ? α∗ − h ? (Ad(gU )∗ ) −

0.

Proof. This can be proved, e.g., by a brute force calculation based on the existence of bifunctorial 2homotopies Hσ,τ (M ) from 4.5.5. The details of the unilluminating calculation are omitted. (8.1.7.3.4) Corollary. The following diagram Cone(sh,sh◦ω,h)

Cone((α∗i ))  Cone(Ad(g),F (g),m) y

Cone(sh,sh◦ω,h)

Cone((α∗i ))

Cone(α∗ ) −−−−−−−−−−→  Cone(Ad(gU ) ,Ad(gU ) ,0) ∗ ∗ y Cone(α∗ )

−−−−−−−−−−→ 154

is commutative up to homotopy. Proof. This follows from 1.1.8 and Lemma 8.1.7.3.3. (8.1.7.4) Restriction. (8.1.7.4.1) Assume that V C G, V ⊂ U is another open normal subgroup of G; put V = α−1 (V ). Fix coset representatives τj ∈ G of [ [ G/V = τj α(G/V ) = α(G/V )τj−1 . j

j

Then G=

[

V τj α(G) =

j

[

U σi α(G)

i

and for each j we have U τj α(G) = U σi α(G) for unique i = π(j). In particular, τj = uij σi α(g ij ),

uij ∈ U, g ij ∈ G

(i = π(j)).

Set Ad(τj )

α

βj : G−→G −−→G and define a morphism of complexes (functorial in X) M M r = (rij ) : C • (U , α∗i X) −→ C • (V , βj∗ X) i

j

by Ad(g −1 ) ij

(σ−1 )∗

res

(τj )∗

i • rij : C • (U , α∗i X)−−−−→C (U , α∗ X)−−−−−−→C • (U , α∗ X)−−→C • (V , α∗ X)−−→C • (V , βj∗ X).

(8.1.7.4.2) The faces of the cubic diagram ωU ◦α∗

C • (G, XU ) FF FF FF FF sh FF FF FF FF F" Tr∗ C • (U, X)

/ L C • (G, (α∗ X) ) i i MMM U MMM MMM MMsh −1 MMM ωV ◦Tr∗ ◦ωU MMM ,4 MM h MM& ∗ (αi ) L ∗ • / i C (U , αi X)

res

 ωV ◦α∗ C • (G, XV ) EE EE EE EE -5 EE k1 sh EEE EE EE "  C • (V, X)

.6

/

`

L j

(βj∗ )

 19 r C • (G, (βj∗ X)V ) MMM MMMsh MM ,4 k2 MMM MMM m M,4 M MMM h0 M L & •  ∗ / j C (V , βj X)

commute up to the following homotopies: 155

`=0 k1 = 0 h = ((α∗i ◦ sh) ? hσi )i h0 = ((βj∗ ◦ sh) ? hτj )j

(as in 8.1.7.2)

m = ((βj ◦ res) ? huij )j

(as r ◦ (α∗i ) = (βj∗ ) ◦ res ◦ Ad(uij ))

k2 = −((r ◦ sh) ? hgij )j

(as sh ◦ ωV,j ◦ Tr∗ = r ◦ sh ◦ ωU,i ◦ Ad(g ij ))

(as in 8.1.7.2)

(above, i = π(j)). (8.1.7.4.3) Lemma. The boundary of the cube in 8.1.7.4.2 is trivialized by a 2-homotopy m ? sh + r ? h − k2 ? α∗ − h0 ? Tr∗ −

0.

Proof. Again, this can be proved by an explicit calculation, the details of which are omitted. (8.1.7.4.4) Corollary. The following diagram Cone(sh,sh,h)

Cone(ωU ◦ α∗ ) −−−−−−−−→   yCone(Tr∗ ,ωV ◦Tr∗ ◦ωU−1 ,0) Cone(sh,sh,h0 )

Cone(ωV ◦ α∗ ) −−−−−−−−→

Cone((α∗i ))  Cone(res,r,m) y Cone((βj∗ ))

is commutative up to homotopy. Proof. This follows from 1.1.8 and Lemma 8.1.7.4.3. (8.1.7.5) Corestriction. In the notation of 8.1.7.4.1, there is a similar construction of ‘corestriction’ morphisms M M c: C • (V , βj∗ X) −→ C • (U , α∗i X), j

i

yielding the cubic diagram ωV ◦α∗

C • (G, XV ) EE EE EE EE sh EE EE EE EE E" pr∗ C • (V, X)

cor

 ωU ◦α∗ C • (G, XU ) FF FF FF FF -5 FF k1 F FF sh FF FF "  C • (U, X)

/

L

C • (G, (βj∗ X)V ) MMM MM MMM MMsh −1 MM ωU ◦pr∗ ◦ωV MMM ,4 MMM h MM ∗ (βj ) L & • ∗ / j C (V , βj X) j

`

.6

 / L C • (G, (α∗ X) ) 19 c i i MMM U MMMsh MMM ,4 k2 MMM MMM m M,4 MM h0 MMM &  ∗ (αi ) L • / C (U , α∗i X) i

and the square 156

Cone(sh,sh,h)

Cone((βj∗ ))  Cone(cor,c,m) y

Cone(sh,sh,h)

Cone((α∗i )),

Cone(ωV ◦ α∗ ) −−−−−−−−→   yCone(pr∗ ,ωU ◦pr∗ ◦ωV−1 ,0) Cone(ωU ◦ α∗ )

−−−−−−−−→

commutative up to homotopy. The details are omitted. (8.1.7.6) The constructions in 8.1.7.1-5 have the following useful variant. As the action of σi−1 defines an isomorphism of G-modules ∼

σi−1 : α∗i X −→ α∗ X, the modified map ω e = (σi−1 ◦ ωi ) yields a G-isomorphism M ∼ ω e : α∗ (XU ) −→ (α∗ X)U . i

If we use consistently ω e instead of ω, then the map F (g) has to be replaced by Fe(g) = (Fe(g)i )i , where −1

Ad(gi Fe(g)i : C • (U , α∗ X)g(i) −−→ C • (U , α∗ X)i ,

where each subscript refers to the corresponding summand in the direct sum M M ∼ α∗i X −→ α∗ X. i

i

Similarly, the map rij has to be replaced by Ad(g−1 ij

res

reij : C • (U , α∗ X)i −−→ C • (U , α∗ X)i −−→C • (V , α∗ X)j . If α is injective, then the morphisms of pairs (αi , σi−1 ) : (σi α(U )σi−1 , X) −→ (U , α∗ X) induces an isomorphism of complexes ∼

C • (σi α(U )σi−1 , X) −→ C • (U , α∗ X). ∼

The latter complex is isomorphic to C • (U , α∗i X), via σi : α∗ X −→ α∗i X. (8.2) Shapiro’s Lemma for ind-admissible modules Recall that R is assumed to have finite residue field of characteristic p. Let U ⊂ G be as in 8.1. (8.2.1) For M ∈ (ind−ad R[G] Mod) we define R[G]-modules MU = M ⊗R R[G/U ],

UM

= HomR (R[G/U ], M )

with the G-action given by the formulas in 8.1.3. These modules have the folowing properties: (i) If the action of G on M is discrete, then MU (resp., U M ) coincides with the coresponding object defined in 8.1.3. ∼ (ii) The formulas in 8.1.3 define an isomorphism MU −→ U M , functorial in M . (iii) The functors M 7→ MU and M 7→ U M commute with arbitrary direct and inverse limits. In particular, if M is of finite type over R, then the canonical maps ∼

∼

MU /mn MU −→ (M/mn M )U , are isomorphisms (and similarly for

MU −→ lim (M/mn M )U ← n−

U M ).

157

These observations imply that MU , U M ∈ (ind−ad R[G] Mod). (8.2.2) Writing M =

lim −→

Mα ∈S(M)

Mα and each Mα as Mα = lim Mα /mn Mα , Shapiro’s Lemma applied to each ← n−

Mα /mn Mα (and the fact that Mα /mn Mα is a surjrctive projective system) gives quasi-isomorphisms ∼

∼

sh

G • • • n Ccont (G, (Mα )U ) −→ lim Ccont (G, (Mα /mn Mα )U ) −→ lim ←− ←− Ccont (G, IndU (Mα /m Mα ))−→ sh n • • −→ ← lim − Ccont (U, Mα /m Mα ) = Ccont (U, Mα )

and ∼

sh

• • • • Ccont (G, MU ) −→ lim −→ Ccont (G, (Mα )U )−→ lim −→ Ccont (U, Mα ) = Ccont (U, M ).

α

The same argument applies to

α

UM.

(8.2.3) The definitions above extend in an obvious way to complexes M • of ind-admissible R[G]-modules. If at least one of the following conditions is satisfied: (A) M • is bounded below; (B) cdp (G) < ∞, then both morphisms • • • Ccont (G, MU• ) −→ Ccont (U, M • ) ←− Ccont (G, U M • )

are quasi-isomorphisms (using the spectral sequence (3.5.3.1)). (8.2.4) Put RU = R[G/U ] and denote by ι : RU −→ RU the R-linear involution induced by the map g 7→ g −1 on G/U . Both MU and U M are (left) RU [G]-modules, with the action of RU given by the Ad-action of G/U described in 8.1.6.3. If G/U is abelian, then the action of x ∈ RU on MU = M ⊗R RU (resp., on by id ⊗ ι(x) (resp., Hom(x, id)).

UM

= HomR (RU , M )) is given

(8.2.5) Lemma. Assume that G/U is abelian. Then (i) RU is an equidimensional (of dimension d) semilocal complete noetherian ring. (ii) If M is an (ind-)admissible R[G]-module, then U M , MU are (ind-)admissible RU [G]-modules. (iii) If M is of finite (resp., co-finite) type over R, then U M , MU are of finite (resp., co-finite) type over RU . Proof. Everything follows from the fact that RU is a free R-module of finite type. (8.2.6) Corollary. If G/U is abelian, then both functors M 7→ MU , M 7→ U M map (ad R[G] Mod)R−∗ to (ad Mod) (and similarly for ind − ad), with ∗ = f t, cof t. Under the canonical quasi-isomorphisms RU −∗ RU [G] sh

sh

• • • Ccont (G, MU )−→Ccont (U, M )←−Ccont (G, U M ) • the action of RU on MU , U M corresponds to the homotopy action of G/U on Ccont (U, M ).

(8.3) Infinite extensions (8.3.1) Let H C G be a closed normal subgroup of G. Put Γ = G/H, U = {U ⊂ G | U open subgroup, U ⊃ H} R = R[[Γ]] = lim ←− R[G/U ]. U ∈U 158

For every M ∈ (ind−ad R[G] Mod), the RU [G]-modules MU (resp., U M ) for variable U ∈ U form a projective (resp., an inductive) system with transition maps induced by the projections pr : RV −→ RU (for V ⊂ U , U, V ∈ U ). Denote by FΓ (M ) = lim ←− MU , U ∈U

FΓ (M ) = lim −→ U ∈U

UM

the corresponding limits; they are both (left) R[G]-modules. We define FΓ (M • ), FΓ (M • ) for a complex M • of ind-admissible R[G]-modules by the same formulas (termwise). (8.3.2) Lemma. (i) If M • is a complex of ind-admissible R[G]-modules, so is FΓ (M • ) and the canonical map • • • • lim −→ Ccont (G, U M ) −→ Ccont (G, FΓ (M )) U

is an isomorphism of complexes. (ii) If M is contained in (ind−ad R[G] Mod){m} , so is FΓ (M ) and the composite morphism sh

res

• • • lim −U → Ccont (G, U M )−→ lim −U → Ccont (U, M )−→Ccont (H, M )

is an quasi-isomorphism. The same is true for complexes M • of such modules, provided they satisfy 8.2.3 (A) or (B). Proof. It is enough to consider only the case M • = M . (i) FΓ (M ) is ind-admissible, since each U M is; then apply 3.4.1.5. (ii) FΓ (M ) is supported at {m}, since each U M is; in particular, both M and FΓ (M ) are discrete G-modules. The map res is an isomorphism by ([Se 2], I.2.2, Prop. 8) and sh is a quasi-isomorphism. (8.3.3) Corollary. If M • is a complex in (ind−ad R[G] Mod){m} satisfying 8.2.3 (A) or (B), then there is a canonical quasi-isomorphism • • Ccont (G, FΓ (M • )) −→ Ccont (H, M • ) such that the R-action on FΓ (M • ) corresponds to the homotopy action of G/H = Γ on the R.H.S. • (8.3.4) For any M ∈ (ind−ad R[G] Mod) the complexes Ccont (G, MU ) form a projective system indexed by U ∈ U , with surjective transition maps. The projective limit • lim Ccont (G, MU ) ← U−

is a complex of R-modules; denote by RΓIw (G, H; M ) ∈ D(R Mod) the corresponding object of the derived category and by i HIw (G, H; M ) = H i (RΓIw (G, H; M ))

its cohomology. The same notation will be used for complexes M • of ind-admissible R[G]-modules (the subscript “Iw” stands for “Iwasawa” - this notation is due to Fontaine). (8.3.5) Proposition. Let M ∈ (ind−ad R[G] Mod). Then (i) There is a spectral sequence j i+j (i) E2i,j = lim (Hcont (U, M )) =⇒ HIw (G, H; M ). ←− U,cor

(ii) If M is of finite type over R and if G satisfies (F), then E2i,j = 0 for i 6= 0 and j j HIw (G, H; M ) = ← lim Hcont (U, M ). − U,cor

159

(iii) Assume that M is of finite type over R, U contains a cofinal chain U1 ⊃ U2 ⊃ U3 ⊃ · · · and the 0 pro-finite order of Γ is divisible by p∞ . Then HIw (G, H; M ) = 0. Proof. (i) Consider the two hyper-cohomology spectral sequences for the functor lim ←− and the projective • system Ccont (G, MU ): I

j (j) i (i) E1i,j = lim Ccont (G, MU ) =⇒ H i+j ⇐= II E2i,j = lim (Hcont (G, MU )). ← − ← − U U

i For each i, the projective system Ccont (G, MU ) is “weakly flabby” in the sense of [Je, Lemma 1.3(ii)],as i i lim Ccont (G, MU ) −→ Ccont (G, MV ) ← U−

is surjective for every V ∈ U . This implies that I E1i,j = 0 for j 6= 0, hence H n = H n (i 7→ I E1i,0 ) = n HIw (G, H; M ). By Lemma 8.1.5, the transition maps in II

∼

E2i,j −→ lim ←− U ∈U

(i)

j (Hcont (U, M ))

are given by the corestriction, as claimed. (ii) The same argument as in (i) yields a spectral sequence j i+j (i) E2i,j = lim (Hcont (U, M/mn M )) =⇒ HIw (G, H; M ). ← − U,n j As G satisfies (F), the projective system Hcont (U, M/mn M ) (indexed by U × N) consists of R-modules of i,j finite length, hence E2 = 0 for i 6= 0 ([Je, Cor. 7.2]). It follows that ∼

∼

j j j HIw (G, H; M ) −→ E20,j = lim lim (Hcont (U, M/mn M )) −→ lim Hcont (U, M ) ← ← n− U− ← U−

(the last isomorphism by Lemma 4.2.2). (iii) As M is of finite type over R, the sequence of invariants M U1 ⊆ M U2 ⊆ · · · stabilizes: M Un = N for all n ≥ n0 . This implies that 0 0 HIw (G, H; M ) = ← lim lim − Hcont (Un , M ) = n≥n ←−0 N, n,cor

with the transition maps given by the multiplication by [Un : Un0 ] (n0 ≥ n ≥ n0 ). For fixed n ≥ n0 , the power of p dividing [Un : Un0 ] tends to infinity as n0 −→ ∞, hence \
0 0 Im HIw (G, H; M ) −→ Hcont (Un , M ) ⊆ [Un : Un0 ] · N = 0, n0 ≥n

proving the claim. (8.3.6) In 8.3.5(i) we can replace M by a complex M • of ind-admissible R[G]-modules, provided 8.2.3 (A) or (B) holds. In 8.3.5(ii) we have to assume, in addition, that M • has cohomology groups of finite type over R (cf. proof of Proposition 4.2.5). This implies that RΓIw (G, H; −) induces an exact functor ∗ RΓIw (G, H; −) : D∗ (ind−ad R[G] Mod) −→ D (R Mod)

for ∗ = + (resp., for ∗ = +, ∅, provided cdp (G) < ∞). (8.4) Infinite abelian extensions We retain the notation and assumptions of 8.3. 160

(8.4.1) In practice one is usually interested in the case when Γ = G/H is a p-adic Lie group. In this paper we consider only the case of abelian Γ. According to Lemma 4.1.4, cohomological invariants do not change if Γ (resp., G) is replaced by its pro-p-Sylow subgroup Γ(p) (resp., by the inverse image of Γ(p) ⊂ G/H in G). Without loss of generality we shall, therefore, assume that Γ = Γ0 × ∆, where ∆ is a finite abelian group and Γ0 is isomorphic to Zrp for some r ≥ 1. In this case R = R[[Γ]] = R0 ⊗R R[∆],

R0 = R[[Γ0 ]].

∼

A choice of an isomorphism Γ0 −→ Zrp (i.e. a choice of a Zp -basis γ1 , . . . , γr of Γ0 ) gives an isomorphism of rings ∼

R0 −→ R[[X1 , . . . , Xr ]],

γi 7→ 1 + Xi .

In particular, R is an equidimensional semilocal complete noetherian ring, of dimension d + r. Let m ⊂ R be the radical of R. (8.4.2) Denote by χΓ : G  Γ ,→ R[Γ]∗

∗

(⊂ R )

the tautological one-dimensional representation of G over R[Γ]. For every (R[Γ])[G]-module M and n ∈ Z we construct new (R[Γ])[G]-modules M < n > resp., M ι as follows: as an R[Γ]-module, M < n > coincides with M , but the action of G is given by gM = χΓ (g)n gM

(g ∈ G).

M ι coincides with M as an R[G]-module, but the action of R[Γ] is given by xM ι = ι(x)M

(x ∈ R[Γ]).

ι

A more functorial definition of M is M ⊗R[Γ],ι R[Γ], as in 6.6.4. With this notation (which applies, in particular, to R[G]-modules), we have ∼

M < n >ι −→ M ι < −n > . (8.4.3) Injective hulls. We first relate IR to IR . (8.4.3.1) Lemma. As an R[G]-module, FΓ (IR ) < 1 > is isomorphic to the Pontrjagin dual of R (the action of G on IR and R being trivial). Proof. As in 2.9, fix an isomorphism of R-modules ∼ b n IR −→ R = lim −→ HomZp (R/m R, Qp /Zp ). n

This induces an isomorphism of R[G/U ]-modules ∼

∼

n n HomR (R[G/U ], IR ) −→ lim −→ HomR/mn R (R/m R[G/U ], HomZp (R/m R, Qp /Zp )) −→ n

∼

n −→ lim −→ HomZp (R/m R[G/U ], Qp /Zp ), n

hence an isomorphism of R-modules between FΓ (IR ) = lim −→ HomR (R[G/U ], IR ) U ∈U 161

(cf. 8.2.4) and lim HomZp (R/mn R[G/U ], Qp /Zp ), − → U,n which is nothing but the Pontrjagin dual of n lim ← − R/m R[G/U ] = R. U,n

The action of g ∈ G on FΓ (IR ) is given by (g ∗ f )(x) = g(f (g −1 (x))) = f (g −1 (x)) = f (χΓ (g)−1 x) = χΓ (g)−1 f (x), hence G acts trivially on FΓ (IR ) < 1 >. (8.4.3.2) Corollary. The R-module I = IR := FΓ (IR ) < 1 > is an injective hull of R/m over R. (8.4.4) The functor FΓ . (8.4.4.1) Proposition. Let M ∈ (ind−ad R[G] Mod). Then (i) If M is of finite type over R, then there are canonical isomorphisms of R[G]-modules ∼

ι

∼

∼

FΓ (M )ι −→ (M ⊗R R ) < −1 >−→ (M ⊗R R) < 1 >,

FΓ (M ) −→ (M ⊗R R) < −1 >,

functorial in M . In particular, FΓ (M ) is of finite type over R. ind−ad (ii) If M ∈ (ind−ad R[G] Mod){m} , then FΓ (M ) ∈ (R[G] Mod){m} . (iii) If M is of co-finite type over R, then FΓ (M ) is of co-finite type over R. Proof. (i) The canonical map of R-modules N ⊗R R = N ⊗R lim RU −→ lim (N ⊗R RU ) ← ← U− U−

(8.4.4.1.1)

is an isomorphism for every R-module N of finite type. For N = M , the natural action of R[G] on FΓ (M ) = ← lim − (M ⊗R RU ) is the following (cf. 8.1.3 and 8.2.4): U

x ∈ R acts by id ⊗ ι(x) g ∈ G acts by g ⊗ χΓ (g). this implies that (8.4.4.1.1) and ι induce the following isomorphisms of R[G]-modules: ι

id⊗ι

FΓ (M ) = (M ⊗R R ) < 1 > −−→(M ⊗R R) < −1 > . The statement about FΓ (M )ι is proved in the same way. (ii) This follows from Lemma 8.3.2. (iii) It is sufficient to treat the case Γ = Γ0 , when R = R0 is local (cf. proof of Lemma 8.2.5). We must show that dimk (FΓ (M )[m]) < ∞ (here k = R/m = R/m). As FΓ (M [m])[m] = FΓ (M )[m], we can assume that M = M [m] and hence, by d´evissage, that M = k. In this case dimk (FΓ (k)[m]) = 1 by Lemma 8.4.3.1 applied to R = k. (8.4.4.2) Proposition. For every M ∈ (ind−ad R[G] Mod)R−f t , the canonical morphism of complexes ∼

• • Ccont (G, FΓ (M )) −→ lim ←− Ccont (G, MU )

U

is an isomorphism, hence inducing an isomorphism ∼

RΓcont (G, FΓ (M )) −→ RΓIw (G, H; M ) 162

in D+ (R Mod). Proof. We must show that the two projective limit topologies on ∼

∼

n n FΓ (M ) −→ ← lim − FΓ (M )/m FΓ (M ) −→ lim ←− MU /m MU n

U,n

coincide (the first (resp., second) isomorphism follows from the fact that FΓ (M ) is m-adically (resp., m∼ adically) complete). We can assume that Γ = Γ0 , in which case R = R0 −→ R[γ1 − 1, . . . , γr − 1] is local, with m = mR + (γ1 − 1, . . . , γr − 1). Put JU = Ker(R −→ R[G/U ]); then MU /mn MU = FΓ (M )/(JU + mn R)FΓ (M ) ∼

and the claim follows from the fact (used in the proof of the isomorphism R −→ R[γ1 − 1, . . . , γr − 1]) that the systems of ideals {mn }n∈N and {JU + mn R}(U,n)∈U ×N of R are cofinal in each other. (8.4.4.3) It follows from Lemma 8.3.2 and Proposition 8.4.4.1 that the functors FΓ and FΓ derive (for each ∗ = ∅, +, −, b) trivially to exact functors ∗ ad FΓ : D∗ ((ad R[G] Mod)R−cof t ) −→ D ((R[G] Mod)R−cof t ) ∗ ad FΓ : D∗ ((ad R[G] Mod)R−f t ) −→ D ((R[G] Mod)R−f t ),

hence, using Proposition 3.2.8, to functors ∗ ad ∗ ad ∗ ad FΓ : DR−cof t (R[G] Mod) ≈ D ((R[G] Mod)R−cof t ) −→ DR−cof t (R[G] Mod)

FΓ :

∗ ad DR−f t (R[G] Mod)

≈D

∗

((ad R[G] Mod)R−f t )

−→

(∗ = +, b)

∗ DR−f (ad Mod) t R[G]

(∗ = −, b).

Proposition 8.4.4.2 implies that there is a canonical isomorphism in D+ (R Mod) ∼

b ad (M ∈ DR−f t (R[G] Mod)).

RΓcont (G, FΓ (M )) −→ RΓIw (G, H; M ),

(8.4.5) Relating the functors FΓ and FΓ . Our next goal is to investigate the relationship between the functors FΓ , FΓ and the duality diagrams 2.8.2 over the rings R and R. We denote the functors (DR , DR , ΦR ) (resp., (DR , DR , ΦR )) appearing in these diagrams by (D, D, Φ) (resp., (D, D, Φ)). (8.4.5.1) Lemma. For every M ∈ (ad R[G] Mod)R−cof t there are canonical isomorphisms of R[G]-modules ∼

∼

∼

FΓ (M ) −→ HomR (D(M ) ⊗R R, I) < −1 >−→ D((D(M ) ⊗R R) < 1 >) −→ D(FΓ (D(M ))ι ) ∼

D ◦ FΓ (M ) −→ ι ◦ FΓ ◦ D(M ). Proof. The first isomorphism is given by ∼

∼

∼

FΓ (M ) −→ lim −→ HomR (R[G/U ], HomR (D(M ), I)) −→ lim −→ HomR (R[G/U ] ⊗R D(M ), I) −→ U

U

∼

∼

−→ lim lim −→ HomR (D(M ), HomR (R[G/U ], I)) −→ HomR (D(M ), − → HomR (R[G/U ], I)) = U

U

∼

= HomR (D(M ), I < −1 >) −→ HomR (D(M ) ⊗R R, I) < −1 >, using Lemma 8.4.3.1 (and the fact that the direct limit can be interchanged with HomR , since D(M ) is of ∼ finite type over R). The second isomorphism follows by applying D and the isomorphism ε : id −→ D ◦ D. (8.4.5.2) Fix γi (i = 1, . . . , r) as in 8.4.1 and put Xi = γi − 1 ∈ R0 . Fix a system of parameters x1 , . . . , xd • of R; then x1 , . . . , xd , X1 , . . . , Xr form a system of parameters of R0 . Let CR (Γ0 ) be the following complex in degrees [0, r] (and differentials as in 2.4.3):   M M R0 −→ (R0 )Xi −→ (R0 )Xi Xj −→ · · · −→ (R0 )X1 ···Xr  ; i

i ⊗R (Y ⊗R R ) < 1 > −−−−−−→(X ⊗R R) < −1 > ⊗R (Y ⊗R R) < 1 >= (X ⊗R Y )⊗R R. (8.4.6.2) Dualizing complexes. For each S = R, R, fix a complex ωS• = σ≥0 ωS• of injective S-modules representing ωS . There exists a quasi-isomorphism (unique up to homotopy) • • ϕ : ωR ⊗R R −→ ωR ,

which we fix, once for all. (8.4.6.3) Given X and Y as in 8.4.6.1 and a morphism of complexes • π : X ⊗R Y −→ ωR (1),

we obtain a morphism of complexes m

id⊗π

ϕ

• • π : FΓ (X) ⊗R FΓ (Y )ι −−→R ⊗R (X ⊗R Y )−−→R ⊗R ωR (1)−−→ωR (1).

165

The corresponding adjoint map is equal to id⊗adj(π)

∼

• adj(π) : FΓ (X) −→ R ⊗R X < −1 > −−−−−−−−→R ⊗R Hom•R (Y, ωR (1)) < −1 >=

ϕ∗

• • • • = HomR (R ⊗R Y < 1 >, R ⊗R ωR (1)) = Hom•R (FΓ (Y )ι , R ⊗R ωR (1))−−→Hom•R (FΓ (Y )ι , ωR (1)).

(8.4.6.4) Lemma. There exist functorial isomorphisms ∼

∼

ι ◦ FΓ ◦ D(X) −→ D ◦ FΓ (X),

b ad (X ∈ DR−f t (R[G] Mod)).

Φ ◦ FΓ (X) −→ FΓ ◦ Φ(X)

Proof. By Proposition 3.2.6 there exists a bounded complex Y in (ad R[G] Mod)R−f t and a quasi-isomorphism • f : Y −→ D(X) = Hom•R (X, ωR ). Applying 8.4.6.3 to the pairing id⊗f

ev

2 • π : X ⊗R Y −−→X ⊗R D(X)−−→ω R,

we obtain a morphism of complexes • adj(π) : FΓ (Y )ι −→ Hom•R (FΓ (X), ωR ) = D ◦ FΓ (X),

b hence a morphism in DR−f (ad Mod) t R[G] (FΓ (f )ι )−1

adj(π)

αX : ι ◦ FΓ ◦ D(X) = FΓ (D(X))ι −−−−−−→FΓ (Y )ι −−−−→D ◦ FΓ (X), which is functorial in X. In order to prove that αX is an isomorphism, we can disregard the action of G, i.e. consider only X ∈ Dfb t (R Mod), FΓ (X) = X ⊗R R and replace ι by the identity map. As we are dealing with “way-out functors” ([RD], I.7), standard d´evissage reduces the claim to the fact that αR = ϕ is an isomorphism in Dfb t (R Mod). As regards the second isomorphism, we combine αX with Lemma 8.4.5.1, obtaining functorial isomorphisms ∼

∼

∼

∼

FΓ ◦ Φ(X) −→ D ◦ ι ◦ FΓ ◦ D ◦ Φ(X) −→ D ◦ ι ◦ FΓ ◦ D(X) −→ D ◦ D ◦ FΓ (X) −→ Φ ◦ FΓ (X). (8.4.6.5) Corollary. If, in the situation of 8.4.6.3, adj(π)

• • X −−−−→HomR (Y, ωR (1)) −→ W −→ X[1]

b ad is an exact triangle in DR−f t (R[G] Mod), then adj(π)

• FΓ (X)−−−−→Hom•R (FΓ (Y )ι , ωR (1)) −→ FΓ (W ) −→ FΓ (X)[1]

b is an exact triangle in DR−f (ad Mod). In particular, t R[G]

π is a perfect duality over R ⇐⇒ π is a perfect duality over R. (8.4.6.6) The isomorphisms from Lemma 8.4.5.1 and 8.4.6.4 can be summarized as follows. b ad ∗ b ad If T, T ∗ ∈ DR−f t (R[G] Mod) and A, A ∈ DR−cof t (R[G] Mod) are related by the duality diagram D

T ←−−−−−−−−→  ←−  −− D −−→  −−−−− Φ −−−−−−− y −− → − ← A

T∗    Φ y A∗

b b over R, then FΓ (T ), FΓ (T ∗ ) ∈ DR−f (ad Mod) and FΓ (A), FΓ (A∗ ) ∈ DR−cof (ad Mod) are related by the t R[G] t R[G] duality diagram D

FΓ (T ) ←−−−−−−−−→ FΓ (T ∗ )ι ←−     −− D −−→ −−−−−   Φ Φ − −− −−−− y y − − → ←− FΓ (A) FΓ (A∗ )ι over R. 166

+ ad (8.4.6.7) Proposition. (i) Let T ∈ D+ ((ad R[G] Mod)R−f t ); put A = Φ(T ) ∈ DR−cof t (R[G] Mod). Then there are canonical isomorphisms ∼

∼

∼

Φ(RΓIw (G, H; T )) −→ Φ(RΓcont (G, FΓ (T ))) −→ RΓcont (G, FΓ (A)) −→ RΓcont (H, A) in D+ (R Mod). b ad b ad (ii) Let T ∈ DR−f t (R[G] Mod); put A = Φ(T ) ∈ DR−cof t (R[G] Mod). If G satisfies (F) (resp., if G satisfies + b (F) and cdp (G) < ∞), then the above isomorphisms take place in Dcof t (R Mod) (resp., in Dcof t (R Mod)) and there is a spectral sequence j i+j i+j E2i,j = ExtiR (D(Hcont (H, A)), ωR ) =⇒ HIw (G, H; T ) = ← lim − Hcont (U, T ) U ∈U

(all terms of which are R-modules of finite type). Proof. Represent T by a bounded below complex T • in (ad R[G] Mod)R−f t . Lemma 8.4.6.4 applied to each component of T • then yields an isomorphism ∼

Φ ◦ FΓ (T ) −→ FΓ ◦ Φ(A)

+ ad in DR−cof t (R[G] Mod). The statement of (i) follows from 4.3.1, applied to the pair FΓ (T ) and FΓ (A) over R, while (ii) follows from (i) and 4.3.1. (8.4.6.8) A variant of the above spectral sequence for R = Zp and T consisting of a single module was constructed by Jannsen [Ja 3], who also considered the case of non-abelian Γ = G/H. (8.4.7) Dihedral case (8.4.7.1) Assume that we are given the following dihedral data. (8.4.7.1.1) An exact sequence of pro-finite groups

1 −→ G −→ G+ −→ {±1} −→ 1 such that (8.4.7.1.2) H is a normal subgroup of G+ , (8.4.7.1.3) The quotient exact sequence of groups 1 −→ Γ −→ Γ+ −→ {±1} −→ 1 (in which Γ+ = G+ /H) is split and −1 ∈ {±1} acts on Γ as ι : g 7→ g −1 . The last condition can be reformulated by saying that there exists an element τ ∈ Γ+ − Γ such that τ 2 = 1,

τ γτ −1 = γ −1

(γ ∈ Γ).

(8.4.7.1.4)

+

If M is an R[Γ ]-module, then the map m 7→ τ (m)

(m ∈ M )

defines an isomorphism of R[Γ]-modules ∼

M −→ M ι . The same is true for complexes of R[Γ+ ]-modules. • In particular, if T ∈ (ad R[G+ ] Mod)R−f t , then RΓIw (G, H; T ) can be represented not just by Ccont (G, FΓ (T )), • but by Ccont (G+ , FΓ+ (T )), where + + FΓ+ (T ) = ← lim − T ⊗R R[G /U ], U+

where U runs through all open subgroups of G containing H. Similarly, for every M ∈ (ind−ad R[G+ ] Mod){m} , • + RΓcont (H, M ) can be represented by Ccont (G , FΓ+ (M )), where +

+

+ + FΓ+ (M ) = lim −→ HomR (R[G /U ], M ). U+

• • Applying the previous discussion to the complexes Ccont (G+ , FΓ+ (T )) and Ccont (G+ , FΓ+ (M )) we obtain the following statement.

167

b ad (8.4.7.2) Proposition. Assume that we are given the dihedral data 8.4.7.1.1-3. If T ∈ DR−f t (R[G+ ] Mod)

and M ∈ D+ ((ind−ad R[G+ ] Mod){m} ), then the action of τ from (8.4.7.1.4) gives isomorphisms ∼

∼

RΓIw (G, H; T )ι −→ RΓIw (G, H; T ),

RΓcont (H, M )ι −→ RΓcont (H, M )

in D+ (R Mod). (8.4.8) Descent (8.4.8.1) Proposition. (i) For every M ∈ D+ ((ind−ad R[G] Mod){m} ) there is a spectral sequence j i+j i E2i,j = Hcont (Γ, Hcont (H, M )) =⇒ Hcont (G, M ). ∼

+ (ii) If Γ = Γ0 −→ Zrp , then for every T ∈ D+ ((ad R[G] Mod)R−f t ) there is a canonical isomorphism in D (R Mod) L

∼

RΓIw (G, H; T )⊗R R −→ RΓcont (G, T ) (where the product is taken with respect to the augmentation map R −→ R), which induces a (homological) spectral sequence −j −i−j 0 2 Ei,j = Hi,cont (Γ, HIw (G, H; T )) =⇒ Hcont (G, T ) 2 (where Hi,cont (Γ, −) was defined in (7.2.7.2)). If G satisfies (F), then each term 0 Ei,j is an R-module of finite type.

Proof. (i) This is just the Hochschild-Serre spectral sequence for discrete G-modules. (ii) Represent T by a bounded below complex in (ad R[G] Mod)R−f t , which will also be denoted by T . The elements γ1 − 1, . . . , γr − 1 form a regular sequence x in R and the augmentation map defines an isomorphism ∼ R/xR −→ R. For each i = 0, . . . , r, let Hi ⊂ Γ be the subgroup topologically generated by γ1 , . . . , γi and put Ri = R[[Γ/Hi ]] (e.g., R0 = R, Rr = R). For each i = 1, . . . , r, the tautological exact sequence γi −1

0 −→ Ri−1 −−→Ri−1 −→ Ri −→ 0   yields, upon tensoring with T , an exact sequence of complexes in ad Mod R[G] γi −1

0 −→ FΓ/Hi−1 (T )−−→FΓ/Hi−1 (T ) −→ FΓ/Hi (T ) −→ 0, hence an exact sequence of complexes of Ri−1 -modules γi −1

• • • 0 −→ Ccont (G, FΓ/Hi−1 (T ))−−→Ccont (G, FΓ/Hi−1 (T )) −→ Ccont (G, FΓ/Hi (T )) −→ 0,

(8.4.8.1.1)

which can be rewritten as an isomorphism of complexes   ∼ γi −1 • • Ccont (G, FΓ/Hi−1 (T )) ⊗Ri−1 Ri−1 −−→Ri−1 −→ Ccont (G, FΓ/Hi (T )) (with the complex [γi − 1 : Ri−1 −→ Ri−1 ] supported in degrees −1, 0). This yields a canonical isomorphism in D(Ri Mod) L

∼

RΓcont (G, FΓ/Hi−1 (T ))⊗Ri−1 Ri −→ RΓcont (G, FΓ/Hi (T )). Iterating this construction r-times, we obtain the desired isomorphism L

L

∼

RΓIw (G, H; T )⊗R R = RΓcont (G, FΓ (T ))⊗R R −→ RΓcont (G, T ) in D(R Mod). 168

The augmentation map f : R −→ R defines an exact functor f ∗ : (R Mod) −→ (R Mod) (f ∗ M = M as an • abelian group, with r ∈ R acting on f ∗ M by f (r)). As the Koszul complex KR (R, x) is an R-free resolution ∗ of f R[−r], we obtain from the previous result an isomorphism in D(R Mod) ∼

• f ∗ RΓcont (G, T ) −→ RΓcont (G, FΓ (T )) ⊗R KR (R, x)[r],

where the R.H.S. is represented by the complex • • • • • • K • = Ccont (G, FΓ (T )) ⊗R KR (R, x)[r] = Ccont (G, FΓ (T ) ⊗R KR (R, x))[r] = KR (Ccont (G, FΓ (T )), x)[r]. • The stupid filtration σ≥i on KR (R, x) induces a finite decreasing filtration F i K • on K • . The corresponding spectral sequence (3.5.2.1) is given by

0

j i+j i+j i+r E1i,j = KR (Hcont (G, FΓ (T )), x) =⇒ Hcont (G, K • ) = Hcont (G, T ).

It follows from (7.2.7.2) that 0

j j E2i,j = H−i,cont (Γ, Hcont (G, FΓ (T ))) = H−i,cont (Γ, HIw (G, H; T )),

m −i,−j which proves the result if we pass to the homological notation 0 Ei,j = 0 Em . j j Finally, if G satisfies (F), then each cohomology group M = Hcont (G, FΓ (T )) is an R-module of finite type, • hence the cohomology groups of the Koszul cmplex KR (M j , x) are R/xR-modules of finite type. ∼

(8.4.8.2) Corollary. Assume that Γ −→ Zrp and T ∈ D+ ((ad R[G] Mod)R−f t ). Then ∼

(i) If Γ −→ Zp , then 0 E 2 = 0 E ∞ degenerates into short exact sequences Γ

j j j+1 0 −→ HIw (G, H; T )Γ −→ Hcont (G, T ) −→ HIw (G, H; T ) −→ 0. ∼

(ii) If cdp (G) = e < ∞ and τ≤n T −→ T , then 0

∼

e+n e+n 2 E0,−e−n = HIw (G, H; T )Γ −→ Hcont (G, T ).

0 (8.4.8.3) Proposition. Assume that T ∈ D+ ((ad R[G] Mod)R−f t ). Let Γ ⊂ Γ = G/H be a closed subgroup 0

of Γ isomorphic to Zrp (r0 ≤ r) and H 0 ⊂ G its inverse image in G. Then there is a canonical isomorphism in D+ (R[[Γ/Γ0 ]] Mod) L

∼

RΓIw (G, H; T )⊗R R[[Γ/Γ0 ]] −→ RΓIw (G, H 0 ; T ) (where the product is taken with respect to the canonical map induced by the projection Γ −→ Γ/Γ0 ), which induces a (homological) spectral sequence 0

−j −i−j 2 Ei,j = Hi,cont (Γ0 , HIw (G, H; T )) =⇒ HIw (G, H 0 ; T ).

Proof. This follows from the same argument as in the proof of Proposition 8.4.8.1(ii) if we replace x = (γ1 − 1, . . . , γr − 1) by x0 = (γ10 − 1, . . . , γr0 0 − 1), where γ10 , . . . , γr0 0 are topological generators of Γ0 , as R/x0 R = R[[Γ/Γ0 ]]. (8.4.8.4) Corollary. Under the assumptions of 8.4.8.3, ∼ (i) If Γ0 −→ Zp , then 0 E 2 = 0 E ∞ degenerates into short exact sequences Γ0

j j+1 0 −→ HIw (G, H; T )Γ0 −→ RΓIw (G, H 0 ; T ) −→ HIw (G, H; T ) ∼

(ii) If cdp (G) = e < ∞ and τ≤n T −→ T , then 0

∼

e+n e+n 2 E0,−e−n = HIw (G, H; T )Γ0 −→ HIw (G, H 0 ; T ).

169

−→ 0.

∼

(8.4.8.5) Proposition. Assume that Γ −→ Zrp , G satisfies (F) and cdp (G) < ∞. Let p ∈ Spec(R); denote by p ∈ Spec(R) the inverse image of p under the augmentation map R −→ R. If T ∈ D b ((ad R[G] Mod)R−f t ) ∼

∼

satisfies RΓcont (G, T )p −→ 0 in Dfb t (Rp Mod), then RΓcont (G, FΓ (T ))p −→ 0 in Dfb t (R Mod). p


Proof. According to Proposition 8.4.8.1(ii), there is a spectral sequence in Rp Mod f t −i−j 2 Ei,j = Hi,cont (Γ, H −j )p =⇒ Hcont (G, T )p = 0,

where we have denoted −j H −j := Hcont (G, FΓ (T )) ∈




R Mod f t

.

2 If, for some j ∈ Z, we have E0,j = 0, then we deduce from 7.2.7 and Lemma 8.10.5 below (applied to −j 2 B = Rp , M = (H )p ) that Ei,j = 0 vanishes for every i ≥ 0. As the spectral sequence has only finitely 2 many non-zero terms and Ei,j = 0 for i < 0, induction on j shows that

(∀j ∈ Z)

2 0 = E0,j = (HΓ−j )p = (Hp−j )Γ = (Hp−j )/J,

where J = Ker(R[[Γ]] −→ R) denotes the augmentation ideal. As J ⊆ p, Nakayama’s Lemma implies that Hp−j = 0 for all j ∈ Z, as claimed. (8.5) Duality theorems in Iwasawa theory Assume that we are in the situation of 5.1, with K a number field (totally imaginary if p = 2, because of the condition (P )). (8.5.1) Notation. Assume that K∞ /K is a Galois extension contained in KS , with Galois group ∼ Gal(K∞ /K) = Γ = Γ0 × ∆, where Γ0 −→ Zrp (r ≥ 1) and ∆ is a finite abelian group. The results of 8.4 then apply to G = GK,S = Gal(KS /K) and H = Gal(KS /K∞ ). We shall use a simplified notation i RΓIw (K∞ /K, −) for RΓIw (GK,S , Gal(KS /K∞ ); −) and HIw (K∞ /K, −) for its cohomology, even though these objects depend, in general, also on S. Let {Kα } be the set of all finite extensions of K contained in K∞ ; put Γα = Gal(K∞ /Kα ). If v is a prime of K, then vα (resp., v∞ ) will usually denote a prime of Kα (resp., of K∞ ) above v. (8.5.2) Let v ∈ Sf . As Γ is abelian, the decomposition (resp., inertia) group of any prime v∞ |v in K∞ /K is independent of v∞ ; denote it by Γv (resp., I(Γv )). Both groups I(Γv ) ⊂ Γv are closed subgroups of Γ and the quotient Γv /I(Γv ) is isomorphic to Zp or Z/pn Z, for some n ≥ 0. The subgroups I(Γv ) (v ∈ Sf ) generate a subgroup of finite index in Γ, since the maximal abelian extension of K unramified at all finite primes is finite over K. If v ∈ Sf and v - p, then I(Γv ) ⊆ ΓZp −tors = ∆, by class field theory. In particular, there is v ∈ Sf such that v|p and |I(Γv )| = ∞. If there is only one prime v above p in K, then (Γ : I(Γv )) < ∞. (8.5.3) Semilocal theory (8.5.3.1) Let K 0 /K be a finite Galois subextension of KS /K. Denote by S 0 the set of primes of K 0 above S; then GK 0 ,S 0 = Gal(KS /K 0 ). For fixed v ∈ Sf we can apply the general discussion in 8.1.7 to the groups U = GK 0 = Gal(K/K 0 ) C G = GK = Gal(K/K) and G = Gv = Gal(K v /Kv ). As in 5.1, the fixed embedding of K ,→ K v defines an injective homomorphism α = αv : G ,→ G and a prime v00 ∈ Sf0 above v; then U = Gv00 = Gal(K v /Kv0 0 ). 0

For each σ ∈ G, the subgroup U σ = σα(U )σ −1 ⊂ U depends only on the double coset U σα(G) ∈ U \G/α(G), and is equal to the decomposition group Gσ(v00 ) ⊂ U of the prime σ(v00 ). Fixing coset representatives σi ∈ G (with σ0 = 1) [ G= U σi α(G) i 0

as in 8.1.7, the set of primes {v |v} in K coincides with {vi0 = σi (v00 )}. For every complex X of discrete G-modules we have, as in 8.1.7.6, isomorphisms of complexes 170

∼

∼

C • (Gσi (v00 ) , X) −→ C • (U , α∗ X) −→ C • (U , α∗i X),

(8.5.3.1.1)

induced by the morphisms of pairs (αi , σi−1 ) resp., (id, σi ). The results of 8.1.7 then give, for every complex X of discrete GK,S -modules, the following: (8.5.3.2) A quasi-isomorphism (8.1.7.2.1)[-1] shc : Cc• (GK,S , XU ) −→ Cc• (GK 0 ,S 0 , X), which is functorial in X, the homotopy class of which is independent of any choices, and which makes the bottom half of the following diagram with exact rows commutative up to homotopy (the top half is commutative in the usual sense): 0

L

−→

0

−→

0

−→

C • (Gv , XU )[−1]  o y

v∈Sf

L

L

v∈Sf

v 0 |v

L

−→

C • (GK,S , XU ) −→

k

C • (Gv , Z[Gv /Gv0 ] ⊗Z X)[−1] −→  sh[−1] y

v 0 ∈Sf0

Cc• (GK,S , XU ) −→

C • (Gv0 , X)[−1]

−→

0

k

Cc• (GK,S , XU ) −→  sh y c

C • (GK,S , XU ) −→   ysh

0

Cc• (GK 0 ,S 0 , X) −→

C • (GK 0 ,S 0 , X) −→ 0

(8.5.3.3) For each g ∈ GK,S , 8.1.7.3 gives a morphism Ad(g)c = (Ad(g), F (g), m)[−1] : Cc• (GK 0 ,S 0 , X) −→ Cc• (GK 0 ,S 0 , X), which is functorial in X and makes the following diagrams commutative up to homotopy: 0 −→

0 −→

L

v 0 ∈Sf0

L

v 0 ∈Sf0

C • (Gv0 , X)[−1] −→  F (g)[−1] y

Cc• (GK 0 ,S 0 , X) −→  Ad(g) c y

C • (GK 0 ,S 0 , X) −→  Ad(g) y

0

C • (Gv0 , X)[−1] −→

Cc• (GK 0 ,S 0 , X) −→

C • (GK 0 ,S 0 , X) −→

0

sh

Cc• (GK 0 ,S 0 , X)  Ad(g) c y

sh

Cc• (GK 0 ,S 0 , X)

c Cc• (GK,S , XU ) −−→  Ad(gU ) ∗ y c Cc• (GK,S , XU ) −−→

(8.5.3.4) If K ⊂ K 0 ⊂ K 00 are two finite Galois subextensions of KS /K, 8.1.7.4-5 give restriction and corestriction morphisms of complexes resc : Cc• (GK 0 ,S 0 , X) −→ Cc• (GK 00 ,S 00 , X) corc : Cc• (GK 00 ,S 00 , X) −→ Cc• (GK 0 ,S 0 , X), which are functorial in X and make the following diagrams commutative up to homotopy: 0

−→

0

−→

L

v 0 ∈Sf0

L

v 00 ∈Sf00

C • (Gv0 , X)[−1]  res[−1] y

−→

C • (Gv00 , X)[−1] −→

Cc• (GK 0 ,S 0 , X)  resc y

−→

Cc• (GK 00 ,S 00 , X) −→ 171

C • (GK 0 ,S 0 , X)  res y

−→

0

C • (GK 00 ,S 00 , X) −→

0,

0 −→

0 −→

L

v 00 ∈Sf00

L

v 0 ∈Sf0

Cc• (GK,S , XU )  Tr y ∗

C • (Gv00 , X)[−1] −→  cor[−1] y

Cc• (GK 00 ,S 00 , X) −→  corc y

C • (GK 00 ,S 00 , X) −→  cor y

0

C • (Gv0 , X)[−1]

Cc• (GK 0 ,S 0 , X)

C • (GK 0 ,S 0 , X)

0

sh

c −−→

−→

Cc• (GK 0 ,S 0 , X)  res y c

sh

c Cc• (GK,S , XV ) −−→ Cc• (GK 00 ,S 00 , X),

−→

−→

sh

Cc• (GK 00 ,S 00 , X)  cor y c

sh

Cc• (GK 0 ,S 0 , X)

Cc• (GK,S , XV )  pr∗ y

c −−→

Cc• (GK,S , XU )

c −−→

(8.5.3.5) As all morphisms in 8.5.3.2-5 are functorial in X, they induce morphisms valid for arbitrary ind-admissible R[GK,S ]-modules X (more generally, for complexes of such modules). ind−ad (8.5.4) For every bounded below complex T (resp., M ) in (ad R[GK,S ] Mod)R−f t (resp., in (R[GK,S ] Mod){m} ), we denote • • Cc,Iw (K∞ /K, T ) = Cc,cont (GK,S , FΓ (T )) • Cc• (KS /K∞ , M ) = Cc,cont (GK,S , FΓ (M )).

The corresponding objects of D+ (R Mod) will be denoted by RΓc,Iw (K∞ /K, T ) (resp., RΓc (KS /K∞ , M )) i and their cohomology by Hc,Iw (K∞ /K, T ) (resp., Hci (KS /K∞ , M )). This is in line with the notation • RΓIw (K∞ /K, T ) (from 8.5.1) for the object of D(R Mod) represented by Ccont (GK,S , FΓ (T )). Similarly, + • let RΓ(KS /K∞ , M ) be the object of D (R Mod) represented by Ccont (GK,S , FΓ (M )) and H i (KS /K∞ , M ) its cohomology. As in 8.5.1, the notation for RΓc,Iw is slightly ambiguous, since S does not appear explicitly. (8.5.5) Proposition. (i) For every bounded below complex M in (ind−ad R[GK,S ] Mod){m} , the morphisms shc induce isomorphisms of R-modules ∼

∼

∼

i i i i 0 0 lim lim − → Hc,cont (GK ,S , M ) ←− U,Tr −→∗ Hc,cont (GK,S , MU ) −→ lim −U → Hc,cont (GK,S , U M ) −→ Hc,cont (KS /K∞ , M ). res

(ii) For every bounded below complex T in (ad R[GK,S ] Mod)R−f t , the canonical morphism of complexes ∼

• • Cc,cont (G, FΓ (M )) −→ lim ←− Cc,cont (G, MU )

U

is an isomorphism, and the morphisms shc induce isomorphisms of R-modules ∼

∼

i i i 0 0 lim lim ← − Hc,cont (GK ,S , T ) ←− U,pr ←−∗ Hc,cont (GK,S , TU ) −→ Hc,Iw (K∞ /K, T ). cor

Above, K 0 /K runs through finite subextensions of K∞ /K and U = Gal(K∞ /K 0 ). • i Proof. This follows from 8.5.3.2-4 and the corresponding statements for Ccont (G, −) and Hcont (G, −) (with G = GK,S , Gv ), proved in Lemma 8.3.2, Proposition 8.3.5 and Proposition 8.4.4.2.

b ad ∗ b ad (8.5.6) Theorem. If T, T ∗ ∈ DR−f t (R[GK,S ] Mod), A, A ∈ DR−cof t (R[GK,S ] Mod) are related by the duality diagram D T ←−−−−−−−−→ T ∗  ←−   −− D −−→  −−−−− Φ Φ   −−−−− − y −→ y −− − ← A A∗

172

over R, then

D

RΓIw (K∞ /K, T ) ←−−−−−−−−→ RΓc,Iw (K∞ /K, T ∗(1))ι [3] ←−     −− D −−→   −−−−− Φ Φ − − − −− − y y −→ −− − ← RΓ(KS /K∞ , A) RΓc (KS /K∞ , A∗ (1))ι [3] b are related in the same way in D(co)f t (R Mod) and there are spectral sequences 3−j i+j E2i,j = ExtiR (D(H j (KS /K∞ , A)), ωR ) = ExtiR (Hc,Iw (K∞ /K, T ∗ (1)), ωR )ι =⇒ HIw (K∞ /K, T ) 0

3−j i+j E2i,j = ExtiR (D(Hcj (KS /K∞ , A)), ωR ) = ExtiR (HIw (K∞ /K, T ∗ (1)), ωR )ι =⇒ Hc,Iw (K∞ /K, T ).

Proof. By Proposition 3.2.8 we can assume that T, T ∗ (resp., A, A∗ ) are represented by bounded complexes of admissible R[GK,S ]-modules, of finite (resp., co-finite) type over R. The statement then follows from Theorem 5.4.5 applied to the duality diagram 8.4.6.6, if we take into account 8.3.2, 8.3.5, 8.4.4.2 and 8.5.5. (8.5.7) The following fact was implicitly used in 8.5.6: for every X ∈ D(R Mod) and i ∈ Z there is a canonical isomorphism ∼

ExtiR (X, ωR )ι −→ ExtiR (X ι , ωR ). ∼

ι

This follows from the fact that the isomorphism ι : R −→ R of R-modules induces isomorphisms in D(R Mod) ∼

ι

ωR = ω ⊗R R −→ ω ⊗R R = (ωR )ι and ∼

D(X)ι −→ D(X ι ). (8.5.8) Over each finite subextension Kα /K of K∞ /K, the pairing ev2 : T ⊗R T ∗ (1) −→ ωR (1) induces cup products (5.4.2.1) j i h , iα : Hc,cont (GKα ,Sα , T ) ⊗R Hcont (GKα ,Sα , T ∗ (1)) −→ H i+j−3 (ωR )

(where Sα denotes the set of primes of Kα above S). The pairing ev2 : FΓ (T ) ⊗R FΓ (T ∗ (1))ι −→ ωR (1) (cf. 8.4.6.3) induces, in turn, products on the projective limits    ι j i ∗ h , i : lim Hc,cont (GKα ,Sα , T ) ⊗R lim Hcont (GKα ,Sα , T (1)) −→ H i+j−3 (ωR ) = ← ← α− α−
= H i+j−3 (ωR ) ⊗R R = lim H i+j−3 (ωR ) ⊗R R[Gal(Kα /K)] ← − α (and similarly for the products in which the roles of cohomology and cohomology with compact support are interchanged). 173

(8.5.9) Proposition. In the situation of 8.5.8, 



X

h(xα ), (yα )i = 

hxα , σyα iα ⊗ [σ]

σ∈Gal(Kα /K)

α

j (where σ acts on Hcont (GKα ,Sα , T ∗ (1)) by conjugation).

Proof. This follows from the proof of the corresponding local statement 8.11.10 and the commutative diagrams 8.5.3.3-4. (8.6) Local conditions and Shapiro’s Lemma Let K and S be as in 8.5. (8.6.1) Assume that K ⊂ K 0 ⊂ K 00 are finite Galois subextensions of KS /K. For each v ∈ Sf , the fixed embedding K ,→ K v induces primes v00 and v000 of K 0 and K 00 , respectively, with completions Kv ⊂ Kv0 0 ⊂ 0

Kv0000 ⊂ K v and absolute Galois groups 0

G = Gal(K v /Kv ) ⊃ U = Gal(K v /Kv0 0 ) ⊃ V = Gal(K v /Kv0000 ). 0

0

Both U and V are normal in G. As in 8.5.3, we have an injective homomorphism α : G ,→ G = Gal(K/K) such that U = α−1 (U ) and V = α−1 (V ), where U = Gal(K/K 0 ) ⊃ V = Gal(K/K 00 ). (8.6.2) Let X = X • be a complex of ind-admissible R[G]-modules. We shall abuse the notation and write X for α∗ X, i.e. for the same object, but viewed as a complex of R[G]-modules via the map α : G ,→ G. As in 8.2, we let XU = X ⊗R R[G/U ] (and similarly for XV ). Assume that we are given the following local conditions (i.e. morphisms of complexes of R-modules) + • i+ v (XU ) : Uv (XU ) −→ Ccont (G, XU ) + • i+ v (XV ) : Uv (XV ) −→ Ccont (G, XV ) + • i+ v 0 (X) : Uv 0 (X) −→ Ccont (U , X) 0

0

i+ v000 (X)

:

Uv+00 (X) 0

• −→ Ccont (V , X)

together with the following data: (8.6.2.1) Morphisms of complexes sh+ : Uv+ (XU ) −→ Uv+0 (X) 0

sh+ : Uv+ (XV ) −→ Uv+00 (X), 0

which are quasi-isomorphisms and make the diagram i+

Uv+ (XU )   + ysh

v −−→

Uv+0 (X)

0 −−→

0

i+0 v

• Ccont (G, XU )   ysh • Ccont (U , X)

(and its variant with U replaced by V and v00 by v000 ) commutative. (8.6.2.2) For each g ∈ G a morphism of complexes Ad+ (g) : Uv+0 (X) −→ Uv+0 (X) 0

0

such that the faces of the following cubic diagram 174

i+ v

/ C • (G, X ) cont HHU HH HH HH HHsh Ad(gU)∗ HH HH -5 HH 0 HH + i 0 $ v 0 / C • (U , X) cont

Uv+ (XU ) BB BB BB BB sh+ BB BB BB BB ! Uv+0 (X) Ad(gU )∗ 0

Ad+ (g)

0

/7

 / C • (G, X ) 3; Ad(g) cont HHU HH HHsh HH -5 k2 HH HH m HH -5 HH 0 HH + i 0 $  v 0 / C • (U , X) cont

 i+ v Uv+ (XU ) BB BB BB BB k1 .6 BB B + BB sh BB B!  Uv+0 (X) 0

commute up to the indicated homotopies and the boundary of the cube is trivialized by a 2-homotopy + + i+ v 0 ? k1 + m ? sh − k2 ? iv − 0

0.

(8.6.2.3) A morphism of complexes res+ : Uv+0 (X) −→ Uv+00 (X) 0

0

such that each face of the following cubic diagram is commutative: i+ v

/ C • (G, X ) cont HHU HH HH HH HHsh Tr∗ HH HH HH HH + i 0 $ v 0 / C • (U , X) cont

Uv+ (XU ) BB BB BB BB sh+ BB BB BB BB ! Uv+0 (X) Tr∗ 0

res+

 Uv+ (XV ) BB BB BB BB sh+ BB BB BB BB !  Uv+00 (X)

i+ v

0

 / C • (G, X ) res cont HHV HH HH HH HHsh HH HH HH HH + i 00 $  v 0 / C • (V , X) cont

(8.6.2.4) A morphism of complexes cor+ : Uv+00 (X) −→ Uv+0 (X) 0

0

175

such that the faces of the cubic diagram i+ v

/ C • (G, X ) cont HHV HH HH HH HHsh pr∗ HH HH -5 HH 0 HH + i 00 $ v 0 / C • (V , X) cont

Uv+ (XV ) BB BB BB BB sh+ BB BB BB BB ! pr∗ Uv+00 (X) 0

cor +

/7

0

 / C • (G, X ) 3; cor cont HHU HH HHsh HH -5 k2 HH HH m HH -5 HH 0 HH + i 0 $  v 0 / C • (U , X) cont

 i+ v Uv+ (XU ) BB BB BB BB k1 .6 BB B + BB sh BB B!  Uv+0 (X) 0

S (in which the vertical arrow cor is defined using fixed coset representatives U = V αi ) are commutative up to the indicated homotopies and the boundary of the cube is trivialized by a 2-homotopy + + i+ v 0 ? k1 + m ? sh − k2 ? iv −

0.

0

(8.6.3) Fixing coset representatives σi , τj ∈ G of [ [ G= U σi α(G) = V τj α(G) i

j

as in 8.1.7 (with σ0 = τ0 = 1), we obtain from 8.5.3 isomorphisms of complexes M M M ∼ ∼ • • • Ccont (Gv0 , X) −→ Ccont (U , α∗ X) −→ Ccont (U , α∗i X) i

v 0 |v

M

∼

Ccont (Gv00 , X) −→ •

M

i

∗

∼

Ccont (V , α X) −→ •

M

j

v 00 |v

• Ccont (V , βj∗ X)

j

(where v 0 ∈ S 0 , v 00 ∈ S 00 ). We define the local condition for X at each v 0 = σi (v00 ) to be the composition of + • • ∗ i+ v 0 : Uv 0 (X) −→ Ccont (U , X) = Ccont (U , α X) 0

0

with the inverse of the isomorphism (8.5.3.1.1) ∼

• • Ccont (Gv0 , X) −→ Ccont (U , α∗ X)

(and similarly for the local condition at each v 00 = τj (v000 )). The isomorphism M ∼ ω eU : α∗ (XU ) −→ (α∗ X)U i

(resp., its analogue ω eV for XV ) together with (resp., XV ).

i+ v (XU ) 176

(resp., i+ v (XV )) define local conditions at v for XU

(8.6.4) Putting together the diagrams from 8.1.7.2-5 (modified in accordance with 8.1.7.6) and 8.6.2.1-4, we obtain the following generalizations of 8.5.3: (8.6.4.1) A quasi-isomorphism ef• (GK,S , XU ; ∆(XU )) −→ C ef• (GK 0 ,S 0 , X; ∆(X)), shf : C which is functorial in X, the homotopy class of which is independent of any choices, and such that the following diagram with exact columns is commutative up to homotopy: 0   y

L v∈Sf

• Ccont (Gv , XU )[−1]   y

e• (GK,S , XU ; ∆(XU )) C f   y L • Ccont (GK,S , XU ) ⊕ v∈Sf Uv+ (XU )   y

L

sh[−1]

−−→ shf

−−→ (sh,sh+ )

−−−−→

0   y

v 0 ∈Sf0

• Ccont (Gv0 , X)[−1]   y

e• (GK 0 ,S 0 , X; ∆(X)) C f   y L • Ccont (GK 0 ,S 0 , X) ⊕ v0 ∈S 0 Uv+0 (X) f   y

0

0

(8.6.4.2) For each g ∈ GK,S a morphism of complexes e• (GK 0 ,S 0 , X; ∆(X)) −→ C e• (GK 0 ,S 0 , X; ∆(X)), Ad(g)f : C f f which is functorial in X and makes the following diagrams commutative up to homotopy:

L

v 0 ∈Sf0

0   y • Ccont (Gv0 , X)[−1]   y

L

F (g)[−1]

−−−−→

v 0 ∈Sf0

Ad(g)f e • (GK 0 ,S 0 , X; ∆(X)) C −−−−→ f   y L (Ad(g),Ad+ (g)) • Ccont (GK 0 ,S 0 , X) ⊕ v0 ∈S 0 Uv+0 (X) −−−−−−−−→ f   y

0

0   y • Ccont (Gv0 , X)[−1]   y

e• (GK 0 ,S 0 , X; ∆(X)) C f   y L • Ccont (GK 0 ,S 0 , X) ⊕ v0 ∈S 0 Uv+0 (X) f   y 0

e • (GK,S , XU ; ∆(XU )) C f  Ad(gU ) ∗ y

−−→

shf

e• (GK 0 ,S 0 , X; ∆(X)) C f  Ad(g) f y

e • (GK,S , XU ; ∆(XU )) C f

−−→

shf

e• (GK 0 ,S 0 , X; ∆(X)) C f

(8.6.4.3) Morphisms of complexes e • (GK 0 ,S 0 , X; ∆(X)) −→ C e • (GK 00 ,S 00 , X; ∆(X)) resf : C f f e• (GK 00 ,S 00 , X; ∆(X)) −→ C e • (GK 0 ,S 0 , X; ∆(X)), corf : C f f 177

which are functorial in X and make the following diagrams commutative up to homotopy:

L

0   y

v 0 ∈Sf0

• Ccont (Gv0 , X)[−1]   y

0   y

L

res[−1]

−−−−→

v 00 ∈Sf00

resf e • (GK 0 ,S 0 , X; ∆(X)) C −−−−→ f   y L (res,res+ ) • Ccont (GK 0 ,S 0 , X) ⊕ v0 ∈S 0 Uv+0 (X) −−−−−−→ f   y

• Ccont (Gv00 , X)[−1]   y

e• (GK 00 ,S 00 , X; ∆(X)) C f   y L • Ccont (GK 00 ,S 00 , X) ⊕ v00 ∈S 00 Uv+00 (X) f   y

0 e • (GK,S , XU ; ∆(XU )) C f  Tr y ∗

0 shf

−−→

shf e • (GK,S , XV ; ∆(XV )) −−→ C f

L

e• (GK 0 ,S 0 , X; ∆(X)) C f  resf y e • (GK 00 ,S 00 , X; ∆(X)) C f

0   y v 00 ∈Sf00

• Ccont (Gv00 , X)[−1]   y

L

cor[−1]

−−−−→

v 0 ∈Sf0

corf e• (GK 00 ,S 00 , X; ∆(X)) C −−−−→ f   y L (cor,cor + ) • Ccont (GK 00 ,S 00 , X) ⊕ v00 ∈S 00 Uv+00 (X) −−−−−−→ f   y

0

• Ccont (Gv0 , X)[−1]   y

e• (GK 0 ,S 0 , X; ∆(X)) C f   y L • Ccont (GK 0 ,S 0 , X) ⊕ v0 ∈S 0 Uv+0 (X) f   y 0

shf e • (GK,S , XV ; ∆(XV )) −−→ C f  pr∗ y

e • (GK,S , XU ; ∆(XU )) C f

0   y

shf

−−→

e • (GK 00 ,S 00 , X; ∆(X)) C f  corf y e • (GK 0 ,S 0 , X; ∆(X)). C f

e• (GK 0 ,S 0 , X; ∆(X)) is homotopic to (8.6.4.4) Lemma. (i) For g, g 0 ∈ GK,S , the action of Ad(gg 0 )f on C f 0 Ad(g)f ◦ Ad(g )f . If g ∈ GK 0 ,S 0 ⊂ GK,S , then Ad(g)f is homotopic to the identity. e• (GK 0 ,S 0 , X; ∆(X)). (ii) corf ◦ resf is homotopic to [K 00 : K 0 ] · id on C f 00 0 e • (GK 00 ,S 00 , X; ∆(X)) is homotopic (iii)P If K /K is a Galois extension, then the action of resf ◦ corf on C f to σ∈Gal(K 00 /K 0 ) Ad(e σ )f , where σ e ∈ GK 0 ,S 0 is any lift of σ. (iv) If K 00 /K 0 is a Galois extension of degree d = [K 00 : K 0 ], then resf induces isomorphisms  Gal(K 00 /K 0 ) ∼ e i (GK 0 ,S 0 , X; ∆(X)) ⊗R R[1/d] −→ e i (GK 00 ,S 00 , X; ∆(X)) ⊗R R[1/d] H H . f f 178

Proof. (i) This follows from the last diagram in 8.6.4.2 and equalities Ad(gg 0 U ) = Ad(gU )Ad(g 0 U ), Ad(U ) = id. (ii), (iii) P Combine the third and the last diagram in 8.6.4.3 with the formulas pr ◦ Tr = [U : V ] · id, Tr ◦ pr = σV ∈U/V Ad(σV ). (iv) This follows from (ii) and (iii). (8.7) Functoriality of the unramified local conditions Before proceeding further we must clarify various functorial properies of the objects studied in Chapter 7. Fix v ∈ Sf not dividing p and consider the group G = Gv = hti o hf i introduced in 7.2.1 (i.e. tf = f tL , where L = `r is a power of the residue characteristic ` 6= p of v); in order to emphasize dependence on f and t, we include them in the notation. Throughout Sect. 8.7, M will be a p-primary torsion discrete G-module and M • a bounded below complex of such modules. (8.7.1) Dependence on f, t. Any change of topological generators of G is given by f 0 = f tA ,

b ` , B ∈ (Z/Z b ` )∗ ). (A ∈ Z/Z

t 0 = tB

If A ∈ N0 and B ∈ N, then the formulas α2 (m) = (1 + t + · · · + tB−1 )m

α0 (m) = m,

α1 (m, m0 ) = (m + f (1 + t + · · · + tA−1 )m0 , (1 + t + · · · + tB−1 )m0 ) define a morphism of complexes α = α(f, t; f 0 , t0 ) : C(M, f, t) −→ C(M, f 0 , t0 ) (where C(M, f, t) = C(M ) was defined in 7.2.1) such that the composition µf 0 ,t0

α

C(M, f, t)−−→C(M, f 0 , t0 )−−→C • (G, M ) is equal to µf,t (defined in 7.2.4). For general A, B one can define α by continuity, or by using the pro-finite differential calculus from 7.4. The sign conventions of 7.6 yield morphisms of complexes ∼

α

∼

α : U (M • , f, t) −→ C(M • , f, t)−→C(M • , f 0 , t0 ) −→ U (M • , f 0 , t0 ). The morphisms α are transitive in the following sense. If 0

f 00 = f 0 (t0 )A , 00

t00 = (t0 )B

0

00

(hence f 00 = f tA , t00 = tB with A00 = A + BA0 , B 00 = BB 0 ), then the matrix equality 1

0

f 0 (1 + t0 + · · · + (t0 )A −1 )

0

1 + t0 + · · · + (t0 )B

0

−1

!

1

f (1 + t + · · · + tA−1 )

0

1 + t + · · · + tB−1

! =

1 f (1 + t + · · · + tA 0

1 + t + · · · + tB

00

00

−1

)

!

−1

implies that α(f 0 , t0 ; f 00 , t00 ) ◦ α(f, t; f 0 , t0 ) = α(f, t; f 00 , t00 ). In particular, each α(f, t; f 0 , t0 ) is an isomorphism of complexes. (8.7.2) Restriction. Let U = htB i o hf A i, where A, B ∈ N and (L, B) = 1. This is an open subgroup of 0 G with a similar structure: f 0 = f A and t0 = tB satisfy t0 f 0 = f 0 (t0 )L with L0 = LA . The formulas res0 (m) = m,

res1 (m, m0 ) = ((1 + f + · · · + f A−1 )m, (1 + t + · · · + tB−1 )m0 ), res2 (m) = (1 + t + · · · + tB−1 )(1 + θ + · · · + θA−1 )m 179

define a morphism of complexes res : C(M, f, t) −→ C(M, f A , tB ) making the following diagram commutative: C(M, f, t)  res y

µf,t

−−−−→ µf A ,tB

C(M, f A , tB ) −−−−→

C • (G, M )  res y C • (U, M ).

(8.7.3) Restriction - unramified case. Assume that B = 1, i.e. U = hti o hf A i U = ht0 i o hf 0 i is an open normal subgroup of G with f 0 = f A,

t0 = t,

t0 f 0 = f 0 (t0 )L

0

(A ∈ N). In this case

(L0 = LA ),

0

θ0 = f 0 (1 + t0 + · · · + (t0 )L −1 ) = θA . The formula res = (id, id, 1 + f + · · · + f A−1 , 1 + θ + · · · + θA−1 ) defines a morphism of complexes res : U (M • , f, t) −→ U (M • , f A , t), which maps the subcomplex U + (M • , f, t) to U + (M • , f A , t), reduces to the map res from 8.7.2 if M • = M , and makes the following diagram commutative: U (M • , f, t)  res y

µf,t

−−→ µf A ,t

U (M • , f A , t) −−→

C • (G, M • )  res y C • (U, M • ).

(8.7.4) Corestriction - unramified case. Under the assumptions of 8.7.3, the formula cor = (1 + f + · · · + f A−1 , 1 + θ + · · · + θA−1 , id, id) defines a morphism of complexes cor : U (M • , f A , t) −→ U (M • , f, t), which maps the subcomplex U + (M • , f A , t) to U + (M • , f, t). A short calculation (based on the formulas in Proposition 7.2.4(i) and its proof) shows that the diagram µf A ,t

U (M • , f A , t) −−→  cor y U (M • , f, t)

µf,t

−−→

C • (U, M • )  cor y C • (G, M • )

commutes, where the right vertical arrow cor is defined using the coset representatives αi = f −i (0 ≤ i < A). (8.7.5) Shapiro’s Lemma - unramified case. Still keeping the notation of 8.7.3, we define a morphism of complexes sh : U (MU• , f, t) −→ U (M • , f A , t) 180

to be the composite map (δU )∗

res

U (MU• , f, t)−−→U (MU• , f A , t)−−→U (M • , f A , t), where δU was defined in 8.1.3. As µ commutes with res and is functorial in M • , it follows from 8.1.3 that the following diagram is commutative: U (MU• , f, t)   ysh

µf,t

−−→ µf A ,t

U (M • , f A , t) −−→

C • (G, MU• )   ysh C • (U, M • ).

We know that both horizontal arrows and the right vertical arrow are quasi-isomorphisms; thus the left vertical arrow is also a Qis. (8.7.6) Shapiro’s Lemma for U + - unramified case. The restriction of the morphism sh from 8.7.5 to the subcomplex U + (MU• , f, t) defines a morphism of complexes (δU )∗

res

sh+ : U + (MU• , f, t)−−→U + (MU• , f A , t)−−→U + (M • , f A , t). We claim that sh+ is also a quasi-isomorphism. By d´evissage and (7.6.3.1), it is enough to consider the following two cases: (i) M • = τ≥−1 M • , in which case U + (M • ) = U (M • ) and we can apply 8.7.5. (ii) M • = M 0 = M : putting N = M t=1 , we have (MU )t=1 = NU , and it remains to check that the composite morphism f −1

(id,1+f +···+f A−1 )

f A −1

f A −1

(δU ,δU )

g : [NU −−→NU ] −−−−−−−−−−→ [NU −−→NU ] −−−−→ [NU −−→NU ] is a quasi-isomorphism. This follows from the fact that the quasi-isomorphism µ from 7.2.2 (with respect to σ = f ) maps g to (δU )∗

res

sh : C • (hf i, NU )−−→C • (hf A i, NU )−−→C • (hf A i, NU ). (8.7.7) Conjugation - unramified case. For each g ∈ G, the morphism of complexes µf A ,t

Ad(g)

λf A ,t

Ad(g) : U (M • , f A , t)−−→C • (U, M • )−−→C • (U, M • )−−→U (M • , f A , t) maps the subcomplex U + (M • , f A , t) to itself. (8.7.8) As all morphisms in 8.7.1-7 are functorial in M • , they extend, by the usual limit procedure, to the case when M • is a complex of ind-admissible R[Gv ]-modules. (8.8) Greenberg’s local conditions in Iwasawa theory (8.8.1) Let Σ ⊂ Sf and Σ0 = Sf − Σ be as in 7.8.1, and K∞ /K as in 8.5.1. Throughout sections 8.8-8.9 we assume that the following condition holds: (U ) Each prime v ∈ Σ0 is unramified in K∞ /K. This condition is automatically satisfied if ∆ = 0. As in 7.8, fix fv and tv for each v ∈ Σ0 . ind−ad (8.8.2) Let T (resp., M ) be a bounded below complex in (ad R[GK,S ] Mod)R−f t (resp., in (R[GK,S ] Mod){m} ). + + Assume that we are given, for each v ∈ Σ, a bounded below complex Tv (resp., Mv ) in (ad R[Gv ] Mod)R−f t + + (resp., in (ind−ad R[Gv ] Mod){m} ) and a morphism of complexes of R[Gv ]-modules Tv −→ T (resp., Mv −→ M ).

(8.8.3) The data from 8.8.2 define Greenberg’s local conditions ( • • Ccont (Gv , Zv+ ) −→ Ccont (Gv , Z), ∆v (Z) = • • Cur (Gv , Z) −→ Ccont (Gv , Z), 181

(v ∈ Σ) (v ∈ Σ0 )

for Z = T, M, TU , MU , FΓ (T ), FΓ (M ) (where U is an open subgroup of G = GK ). Recall that we have fixed an embedding K ,→ K v , for each v ∈ Sf . If K 0 /K is a finite subextension of K∞ /K and v00 |v the prime of K 0 induced by K ,→ K v , then Gv00 ⊂ Gv and we can define Greenberg’s local conditions at v00 for Z = T, M by ( • • Ccont (Gv00 , Zv+ ) −→ Ccont (Gv00 , Z), (v ∈ Σ) ∆v00 (Z) = • • Cur (Gv00 , Z) −→ Ccont (Gv00 , Z), (v ∈ Σ0 ). (8.8.4) We must check that the local conditions for Z = T, M defined in 8.8.3 satisfy the properties 8.6.2.1-4 (for K ⊂ K 0 ⊂ K 00 ⊂ K∞ ). (8.8.4.1) This is straightforward for v ∈ Σ: in this case Ad+ (g) = Ad(g), sh+ = sh, res+ = Tr∗ , cor+ = pr∗ , all homotopies in 8.6.2.4 are zero, while in 8.6.2.2 we have m = 0 and k1 , k2 are as in 8.1.6.3 (bi-functorial). Finally, the boundaries of the cubes in 8.6.2.2 and 8.6.2.4 are both trivialized by the zero 2-homotopy, as + + i+ v 0 ? k1 + m ? sh − k2 ? iv = 0 0

in both cases. (8.8.4.2) Assume that v ∈ Σ0 and write f = fv , t = tv . Let X = X • be a complex in (ind−ad R[Gv ] Mod). It will be enough to verify 8.6.2.1-4 in the case when X is a complex of discrete p-primary torsion Gv -modules, provided that all morphisms, homotopies and 2-homotopies are functorial in X. As in 7.5.7, we can assume that the wild inertia Ivw acts trivially on X; it is then sufficient to consider a variant of 8.6.2.1-4, in which G ⊃ U ⊃ V are replaced by their quotients by Ivw . We shall abuse the notation and denote these quotients again by G, U , V . Note that the assumption (U ) implies that U = hti o hf A i,

G = hti o hf i,

0

V = hti o hf AA i

for some A, A0 ∈ N. The morphisms sh+ (resp., res+ )) satisfying 8.6.2.1 (resp., 8.6.2.3) were defined in 8.7.6 (resp., 8.7.3). (8.8.4.3) Conjugation. For g ∈ G the morphism Ad+ (g) = Ad(g) : U + (X, f A , t) −→ U + (X, f A , t) was defined in 8.7.7. As the inclusion U + (X, f A , t) ,→ U (X, f A , t) commutes with Ad(g) and sh (and everything in sight is functorial in X), it is enough to consider the following cubic diagram, which involves U (X) instead of U + (X) µf,t

U (XU , f, t) GG GG GG GG GGsh GG GG GG GG # Ad(gU)∗ U (X, f A , t)

/ C • (G, X ) cont IIU II II II IIsh Ad(gU )∗ II II ,4 II 0 II $ µf A ,t / C • (U , X) cont

Ad(g)

 µf,t U (XU , f, t) GG GG GG GG -5 GG k1 GG sh GG GG GG #  U (X, f A , t)

0

.6

 / C • (G, X ) 2: Ad(g) cont IIU II IIsh II ,4 k2 II II m II ,4 II 0 II $  µf A ,t / C • (U , X) cont 182

(and also check that k1 preserves U + (−)). The faces of the cube are commutative up to the following homotopies: m = b ? (Ad(g) ◦ µf A ,t )

(where b : µf A ,t ◦ λf A ,t −

k1 = (λf A ,t ◦ sh) ? hg ? (µf,t ◦ Ad(gU )∗ )

(as Ad(g) ◦ sh = λf A ,t ◦ Ad(g) ◦ µf,t ◦ sh) (as in 8.1.6.3).

k2 = (sh ◦ Ad(gU )∗ ) ? hg

id is as in 7.4.9)

We now check that the homotopy k1 maps U + (XU , f, t)i ⊂ U (XU , f, t)i to U + (XU , f A , t)i−1 ⊂ U (XU , f A , t)i−1 . As σ≥2 U + (−) = 0 and σ≤−1 U + (−) = σ≤−1 U (−), we only have to consider the case i = 1. However, the inclusion k1 (U + (−)1 ) ⊆ U + (−)0 follows from dk1 (U + (−)1 ) = (dk1 + k1 d)U + (−)1 ⊆ U + (−)0 . Finally, it remains to verify that the homotopy h := µf A ,t ? k1 − k2 ? µf,t + m ? sh is 2-homotopic to zero. We have 2-homotopies µf A ,t ? k1 − k2 ? µf,t = ((µf A ,t ◦ λf A ,t − id) ◦ sh) ? hg ? (Ad(gU)∗ ◦ µf,t ) = = −((db + bd) ◦ sh) ? hg ? (Ad(gU )∗ ◦ µf,t ) −

−b ? (d(sh ? hg ) + (sh ? hg )d) ◦ Ad(gU )∗ ◦ µf,t =

b ? (sh ◦ (id − Ad(g)) ◦ Ad(gU )∗ ◦ µf,t ), hence h−

b ? (sh ◦ Ad(gU )∗ ◦ µf,t ) = b ? (µf A ,t ◦ sh ◦ Ad(gU )∗ ),

which is 2-homotopic to zero, by Lemma 7.4.9. This completes the verification of 8.6.2.2. (8.8.4.4) Corestriction. In order to verify 8.6.2.4, it is enough, as in 8.8.4.3, to consider the following 0 cubic diagram (in which f 0 = f A , f 00 = f AA , µ = µf,t , µ0 = µf 0 ,t , µ00 = µf 00 ,t ) µ

U (XV , f, t) GG GG GG GG GGsh GG GG GG GG # pr∗ U (X, f 00 , t)

/ C • (G, X ) cont IIV II II II IIsh pr∗ II II -5 II 0 II $ 00 µ / C • (V , X) cont

cor

 µ U (XU , f, t) GG GG GG GG -5 GG k1 G G sh GG GG GG #  U (X, f 0 , t)

0

.6

 / C • (G, X ) 2: cor cont IIU II IIsh II ,4 k2 II II m II -5 II 0 II $  0 µ / C • (U , X), cont 183

in which the morphism • • cor : Ccont (V , X) −→ Ccont (U , X)

is defined in terms of the coset representatives (f 00 )−i , 0 ≤ i < A0 . We shall give the details only in the case when G = U (i.e. A = 1, f = f 0 ), and leave the general case to the reader. The faces of the cube are commutative up to the following homotopies: m=0

(by 8.7.4)

k1 = h ? res k2 = defined as in 8.1.5

(with h defined below)

In order to define k1 , it is enough to consider the case when X is concentrated in degree zero. The first step is to construct a homotopy h : (δU )∗ ◦ pr∗ −

cor ◦ (δV )∗ ◦ res

between (δ )∗

pr∗

U C(XV , f 0 , t)−−→C(XU , f 0 , t)−−→C(X, f 0 , t)

and (δ )∗

res

cor

V C(XV , f 0 , t)−−→C(XV , f 00 , t)−−→C(X, f 00 , t)−−→C(X, f 0 , t).

Writing an arbitrary element of XV as x=

0 A −1 X

[(f 0 )j V ] ⊗ xj ,

xj = δ(f 0 )j V (x) ∈ X,

j=0

the homotopy h is given by the formulas

0

0

h (x, x ) =

0 A −1 X

0

0

0 j−1

δ(f 0 )j V (1 + f + · · · + (f )

−1
AX )x = (xj + f 0 xj−1 + · · · + (f 0 )j−1 x1 )

j=0

j=0

h1 (x) = (0,

0 A −1 X

(1 + θ0 + · · · + (θ0 )j−1 )xA0 −j )

j=0

(note that we are considering only the special case when f 0 = f , hence θ0 = θ). Having constructed h, we define k1 = h ? res : sh ◦ pr∗ −

cor ◦ sh,

where res : C(XV , f, t) −→ C(XV , f 0 , t) was defined in 8.7.3-4. The same argument as in 8.8.4.3 shows that k1 maps U + (−)i to U + (−)i−1 . We claim that we can take zero for the 2-homotopy in 8.6.2.4, i.e. µ0 ? k1 − k2 ? µ = 0. Indeed, in degree 1, 184

k20 (c)

=

0 A −1 X


δf j V c(f j ) ,

j=0 0

hence we have (for x, x ∈ XV ) k20

0

◦ µ1 (x, x ) =

0 A −1 X


δf j V (1 + f + · · · + f j−1 )x = h0 (x, x0 ).

j=0

In degree 2, let g = f a tb with a, b ∈ N0 . Then


k21 (z)

(g) =

0 A −1 X

  δf i V z(f i , f −i g(f −i g)−1 ) − z(g, (f −i g)−1 ) ,

i=0

where (f −i g)−1 = f j(i) ,

0 ≤ j(i) < A0 ,

j(i) ≡ i − a (mod A0 ).

If z = µ2 (x) (for x ∈ XV ), then the formulas in 7.2.4(i) give

k21

0 A −1   X
a b ◦ µ2 (x) (f t ) = δf i V f a (1 + t + · · · + tb−1 )(1 + θ + · · · + θj(i)−1 )x =

i=0 a

= f (1 + t + · · · + t

b−1

)

0 A −1 X


(1 + θ + · · · + θj−1 )xA0 −j = (µ01 ◦ k11 )(x) (f a tb ),

j=1

as claimed. This concludes the verification of 8.6.2.4. (8.8.5) In analogy to 8.5.4, we define • ef,Iw ef• (GK,S , FΓ (T ); ∆(FΓ (T ))) C (K∞ /K, T ) = C

ef• (KS /K∞ , M ) = C ef• (GK,S , FΓ (M ); ∆(FΓ (M ))). C gf,Iw (K∞ /K, T ) (resp., RΓ gf (KS /K∞ , M )) The corresponding objects of D(R Mod) will be denoted by RΓ e i (K∞ /K, T ) (resp., H e i (KS /K∞ , M )). and their cohomology by H f,Iw f (8.8.6) Proposition. (i) For every bounded below complex M in (ind−ad Mod){m} , the morphisms shf R[GK,S ] induce isomorphisms of R-modules ∼ ∼ e i (GK 0 ,S 0 , M ; ∆(M )) ←− e i (GK,S , MU ; ∆(MU )) −→ lim H lim H f f − → − → res U,Tr∗ ∼ ∼ ei ei −→ lim −→ Hf (GK,S , U M ; ∆(U M )) −→ Hf (KS /K∞ , M ). U

(ii) For every bounded below complex T in (ad R[GK,S ] Mod)R−f t , the canonical morphism of complexes ∼ ef• (GK,S , FΓ (T ); ∆(FΓ (T ))) −→ e• C lim ←− Cf (GK,S , TU ; ∆(TU )) U

is an isomorphism, and the morphisms shf induce isomorphisms of R-modules ∼ ∼ ei e i (GK 0 ,S 0 , T ; ∆(T )) ←− e i (GK,S , TU ; ∆(TU )) −→ lim H lim H Hf,Iw (K∞ /K, T ). f f ← − ← − cor U,pr∗

185

Above, K 0 /K runs through finite subextensions of K∞ /K and U = Gal(K∞ /K 0 ). Proof. This will follow from 8.6.4 (which applies in our case, thanks to 8.8.4) in the same way as Proposition 8.5.5 does from 8.5.3.2-4, once we check that the canonical maps + + n Uv+ (FΓ (T )) −→ ← lim − Uv (TU ) −→ lim ←− Uv ((T /m T )U ) U

U,n

(v ∈ Sf )

are all isomorphisms (the local conditions for Z = (T /mn T )U are defined as in 8.8.3). For v ∈ Σ this follows from Proposition 8.3.5 and 8.4.4.2. For v ∈ Σ0 , it is sufficient to observe that the condition (U ) from 8.8.1 implies that L(TU ) = L(T )U ,

L(FΓ (T )) = FΓ (L(T )).

(8.8.7) Proposition. Let X be a bounded below complex in (ad R[GK,S ] Mod) equipped with Greenberg’s local + + conditions defined by Xv −→ X (where Xv is a bounded below complex in (ad R[Gv ] Mod), for each v ∈ Σ). Let K 0 /K be a finite Galois subextension of KS /K, of degree d = [K 0 : K] which is not a zero divisor on R, such that all primes v ∈ Σ0 are unramified in K 0 /K. Then, for each homomorphism χ : Gal(K 0 /K) −→ R[1/d]∗ , + the maps i+ v ⊗ id : Xv ⊗ χ −→ X ⊗ χ define Greenberg’s local conditions for X ⊗ χ and there are canonical isomorphisms  χ−1 ∼ e fi (GK,S , X ⊗ χ; ∆(X ⊗ χ)) ⊗R R[1/d] −→ e fi (GK 0 ,S 0 , X; ∆(X)) ⊗R R[1/d] H H , where we denote, for any R[1/d][Gal(K 0 /K)]-module M , −1

Mχ

−1

= M (χ

)

= {x ∈ M | (∀g ∈ Gal(K 0 /K)) g(x) = χ−1 (g)x}.

Proof. This follows from Lemma 8.6.4.4(iv) applied to X ⊗R ⊗R[1/d]⊗χ, as there is a canonical isomorphism of Gal(K 0 /K)-modules ∼ ei e i (GK 0 ,S 0 , X ⊗R ⊗R[1/d] ⊗ χ; ∆(X ⊗R ⊗R[1/d] ⊗ χ)) −→ H Hf (GK 0 ,S 0 , X; ∆(X)) ⊗R ⊗R[1/d] ⊗ χ f

e i ). (with respect to the Adf -action on the cohomology groups H f (8.8.8) Proposition (self-dual case). Assume that we are in the situation of 7.8.11 (in particular, R is an integral domain with fraction field K of characteristic zero, all complexes X and X + are concentrated in degree zero, and there exists a skew-symmetric isomorphism between V := X ⊗R K and V ∗ (1) := HomK (V, K)(1), for which each subspace Vv+ := Xv+ ⊗R K ⊂ V (v ∈ Σ) is maximal isotropic). Let K 0 /K be as in 8.8.7. Then: (i) For each homomorphism χ : Gal(K 0 /K) −→ K∗ , we have (using the notation from 7.8.9 and 8.8.7) e f1 (K 0 , V )χ = dimK H e f1 (K 0 , V )χ−1 = dimK H e f2 (K 0 , V )χ = dimK H e f2 (K 0 , V )χ−1 . dimK H (ii) Assume that there exists an intermediate field K ⊂ K1 ⊂ K 0 such that 2 - [K 0 : K1 ] and Gal(K1 /K) is an abelian group of order 2t (t ≥ 0). If K1 = K, set U = {K}. If K1 6= K, set U = {L | L ⊂ K1 , [L : K] = 2}. Then X X e 1 (K 0 , V ) ≡ e 1 (K, V ⊗ χ) ≡ e 1 (L, V ) (mod 2). dimK H dimK H dimK H f f f L∈U

χ:Gal(K1 /K)−→{±1}

∼ e i (K, V ⊗ χ±1 ) −→ e i (K 0 , V )χ∓1 from Proof. (i) Combine Proposition 7.8.11 with the isomorphisms H H f f Proposition 8.8.7.

186

(ii) As every finite group of odd order is solvable, there exist fields K ⊂ K0 ⊂ K1 ⊂ K2 ⊂ · · · ⊂ Kn = K 0 such that each Galois group Gi = Gal(Ki+1 /Ki ) (i = 0, . . . , n − 1) is abelian (and 2 - |Gi | for i > 0). After adjoining to R suitable roots of unity, we can assume that all characters of G0 , . . . , Gn−1 have values in K∗ . In the sum X e 1 (Ki+1 , V ) = e 1 (Ki+1 , V )χ , dimK H dimK H f f χ:Gi −→K∗

the contributions of χ and χ−1 are the same, by (i); thus X e 1 (Ki+1 , V ) ≡ e 1 (Ki+1 , V )χ (mod 2). dimK H dimK H f f χ:Gi −→K∗ χ=χ−1

e 1 (Ki+1 , V )1 = If i > 0, then the only character of Gi satisfying χ = χ−1 is the trivial character, for which H f 1 e (Ki , V ), by Lemma 8.6.4.4. It follows that H f

e 1 (Kn , V ) ≡ · · · ≡ dimK H e 1 (K1 , V ) (mod 2), dimK H f f which proves the claim if K1 = K. If K1 6= K, then the set of characters of G0 satisfying χ = χ−1 consists ∼ of the trivial character and the characters ηL/K : G0  Gal(L/K) −→ {±1} (L ∈ U ), hence e f1 (K1 , V ) ≡ dimK H

X

e f1 (K, V ⊗ χ) = dimK H e f1 (K, V ) + dimK H

χ:G0 −→{±1}

e f1 (K, V ) + = (1 − |U |) dimK H

X

X

e f1 (K1 , V )ηL/K = dimK H

L∈U

e f1 (L, V ) (mod 2), dimK H

L∈U

which proves the claim, since r ≥ 1 and 1 − |U | = 2 − 2r ≡ 0 (mod 2). (8.8.9). Let X, Y be bounded below complexes in (ad R[GK,S ] Mod) equipped with Greenberg’s local conditions defined by Zv+ −→ Z (Z = X, Y ), where Zv+ is a bounded below complex in (ad R[Gv ] Mod), for each v ∈ Σ. 0 Let K /K be a finite Galois subextension of KS /K such that all primes v ∈ Σ0 are unramified in K 0 /K. Let π : X ⊗R Y −→ ω • (1) be a morphism of complexes such that (∀v ∈ Σ) Xv ⊥π Yv . Then, for each g ∈ Gal(K 0 /K), the diagram ∪π,r,0

e • (GK 0 ,S 0 , X; ∆(X)) ⊗R C e • (GK 0 ,S 0 , Y ; ∆(Y )) C f f  Ad (g)⊗Ad (g) f y f

−−−−→

e • (GK 0 ,S 0 , X; ∆(X)) ⊗R C e • (GK 0 ,S 0 , Y ; ∆(Y )) C f f

∪π,r,0

ω • [−3] k

−−−−→

ω • [−3]

is commutative up to homotopy. Proof. Replacing K 0 by the fixed field of the cyclic group generated by g, we can assume that Gal(K 0 /K) is abelian. In this case the statement follows from Lemma 8.1.6.5 combined with 8.8.4 and the fact that the maps invv commute with corestrictions (cf. 9.2.2 below). (8.9) Duality theorems in Iwasawa theory revisited Assume that we are in the situation of 8.8.1. In particular, each prime v ∈ Σ0 is unramified in K∞ /K. • • • • (8.9.1) As in 8.4.6.2, fix a bounded complex ωR = σ≥0 ωR (resp., ωR = σ≥0 ωR ) of injective R-modules (resp., injective R-modules) representing ωR (resp., ωR ) and a quasi-isomorphism • • ϕ : ωR ⊗R R −→ ωR .

Assume that we are given the following data: 187

+ ∗ + ad (8.9.1.1) Bounded complexes T, T ∗(1) in (ad R[GK,S ] Mod)R−f t and Tv , T (1)v (v ∈ Σ)) in (R[Gv ] Mod)R−f t . (8.9.1.2) Morphisms of complexes of R[Gv ]-modules

jv+ (Z) : Zv+ −→ Z

(v ∈ Σ; Z = T, T ∗(1)).

(8.9.1.3) A morphism of complexes of R[GK,S ]-modules • π : T ⊗R T ∗ (1) −→ ωR (1),

which is a perfect duality in the sense of 6.2.6, i.e. such that its adjoint • • adj(π) : T −→ HomR (T ∗ (1), ωR (1))

is a quasi-isomorphism. (8.9.2) For Z = T, T ∗ (1) consider FΓ (Z) = (Z ⊗R R) < −1 >,

+ + FΓ (Z)+ v = FΓ (Zv ) = (Zv ⊗R R) < −1 >

ad Then FΓ (Z) (resp., FΓ (Z)+ v ) is a complex in (R[G

K,S ]

(v ∈ Σ).

Mod)R−f t (resp., in (ad Mod)R−f t ). The morphisms R[G ] v

jv+ (Z) induce morphisms of complexes of R[Gv ]-modules jv+ (FΓ (Z)) : FΓ (Z)+ v −→ FΓ (Z)

(v ∈ Σ; Z = T, T ∗ (1)).

As in 8.4.6.1,3, π induces a morphism of complexes of R[GK,S ]-modules id⊗(id⊗ι)

ι

π : FΓ (T ) ⊗R FΓ (T ∗ (1)) = (T ⊗R R) < −1 > ⊗R ((T ∗ (1) ⊗R R) < −1 >)ι −−−−−−→ ∼

−→ (T ⊗R R) < −1 > ⊗R (T ∗ (1) ⊗R R) < 1 >= (T ⊗R R) ⊗R (T ∗ (1) ⊗R R) −→ ∼

π⊗id

ϕ

• • −→ (T ⊗R T ∗ (1)) ⊗R R−−−−→ωR (1) ⊗R R−−→ωR (1),

the adjoint of which is equal to (adj(π)⊗id)

• adj(π) : FΓ (T ) = (T ⊗R R) < −1 > −−−−−−−−−−→Hom•R (T ∗ (1) ⊗R R, ωR ⊗R R(1)) < −1 >−→ • ∗ • −→ HomR (T (1) ⊗R R, ωR (1)) < −1 > .

By Corollary 8.4.6.5, π is again a perfect duality. In other words, the data (8.9.1.1-3) induce the same kind of data for FΓ (T ) and FΓ (T ∗ (1))ι , this time over R. (8.9.3) Lemma. Fix v ∈ Σ. Then: + ∗ ι + (i) Tv+ ⊥π T ∗ (1)+ v =⇒ FΓ (T )v ⊥π (FΓ (T (1)) )v . + ∗ + + ∗ (ii) Tv ⊥⊥π T (1)v =⇒ FΓ (T )v ⊥⊥π (FΓ (T (1))ι )+ v. Proof. The statement (i) is trivial and (ii) follows from Corollary 8.4.6.5. (8.9.4) Let Tv− , T ∗ (1)− v (v ∈ Σ) be as in 6.7.1. We apply the construction from 6.7.9 and define bounded ∗ + ad ad complexes A, A∗ (1) (resp., A+ v , A (1)v (v ∈ Σ)) in (R[GK,S ] Mod)R−cof t (resp., in (R[Gv ] Mod)R−cof t ) by A = D(T ∗ (1))(1),

A∗ (1) = D(T )(1),

∗ − A+ v = D(T (1)v )(1),

− A∗ (1)+ v = D(Tv )(1).

− Applying D to the canonical morphisms T ∗ (1) −→ T ∗ (1)− v and T −→ Tv we obtain morphisms of complexes of R[Gv ]-modules

188

A+ v −→ A,

∗ A∗ (1)+ v −→ A (1).

In the notation of 6.7.4, we have Tv+ ⊥⊥ev2 A∗ (1)+ v,

∗ + A+ v ⊥⊥ev1 T (1)v ,

(8.9.4.1)

with respect to the evaluation morphisms ev2 : T ⊗R A∗ (1) −→ IR (1)

ev1 : A ⊗R T ∗ (1) −→ IR (1). (8.9.5) For Z = A, A∗ (1) and v ∈ Σ, put + FΓ (Z)+ v = FΓ (Zv ).

It follows from Lemma 8.4.5.1 that the adjoint of the evaluation morphism • ∗ ι ev2 : FΓ (T ) ⊗R FΓ (A∗ (1))ι = (T ⊗R R) < −1 > ⊗R lim −→ HomR (R[G/U ] < 1 >, A (1)) −→

U

• −→ lim −→ HomR (R[G/U ], IR )(1) = IR (1)

U

is a quasi-isomorphism adj(ev2 ) : FΓ (T ) −→ Hom•R (FΓ (A∗ (1))ι , IR (1)); a similar statement holds for ev1 : FΓ (A) ⊗R FΓ (T ∗ (1))ι −→ IR (1). Combining Lemma 8.4.5.1 with (8.9.4.1) shows that, for each v ∈ Σ, ∗ ι + FΓ (T )+ v ⊥⊥ev2 (FΓ (A (1)) )v ,

∗ ι + FΓ (A)+ v ⊥⊥ev1 (FΓ (T (1)) )v .

(8.9.6) Assume, from now on until the end of Section 8.9, that Tv+ ⊥π T ∗ (1)+ v

(∀v ∈ Σ).

For each v ∈ Σ, define b ad Wv ∈ DR−f t (R[Gv ] Mod)

as in Proposition 6.7.6(iv); it sits in an exact triangle
adj(π) Wv −→ Tv− −−−−→D (T ∗ (1))+ v (1) −→ Wv [1]. According to Corollary 8.4.6.5, adj(π)

FΓ (Wv ) −→ FΓ (Tv− )−−−−→D FΓ (T ∗ (1))+ v is an exact triangle in


ι


(1) −→ FΓ (Wv )[1]

b DR−f (ad Mod). t R[Gv ]

Applying the discussion in 7.8 and 8.8 to FΓ (T ), FΓ (T ∗ (1))ι , FΓ (A), FΓ (A∗ (1))ι (for which we consider the 0 unramified local conditions ∆ur v at all v ∈ Σ ), we obtain the following (below, D = DR and D = DR ): b (8.9.6.1) Isomorphisms in Dfb t (R Mod) (resp., Dcof t (R Mod))   ∼ gf,Iw (K∞ /K, T ) −→ gf (KS /K∞ , A∗ (1))ι [−3] RΓ D RΓ  ι ∼ gf (KS /K∞ , A) −→ gf,Iw (K∞ /K, T ∗(1)) [−3]. RΓ D RΓ 189

(8.9.6.2) Induced isomorphisms on cohomology

  ∼ e i (K∞ /K, T ) −→ e 3−i (KS /K∞ , A∗ (1))ι H D H f,Iw f   ι ∼ 3−i ∗ e fi (KS /K∞ , A) −→ D H e H . f,Iw (K∞ /K, T (1))

(8.9.6.3) A pairing in Dfb t (R Mod)

gf,Iw (K∞ /K, T ) ⊗L RΓ gf,Iw (K∞ /K, T ∗(1))ι −→ ω [−3], RΓ R R

(8.9.6.3.1)

Dfb t (R Mod)

whose adjoint map sits in an exact triangle in  M ι gf,Iw (K∞ /K, T ) −→ D RΓ gf,Iw (K∞ /K, T ∗ (1)) [−3] −→ RΓ Errv ,

(8.9.6.3.2)

v∈Sf

where the error terms Errv are as follows: ur ur ∗ ι Errv = Errur v (D, FΓ (T )) = Errv (∆v (FΓ (T )), ∆v (FΓ (T (1)) ), π)

(∀v ∈ Σ0 )

are as in Proposition 7.6.7(ii) and Corollary 7.6.8; (∀v ∈ Σ)

Errv = RΓcont (Gv , FΓ (Wv )),

by Proposition 6.7.6(iv). (8.9.6.4) If Tv+ ⊥⊥π T ∗ (1)+ v for all v ∈ Σ, and q ∈ Spec(R) with ht(q) = 0, then the localization of (8.9.6.3.2) at q induces isomorphisms ∼ e j (K∞ /K, T )q −→ H DR f,Iw

It follows that `R

q

 q



e 3−j (K∞ /K, T ∗ (1))ι H f,Iw

  q

= DR

q



e 3−j (K∞ /K, T ∗(1))ι H f,Iw

    j 3−j ∗ ι e e Hf,Iw (K∞ /K, T )q = `R Hf,Iw (K∞ /K, T (1)) . q

q

  q

. (8.9.6.4.1) (8.9.6.4.2)

∼

In particular, if Γ = Γ0 −→ Zrp and R is a domain, then ∗ e j (KS /K∞ , A) = rk H e 3−j ej e 3−j (KS /K∞ , A∗ (1)). corkR H f R f,Iw (K∞ /K, T (1)) = rkR Hf,Iw (K∞ /K, T ) = corkR Hf (8.9.6.4.3) (8.9.7) Error terms and local Tamagawa factors for FΓ (T ) We are going to investigate the error terms in (8.9.6.3) under the assumption (U ) from 8.8.1 (which is ∼ ∼ automatically satisfied if ∆ = 0). As Γ = Γ0 × ∆, Γ0 −→ Zrp (r ≥ 1), we have R −→ R[∆][[X1 , . . . , Xr ]] ur (Xi = γi − 1). Our first goal is to show that the cohomology groups of Errv (for v - p) are very often pseudo-null over R. We use repeatedly the fact that FΓ (T ) = FΓ0 (F∆ (T )).

(8.9.7.1) Lemma. (i) {p ∈ Spec(R) | ht(p) = 0} = {pR | p ∈ Spec(R[∆]), ht(p) = 0}. (ii) Let M ∈ (R[∆] Mod)f t and p ∈ Spec(R[∆]), ht(p) = 0. Then: p ∈ supp(M ) ⇐⇒ p := pR ∈ supp(M ⊗R[∆] R) and `R ((M ⊗R[∆] R)p ) = `R[∆]p (Mp ). p

∼

Proof. (i) Note that, for every ideal I ⊂ R[∆], R/IR −→ (R[∆]/I)[[X1 , . . . , Xr ]]. If p ∈ Spec(R) has ht(p) = 0, put p = R[∆] ∩ p ∈ Spec(R[∆]). Then pR ⊂ p, hence pR = p. If there is q ∈ Spec(R[∆]) with q ( p, then qR ( pR = p, which contradicts ht(p) = 0; thus ht(p) = 0. Conversely, if p ∈ Spec(R[∆]) has ht(p) = 0 and p = pR, then R/p = (R[∆]/p)[[X1 , . . . , Xr ]] is a domain, hence p ∈ Spec(R). If ht(p) 6= 0, then there is q ∈ Spec(R), q ( p, ht(q) = 0. By the above argument, q = qR for q = q ∩ R[∆], ht(q) = 0. As q ⊂ p, we have q = p, hence q = qR = pR = p; this contradiction proves that ht(p) = 0. (ii) We have Mp 6= 0 ⇐⇒ M/pM 6= 0 ⇐⇒ (M ⊗R[∆] R)/(pR)(M ⊗R[∆] R) 6= 0 ⇐⇒ (M ⊗R[∆] R)pR 6= 0, which proves the first statement. By d´evissage, it is enough to check the equality of lengths for M = R[∆]/p, in which case both sides are equal to one. 190

(8.9.7.2) Proposition. Let j ≥ 0. Given M ∈ (R Mod)f t with codimR (supp(M )) ≥ j, f ∈ AutR (M ) and ∗ u ∈ R , let N0 , N1 be the R-modules defined by the exact sequence f ⊗u−1

0 −→ N0 −→ M ⊗R R−−−−→M ⊗R R −→ N1 −→ 0. For i = 0, 1, let Bi = supp(Ni ) ∩ {p ∈ Spec(R) | ht(p) ≤ j}. (i) If u ∈ R[∆]∗ , then Ni = Mi ⊗R[∆] R and Bi = {pR | p ∈ Spec(R[∆]), ht(p) = j, p ∈ supp(Mi )}, where M0 , M1 are the R[∆]-modules defined by the exact sequence f u−1

0 −→ M0 −→ M ⊗R R[∆]−−→M ⊗R R[∆] −→ M1 −→ 0. For each p = pR ∈ Bi ,

`R ((Ni )p ) = `R[∆]p ((Mi )p ). p

(ii) If, for each maximal ideal m∆ of R[∆], u (mod m∆ R) ∈ (R/m∆ R)∗ = (R[∆]/m∆ )[[X1 , . . . , Xr ]]∗ is not contained in (R[∆]/m∆ )∗ , then both sets B0 , B1 are empty. Proof. If j > d, then M = 0 and there is nothing to prove. If j ≤ d, then there is an ideal I ⊂ AnnR (M ) with dim(R/I) = d − j. Replacing R by R/I, we can assume that j = 0. (i) As R is flat over R, we have Ni = Mi ⊗R[∆] R; the rest of the statement follows from Lemma 8.9.7.1. (ii) Let p ∈ Spec(R[∆]), ht(p) = 0; put p = pR. We must show that (Ni )p = 0 for i = 0, 1 (again using Lemma 8.9.7.1(i)). The R[∆]-module M [∆] := M ⊗R R[∆] admits a filtration 0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn = ∼ M [∆] with graded quotients Mt+1 /Mt −→ R[∆]/qt , for some qt ∈ Spec(R[∆]). By d´evissage, we can assume ∼ that M [∆] −→ R[∆]/qt . If qt 6= p, then (M ⊗R R)p = (M [∆] ⊗R[∆] R)p = 0 by Lemma 8.9.7.1(ii). If qt = p, then we can replace R[∆] by R[∆]/p, hence assume that R[∆] is a domain and p = (0). In this case f ∈ R[∆]∗ , N0 = 0 and N1 = R/(u − f −1 )R, which is R-torsion; thus (N1 )(0) = 0. (8.9.7.3) Proposition. (i) For each non-archimedean prime v - p of K which is unramified in K∞ /K, there is an isomorphism in Db ((R Mod)/(pseudo − null)) (using the notation of (7.6.5.1)) ( D(Errur v (D, FΓ (T )))

∼

−→

FΓ0 (DR[∆] (Errur v (DR[∆] , F∆ (T )))),

|Γv | < ∞ ( ⇐⇒ Γv ⊂ ∆)

0,

|Γv | = ∞.

(ii) If p ∈ Spec(R), ht(p) = 1 and Errur v (D, FΓ (T ))p is not acyclic (i.e. not isomorphic to 0 in D(Rp Mod)), then Γv ⊂ ∆, p = pR, p ∈ Spec(R[∆]), ht(p) = 1 and Tamv (FΓ (T ), p) = Tamv (F∆ (T ), p) 6= 0. Proof. As v is unramified in K∞ /K, we have ∼

1 1 Hcont (Iv , FΓ (T )) −→ FΓ (Hcont (Iv , T )).

(8.9.7.3.1)

Let N be an R-module of finite type and P• −→ N a resolution of N by free R-modules of finite type. Then P• ⊗R R is a free resolution of N ⊗R R, hence 191

∼

• • • HomR (P• , ωR ) ⊗R R −→ Hom•R (P• ⊗R R, ωR ⊗R R)

gives a canonical isomorphism in D(R Mod) ∼

D(N ) ⊗R R −→ D(N ⊗R R).

(8.9.7.3.2)

1 Applying this observation to N = Hcont (Iv , T ), we obtain isomorphisms of R[Gv /Iv ]-modules

∼

∼

∼

Ext1R (FΓ (N ), ωR ) −→ Ext1R (N ⊗R R, ωR ) < 1 >−→ (Ext1R (N, ωR ) ⊗R R) < 1 >−→ ∼

−→ FΓ (Ext1R (N, ωR ))ι . ∗

Let M = Ext1R (N, ωR ), f = fv−1 ∈ AutR (M ) and u = the image of fv−1 ∈ Γv under Γv ,→ Γ ,→ R . It follows from the above observation and Proposition 7.6.7(ii) that ur ur ∗ ι D (Errur v (D , FΓ (T ))) = Errv (∆v (FΓ (T )), ∆v (FΓ (T (1)) ), π)

is isomorphic in D((R Mod)/(pseudo − null)) to the following complex in degrees −1, 0:   f ⊗u−1 K • = M ⊗R R−−−−→M ⊗R R . If |Γv | < ∞, then Γv ⊂ ∆, u ∈ R[∆]∗ and K • is isomorphic to   fv−1 u−1 M ⊗R R[∆]−−−−→M ⊗R R[∆] ⊗R[∆] R. Both statements (i), (ii) in this case follow from Proposition 8.9.7.2(i) (for j = 1) and Proposition 7.6.7(ii), this time applied to R[∆] and F∆ (T ). n n If |Γv | = ∞, then there is a Zp -basis γ1 , . . . , γr ∈ Γ and n ≥ 0 such that u = δu0 γ1p = δu0 (1 + X1 )p , with δ ∈ ∆, u0 ∈ Z∗p . Applying Proposition 8.9.7.2(ii) with j = 1, we deduce that the cohomology groups of K • are pseudo-null over R. (8.9.7.4) Corollary. For each non-archimedean prime v - p of K which is unramified in K∞ /K, there is an isomorphism in the category Db ((R Mod)/(pseudo − null)) ( Errur v (D , FΓ (T ))

∼

−→

FΓ0 (Errur v (DR[∆] , F∆ (T ))),

|Γv | < ∞

0,

|Γv | = ∞.

Proof. Apply D to the statement (i) in Proposition 8.9.7.3 (and use (8.9.7.3.2)).
(8.9.7.5) For each q ∈ Spec(R[∆]) and M ∈ R Mod , we put Mq = M ⊗R[∆] R[∆]q ; this is an Rq -(= R ⊗R[∆] R[∆]q )-module. (8.9.7.6) Proposition. Assume that T = σ≤0 T and v - p is a non-archimedean prime of K, unramified in K∞ /K. (i) If |Γv | < ∞, then there is an isomorphism in Dfb t (R Mod) ∼

ur D(Errur v (D, FΓ (T ))) −→ FΓ0 (DR[∆] (Errv (DR[∆] , F∆ (T )))). ∼

1 1 (ii) If Hcont (Iv , F∆ (T )) = Hcont (Iv , H 0 (F∆ (T ))) −→ H 0 (F∆ (T ))Iv (−1) is zero or a Cohen-Macaulay R[∆]-module of dimension d = dim(R) = dim(R[∆]), then ∼

b Errur v (D, FΓ (T )) −→ 0 in Df t (R Mod).

(iii) If q ∈ Spec(R[∆]), ht(q) = 0, then ∼

b Errur v (D, FΓ (T ))q −→ 0 in Df t (Rq Mod).

192

(iv) The following statements are equivalent (cf. Corollary 7.6.12(iii)): ∼

Errur v (D, FΓ (T )) −→ 0

∼

in Dfb t (R Mod) ⇐⇒ Errur v (DR[∆] , F∆ (T )) −→ 0 in

Dfb t (R[∆] Mod).

(v) If dim(R) = 1, then ∼

Errur v (D, FΓ (T )) −→ 0

in Dfb t (R Mod) ⇐⇒ (∀m∆ ∈ Max(R))

Tamv (F∆ (T ), m∆ ) = 0.

Proof. We apply Proposition 7.6.11 and Corollary 7.6.12 to FΓ (T ) = FΓ0 (F∆ (T )) = σ≤0 FΓ (T ) over R. Combining (8.9.7.3.1-2), we obtain
∼ 1 1 τ≥1 D Hcont (Iv , FΓ (T )) −→ FΓ (τ≥1 D(Hcont (Iv , T )))ι ; Corollary 7.6.12(i) then yields an isomorphism in Dfb t (R Mod)
∼
−1 −1 D Errur − 1 : τ≥1 DR[∆] (H) ⊗R[∆] R −→ τ≥1 DR[∆] (H) ⊗R[∆] R [1] v (D, FΓ (T )) −→ Cone fv ⊗ χΓ0 (fv ) (8.9.7.6.1) 1 (where H = Hcont (Iv , F∆ (T ))). (i) If |Γv | < ∞, then χΓ0 (fv ) = 1, hence (8.9.7.6.1) and the flatness of R over R imply (using Corollary 7.6.12(i) again, this time for F∆ (T ) over R[∆]) that
∼ ur D Errur v (D, FΓ (T )) −→ Errv (DR[∆] , F∆ (T )) ⊗R[∆] R, as required. 1 (ii) In this case the complex τ≥1 DR[∆] (Hcont (Iv , F∆ (T ))) is acyclic. 1 (iii) The complex τ≥1 DR[∆] (Hcont (Iv , F∆ (T )))q is acyclic, by Lemma 2.4.7(iv) and local duality 2.5. (iv) Put, for each q ≥ 0, 1 Mq := ExtqR[∆] (Hcont (Iv , F∆ (T ))), ωR[∆] ) ∈




R[∆] Mod f t

.

According to Corollary 7.6.12(iii), we have ∼

Errur v (DR[∆] , F∆ (T )) −→ 0 ⇐⇒ (∀q = 1, . . . , d) ∼

Errur v (D, FΓ (T )) −→ 0 ⇐⇒ (∀q = 1, . . . , d)

Coker(fv−1 − 1 : Mq −→ Mq ) = 0,

Coker(fv−1 ⊗χΓ0 (fv )−1 −1 : Mq ⊗R[∆] R −→ Mq ⊗R[∆] R) = 0.

However, as χΓ0 (fv ) − 1 is contained in the radical m of R, Nakayama’s Lemma implies that Coker(fv−1 ⊗ χΓ0 (fv )−1 − 1 : Mq ⊗R[∆] R −→ Mq ⊗R[∆] R) = 0 ⇐⇒ 0 = Coker(fv−1 ⊗ 1 − 1 : Mq ⊗R[∆] R/m −→ Mq ⊗R[∆] R/m) = = Coker(fv−1 − 1 : Mq ⊗R[∆] R[∆]/m −→ Mq ⊗R[∆] R[∆]/m) ⇐⇒ Coker(fv−1 − 1 : Mq −→ Mq ) = 0, where we have denoted by m the radical of R[∆]. (v) Combine (iv) with 7.6.10.8 (for p = m∆ ). (8.9.7.7) Proposition. Assume that v ∈ Σ and q0 ∈ Spec(R[∆]); denote by p ∈ Spec(R) the inverse image of q0 under the projection map R = R[∆][[Γ0 ]] −→ R[∆]. ∼ ∼ (i) If RΓcont (Gv , F∆ (Wv ))q0 −→ 0 in Dfb t (R[∆]q0 Mod), then RΓcont (Gv , FΓ (Wv ))p −→ 0 in Dfb t (R Mod). ∼

p

(ii) If Wv = H 0 (Wv ) is concentrated in degree 0 and RΓur (Gv , F∆ (Wv ))q0 −→ 0 in Dfb t (R[∆]q0 Mod), then ∼ RΓur (Gv , FΓ (Wv ))p −→ 0 in Dfb t (R Mod). p

Proof. We use the fact that FΓ (Wv ) = FΓ0 (F∆ (Wv )). The statement (i) follows from Proposition 8.4.8.5 applied to R[∆] instead of R, p = q0 , G = Gv and T = F∆ (Wv ). 193

(ii) As Wv is concentrated in degree 0, we have RΓur (Gv , F∆ (Wv )) = RΓcont (Gv /Iv , F∆ (Wv )Iv ),

RΓur (Gv , FΓ (Wv )) = RΓcont (Gv /Iv , FΓ (Wv )Iv ).

On the other hand, the assumption (U) from 8.8.1 implies that FΓ (Wv )Iv = FΓ (WvIv ) = FΓ0 (F∆ (Wv )Iv ) = FΓ0 (F∆ (WvIv )); we apply again Proposition 8.4.8.5 with G = Gv /Iv and T = F∆ (WvIv ). (8.9.8) Theorem. Denote Sbad = {pR | p ∈ Spec(R[∆]), ht(p) = 1, (∃v ∈ Σ0 ) |Γv | < ∞, Tamv (F∆ (T ), p) 6= 0}. Let p ∈ Spec(R), ht(p) = 1, p 6∈ Sbad . Assume that, either: (i) (∀v ∈ Σ) Tv+ ⊥⊥π T ∗ (1)+ v ; or: ∼ (ii) Γ = Γ0 × ∆ = Γ00 × Γ000 × ∆, Γ00 −→ Zp , q0 ∈ Spec(R[∆]), ht(q0 ) = 0, p = q0 R + (γ − 1)R ∈ Spec(R), ∼ (∀v ∈ Σ) RΓcont (Gv , F∆ (Wv ))q0 −→ 0 in Dfb t (R[∆]q0 Mod). Then the localized duality map gf,Iw (K∞ /K, T ) −→ D RΓ p R

p



 gf,Iw (K∞ /K, T ∗ (1))ι )p [−3] (RΓ

is an isomorphism in Dfb t (R Mod), inducing exact sequences of Rp -modules of finite type p

  e 4−j (K∞ /K, T ∗(1)), ω )ι −→ H e j (K∞ /K, T ) −→ 0 −→ Ext1R (H f,Iw f,Iw R p p   3−j ∗ ι e −→ Ext0R (H −→ 0. f,Iw (K∞ /K, T (1)), ωR ) p

Proof. We apply 7.8.4.4 with X = FΓ (T ) over R; for each v ∈ Σ (resp., v ∈ Σ0 ), the localization at p of the error term Errv vanishes, by Proposition 8.9.7.7 (resp., Proposition 8.9.7.3(ii)). 0 (8.9.9) Theorem. Assume that Tv+ ⊥⊥π T ∗ (1)+ v (∀v ∈ Σ). If Sbad = ∅ (e.g., if (∀v ∈ Σ ) |Γv | = ∞), then D gf,Iw (K∞ /K, T ) ←−−−− gf,Iw (K∞ /K, T ∗ (1))ι [3] RΓ −−−−→ RΓ ←−     −− D −−→   −−−−− Φ Φ − − −− −−− y y − − → − ← gf (KS /K∞ , A) gf (KS /K∞ , A∗ (1))ι [3] RΓ RΓ b is a duality diagram in Dfb t ((R Mod)/(pseudo − null)) (top row) resp., Dcof t ((R Mod)/(co − pseudo − null)) (bottom row), inducing exact sequences in (R Mod)/(pseudo − null)

e 4−j (K∞ /K, T ∗(1)), ω )ι −→ H e j (K∞ /K, T ) −→ Ext0 (H e 3−j (K∞ /K, T ∗ (1)), ω )ι −→ 0. 0 −→ Ext1R (H f,Iw f,Iw f,Iw R R R These sequences yield monomorphisms in (R Mod)/(pseudo − null) e j (K∞ /K, T ) e 4−j (K∞ /K, T ∗(1)), ω )ι , H −→ Ext1R (H f,Iw f,Iw R R−tors which are isomorphisms if R has no embedded primes. Proof. This follows from Theorem 8.9.8. (8.9.10) Recall that we have established in (8.9.6.2) isomorphisms of R-modules 194

  ∼ e j (K∞ /K, T ) −→ e 3−j (KS /K∞ , A∗ (1))ι H D H f,Iw f   ∼ e j (K∞ /K, T ∗(1)) −→ e 3−j (KS /K∞ , A)ι . H D H f,Iw f (8.9.11) Theorem. Assume that Tv+ ⊥⊥π T ∗ (1)+ v (∀v ∈ Σ) and T = σ≤0 T . Then, for each q ∈ Spec(R[∆]) with ht(q) = 0, the localized duality map   gf,Iw (K∞ /K, T ) −→ D RΓ gf,Iw (K∞ /K, T ∗(1))ι [−3] RΓ R q q

is an isomorphism in Dfb t (Rq Mod), inducing a spectral sequence   e 3−j (K∞ /K, T ∗ (1)), ω )ι =⇒ H e i+j (K∞ /K, T ) . E2i,j = ExtiR (H f,Iw f,Iw R q q

Proof. We apply 7.8.4.4 with X = FΓ (T ) over R; by Proposition 8.9.7.6(iii), the error terms vanish after tensoring with R[∆]q . The spectral sequence is obtained from 2.8.6, by applying the same tensor product. (8.9.12) Theorem. Assume that T = σ≤0 T and (∀v ∈ Σ) Tv+ ⊥⊥π T ∗ (1)+ v . For v ∈ Σ, set Hv = 1 Hcont (Iv , F∆ (T )). If (∀v ∈ Σ0 ) (∀q = 1, . . . , d)

Coker(fv−1 − 1 : ExtqR[∆] (Hv , ωR[∆] ) −→ ExtqR[∆] (Hv , ωR[∆] )) = 0

(if d = dim(R) = 1, then this condition is equivalent to (∀v ∈ Σ0 ) (∀m∆ ∈ Max(R[∆])) Tamv (F∆ (T ), m∆ ) = 0), then D gf,Iw (K∞ /K, T ) ←−−−− gf,Iw (K∞ /K, T ∗ (1))ι [3] RΓ −−−−→ RΓ ←−   →   −− D −− −−−−−   Φ Φ − − − − − y y − −→ −− − ← gf (KS /K∞ , A) gf (KS /K∞ , A∗ (1))ι [3] RΓ RΓ b is a duality diagram in D(co)f t (R Mod), inducing a spectral sequence

e 3−j (K∞ /K, T ∗ (1)), ω )ι =⇒ H e i+j (K∞ /K, T ). E2i,j = ExtiR (H f,Iw f,Iw R Proof. This time the error terms vanish in Dfb t (R Mod), thanks to Proposition 8.9.7.6(iv) and Corollary 7.6.12(iii). (8.9.13) As in 8.5.8, the pairings • π : T ⊗R T ∗ (1) −→ ωR (1),

• π : FΓ (T ) ⊗R FΓ (T ∗ (1))ι −→ ωR (1)

define – for each finite subextension Kα /K of K∞ /K – cup products (6.3.2.2) e fi (GKα ,Sα , T ) ⊗R H e j (GKα ,Sα , T ∗ (1)) −→ H i+j−3 (ωR ) h , iα : H f and    ι j i ∗ e e h , i : lim Hf (GKα ,Sα , T ) ⊗R lim Hf (GKα ,Sα , T (1)) −→ H i+j−3 (ωR ) = ← ← α− α−
= H i+j−3 (ωR ) ⊗R R = lim H i+j−3 (ωR ) ⊗R R[Gal(Kα /K)] . ← − α (8.9.14) Proposition. In the situation of 8.9.13,  h(xα ), (yα )i = 

X

σ∈Gal(Kα /K)

(where σ acts on

e j (GKα ,Sα , T ∗ (1)) H f

 hxα , σyα iα ⊗ [σ] α

by the conjugation action from 8.6.4.2).

Proof. As in the proof of 8.5.9, one reduces to the proof of the corresponding local statement 8.11.10, this time using 8.6.4.2-3. 195

gf,Iw (K∞ /K, T ) is equal to (8.9.15) Theorem. The Euler-Poincar´e characteristic of RΓ X q

  XX X
X
e q (K∞ /K, T ) = (−1)q eR H (−1)q eR (T q )Gv − [Kv : Qp ] (−1)q eR (Tv+ )q . f,Iw v|∞

q

q

v|p

Proof. This follows from Theorem 7.8.6 and the fact that eR (M ⊗R R) = eR (M ), for every R-module M of finite type. (8.9.16) Corollary. Under the assumptions (i)-(ii) of Corollary 7.8.7, X q

  + e q (K∞ /K, T ) = [K : Q](d+ (−1)q eR H ∞ − dp ). f,Iw

(8.10) Control Theorems The goal of this section is to prove analogues of Mazur’s control theorem for classical Selmer groups in our ∼ context, generalizing the results of 8.4.8. We assume that we are in the situation of 8.8.1 with Γ = Γ0 −→ Zrp + (hence the condition (U ) from 8.8.1 is automatically satisfied). Let T and Tv be as in 8.8.2; throughout Section 8.10 we assume that the complexes T and Tv+ are bounded. The key point is the following exact control theorem for Selmer complexes, from which we deduce in Proposition 8.10.4(ii) and 8.10.8(ii) somewhat less precise control results for cohomology groups. (8.10.1) Proposition. There is a canonical isomorphism in Db (R Mod) L

∼ g gf,Iw (K∞ /K, T )⊗ R −→ RΓ RΓf (T ) R

(where the product is taken with respect to the augmentation map R −→ R) and a (homological) spectral sequence 2 e −j (K∞ /K, T )) =⇒ H e −i−j (T ), Ei,j = Hi,cont (Γ, H f,Iw f 2 in which each term Ei,j is an R-module of finite type.

Proof. The same argument as in the proof of Proposition 8.4.8.1(ii) applies: the exact sequences (8.4.8.1.1) for G = GK,S , Gv (v ∈ Sf ) and analogous sequences for the local conditions Uv+ (−) (v ∈ Sf ) yield an exact sequence γi −1 • e• (FΓ/H (T ))−−→ e (FΓ/H (T )) −→ C e • (FΓ/H (T )) −→ 0, 0 −→ C C f f f i−1 i−1 i

and we conclude as in the proof of loc. cit. ∼

(8.10.2) Corollary. If Γ −→ Zp , then there are natural exact sequences Γ

e j (K∞ /K, T ) −→ H e j (T ) −→ H e j+1 (K∞ /K, T ) −→ 0. 0 −→ H f,Iw f f,Iw Γ (8.10.3) Consider the following condition: (8.10.3.1) The canonical maps τ≤0 T −→ T τ≤0 Tv+ −→ Tv+

(v ∈ Σ)

are all quasi-isomorphisms. If satisfied, then the same holds for FΓ (T ) and FΓ (Tv+ ). This implies, by 7.8.5, that the maps 196

gf (T ) −→ RΓ gf (T ) τ≤3 RΓ gf,Iw (K∞ /K, T ) −→ RΓ gf,Iw (K∞ /K, T ) τ≤3 RΓ

(8.10.3.2)

are also quasi-isomorphisms, hence   ∼ e 3 (T ) = H e 3 (H 0 (T )) −→ e 0 (D(H 0 (T ))(1)) H D H f f f  ι ∼ 3 3 e f,Iw e f,Iw e f0 (KS /K∞ , D(H 0 (T ))(1)) H (K∞ /K, T ) = H (K∞ /K, H 0 (T )) −→ D H e 3 (K∞ /K, T ) is (for the induced Greenberg’s local conditions H 0 (Tv ) −→ H 0 (T ) (v ∈ Σ). In particular, H f,Iw an R-module of finite type. The spectral sequence from 8.10.1 yields an isomorphism ∼ e3 3 e f,Iw H (K∞ /K, T )Γ −→ H f (T ).

(8.10.3.3)

(8.10.4) Proposition. Assume that (8.10.3.1) is satisfied. e 3 (T ) = 0, then: (1) If S ⊂ R is a multiplicative set such that H f S e 3 (K∞ /K, T )) = 0. (i) (∀i ≥ 0) (E 2 )S = Hi,cont (Γ, H i,−3

f,Iw

S

(ii) After localizing at S , the canonical map

e 2 (K∞ /K, T ) −→ H e 2 (T ) edge : H f,Iw f Γ (which is an edge map in the spectral sequence E m ) becomes an isomorphism, i.e. Ker(edge)S = Coker(edge)S = 0. (2) In particular, taking S = R − p with p ∈ Spec(R), then 2 e 3 (T )). suppR (Ker(edge)) ∪ suppR (Coker(edge)) ∪ suppR (Ei,−3 ) ⊆ suppR (H f

Proof. It is sufficient to prove (1). By (8.10.3.2), the assumption (8.10.3.1) implies that the spectral sequence 2 E r from Proposition 8.10.1 satisfies Ei,j = 0 for j < −3 or i < 0. As ∼ e2 2 E0,−2 −→ H f,Iw (K∞ /K, T )Γ

and the map in (ii) is an edge homomorphism for Er , it follows that the statement (ii) is a consequence of (i) (for i = 1, 2). In order to prove (i), consider 3 e f,Iw M =H (K∞ /K, T )S .

This is an RS -module of finite type satisfying ∼

2 MΓ −→ E0,−3


S

∼ e3 −→ H f (T )S = 0.

The claim (i) then follows from 7.2.7 and the following Lemma, applied to M , B = RS and ti = γi − 1 (1 ≤ i ≤ r). (8.10.5) Lemma. Let B be a commutative ring, M a B-module of finite type and t1 , . . . , tr ∈ EndB (M ) mutually commuting endomorphisms (i.e. ti tj = tj ti for all i, j). View M as a module over B 0 = B[T1 , . . . , Tr ], r • • with Ti acting as ti . Then: the Koszul complex K • = KB 0 (M, (T1 , . . . , Tr )) is acyclic ⇐⇒ H (K ) = 0. Proof. The implication ‘=⇒’ is trivial. In order to prove ‘⇐=’, it is enough to show that the localiza• tion Km is acyclic, for each maximal ideal m ⊂ B 0 . If T1 , . . . , Tr ∈ m, then the assumption H r (K • ) = M/(T1 , . . . , Tr )M = 0 implies that M/mM = 0, hence M = 0 (=⇒ K • = 0) by Nakayama’s Lemma. If, for 0 • some i = 1, . . . , r, Ti 6∈ m, then Ti is invertible in Bm , hence Km is acyclic ([Br-He], Prop. 1.6.5(c)). 197

(8.10.6) The above results admit a dual formulation. Assume that A (resp., A+ v , v ∈ Σ) is a bounded ad + complex in (ad R[GK,S ] Mod) (resp., in (R[Gv ] Mod)), with cohomology of co-finite type over R, and Av −→ A − (v ∈ Σ) is a morphism of complexes of R[Gv ]-modules. Define Av as in 6.7.1. According to Proposition 3.2.6 there exists a bounded complex T ∗ in (ad R[GK,S ] Mod)R−f t and a quasiisomorphism T ∗ −→ D(A). For each v ∈ Σ, the complex Fv = Cone(T ∗ ⊕ D(A− v ) −→ D(A))[−1] in (ad R[Gv ] Mod) is bounded, has cohomology of finite type over R and sits in a diagram Fv   y

−→

D(A− v )   y

T∗

−→

D(A),

in which both horizontal arrows are quasi-isomorphisms and the cones of the two vertical arrows are canonically isomorphic. Applying Proposition 3.2.6 once again, there exists a bounded complex (T ∗ (1))+ v in ∗ + (ad Mod) and a quasi-isomorphism (T (1)) −→ F (1). R−f t v v R[Gv ] The duality results (7.8.4.3) and (8.9.6.1) give isomorphisms ∼ gf (A) −→ gf (T ∗ (1)))[−3] RΓ DR (RΓ ∼ gf (KS /K∞ , A) −→ gf,Iw (K∞ /K, T ∗ (1)))ι [−3] RΓ DR (RΓ

(8.10.6.1)

in Db (R Mod) and Db (R Mod), respectively. • e• (FΓ (T ∗ (1))), x) appearing implicitly in the proof of Proposition Applying DR to the Koszul complex KR (C f 8.10.1 and using (7.2.7.3) together with (8.10.6.1), we obtain isomorphisms in Db (R Mod) ∼ ∼ • gf (A) −→ ef• (KS /K∞ , A)) −→ ef• (KS /K∞ , A)), RΓ KR (C RΓcont (Γ, C

hence a spectral sequence i,j i e j (KS /K∞ , A)) =⇒ H e i+j (A) E 2 = Hcont (Γ, H f f

(8.10.6.2)

i,j

m dual to Ei,j (in particular, each E 2 is an R-module of co-finite type). By construction, the edge homomorphisms of E r and E r are dual to each other, i.e. the diagram ∼

e i (A) H f   y

−→

e i (KS /K∞ , A)Γ H f

−→

∼



e 3−i (T ∗ (1))) DR (H f   y ι

e 3−i (K∞ /K, T ∗(1)) )Γ DR (H f,Iw

is commutative. (8.10.7) Consider the following condition: (8.10.7.1) The canonical maps A −→ τ≥0 A

+ A+ v −→ τ≥0 Av

are all quasi-isomorphisms. If (8.10.7.1) holds, then the maps 198

(v ∈ Σ)



(8.10.6.3)

gf (A) −→ τ≥0 RΓ gf (A) RΓ gf (KS /K∞ , A) −→ τ≥0 RΓ gf (KS /K∞ , A) RΓ

(8.10.7.2)

are quasi-isomorphisms, T ∗ (1) and (T ∗ (1))+ v satisfy (8.10.3.1) and e f0 (A) = H e f0 (H 0 (A)), H

e f0 (KS /K∞ , A) = H e f0 (KS /K∞ , H 0 (A)). H

e 0 (KS /K∞ , A) is an R-module of co-finite type. Combining (8.10.3.3) with (8.10.6.3), we In particular, H f obtain an isomorphism Γ ∼ e0 e 0 (A) −→ H Hf (KS /K∞ , A) . f

(8.10.7.3)

(8.10.8) Proposition. Assume that (8.10.7.1) is satisfied. e 0 (A))S = 0, then: (1) If S ⊂ R is a multiplicative set such that D(H f i,0 i e 0 (KS /K∞ , A)))S = 0. (i) (∀i ≥ 0) (D(E 2 ))S = D(Hcont (Γ, H f (ii) The kernel and cokernel of the canonical restriction map e f1 (A) −→ H e f1 (KS /K∞ , A) resK∞ /K : H

Γ

(which is an edge map in the spectral sequence E m ) satisfy D(Ker(resK∞ /K ))S = D(Coker(resK∞ /K ))S = 0. (2) In particular, taking S = R − p with p ∈ Spec(R), then i,0 e 0 (A))). suppR (D(Ker(resK∞ /K ))) ∪ suppR (D(Coker(resK∞ /K ))) ∪ suppR (D(E 2 )) ⊆ suppR (D(H f

Proof. This follows from Proposition 8.10.4 applied to T ∗ (1) and the commutative diagram (8.10.6.3). (8.10.9) The results in 8.10.1,4,8 can be further generalized as follows: In 8.10.9-12, we do not require that ∆ = 0, but instead assume that the condition (U) from 8.4.8 is satisfied, and that Γ0 ⊂ Γ is a closed 0 0 0 ∼ 0 subgroup isomorphic to Γ0 −→ Zrp for some r0 ≤ r. Put K∞ = (K∞ )Γ and R = R[[Γ/Γ0 ]]. (8.10.10) Proposition. There is a canonical isomorphism in Db (R0 Mod) ∼ g 0 gf,Iw (K∞ /K, T )⊗ R0 −→ RΓ RΓf,Iw (K∞ /K, T ) R L

(8.10.10.1) 0

(where the product is taken with respect to the canonical projection R = R[[Γ]] −→ R = R[[Γ/Γ0 ]]) and a (homological) spectral sequence 0

2 0 e −j (K∞ /K, T )) =⇒ H e −i−j (K∞ Ei,j = Hi,cont (Γ0 , H /K, T ), f,Iw f,Iw 0

2 in which each term 0 Ei,j is an R -module of finite type. The pairings (8.9.6.3.1) are compatible with respect to the isomorphisms (8.10.10.1).

Proof. The proof of Proposition 8.10.1 applies, with the same modification as in the proof of Proposition 8.4.8.3. The compatibility with the pairings (8.9.6.3.1) follows from the definitions. (8.10.11) Proposition. Assume that (8.10.3.1) is satisfied. If S ⊂ R is a multiplicative set such that 0 0 e 3 (K∞ e 3 (K∞ H /K, T )S = 0 (recall from 8.10.3 that H /K, T ) is an R-module of finite type), then: f,Iw f,Iw 0 2 0 e3 (i) (∀i ≥ 0) ( Ei,−3 ) ⊗R RS = Hi,cont (Γ , Hf,Iw (K∞ /K, T )) ⊗R RS = 0. (ii) The canonical map 2 2 0 e f,Iw e f,Iw edge : H (K∞ /K, T )Γ0 −→ H (K∞ /K, T ) (which is an edge map in the spectral sequence 0 E m ) satisfies

Ker(edge) ⊗R RS = Coker(edge) ⊗R RS = 0. e 3 (K∞ /K, T )⊗R RS Proof. As in the proof of Proposition 8.10.4, apply Lemma 8.10.5 to B = RS , M = H f,Iw 0 0 and ti = γi − 1 (1 ≤ i ≤ r ). 199

(8.10.12) Proposition. Assume that (8.10.7.1) is satisfied. If S ⊂ R is a multiplicative set such that    0 e f0 (KS /K∞ DR H , A) ⊗R RS = 0, then: i,0 i e 0 (KS /K∞ , A)))) ⊗R RS = 0. (i) (∀i ≥ 0) (DR0 (0 E 2 )) ⊗R RS = (DR0 (Hcont (Γ0 , H f (ii) The kernel and cokernel of the canonical map e 1 (KS /K 0 , A) −→ H e 1 (KS /K∞ , A) 0 resK∞ /K∞ :H f ∞ f satisfy

Γ0





0 ) 0 ) DR0 Ker(resK∞ /K∞ ⊗R RS = DR0 Coker(resK∞ /K∞ ⊗R RS = 0.

Proof. As in 8.10.6 there is a spectral sequence 0

i,j i e j (KS /K∞ , A)) =⇒ H e i+j (KS /K 0 , A) E 2 = Hcont (Γ0 , H ∞ f f

such that the edge homomorphisms of 0 E m and 0 E m are related by the following commutative diagram:   ∼ e i (KS /K 0 , A) −→ e 3−i (K 0 /K, T ∗(1))ι H DR0 H ∞ ∞ f f,Iw     y y   0 ∼ e i (KS /K∞ , A)Γ −→ e 3−i (K∞ /K, T ∗ (1))ι )Γ0 . H DR (H f f,Iw The statement follows from Proposition 8.10.11 applied to T ∗ (1). e 3 (T ) = 0. Then: (8.10.13) Proposition. Assume that (8.10.3.1) is satisfied and H f 3 e (i) H (K /K, T ) = 0. ∞ f,Iw (ii) The canonical map e 2 (K∞ /K, T ) 0 −→ H e 2 (K 0 /K, T ) H f,Iw f,Iw ∞ Γ is an isomorphism. e 3 (K∞ /K, T ) is an R-module of finite type satisfying MΓ = H e 3 (T ) = 0 (using (8.10.3.3)), Proof. (i) M = H f,Iw f hence M = 0 by Nakayama’s Lemma. (ii) Apply Proposition 8.10.11 with S = {1}. e 0 (A) = 0. Then: (8.10.14) Proposition. Assume that (8.10.7.1) is satisfied and H f e 0 (KS /K∞ , A) = 0. (i) H f (ii) The canonical map 0 e f1 (KS /K∞ e f1 (KS /K∞ , A)Γ 0 resK∞ /K∞ :H , A) −→ H

0

is an isomorphism. e 0 (KS /K∞ , A) is an R-module of co-finite type satisfying D(N )Γ = D(N Γ ) = D(H e 0 (A)) = 0 Proof. (i) N = H f f (by (8.10.7.3)); Nakayama’s Lemma implies that D(N ) = 0, hence N = 0. (ii) Apply Proposition 8.10.12 with S = {1}. (8.11) Iwasawa theory over local fields (8.11.1) Assume that F is a local field of characteristic char(F ) 6= p, with finite residue field kF . Let ∼ F∞ /F be a Galois extension with Γ = Gal(F∞ /F ) = Γ0 × ∆, where Γ0 −→ Zrp (r ≥ 1) and ∆ is a finite abelian group. If char(kF ) 6= p, then r = 1 and F∞ is a finite abelian extension of the unique unramified 200

Zp -extension of F ; thus the only ‘interesting’ case is when F is a finite extension of Qp . As usual, put GF = Gal(F sep /F ) and GF∞ = Gal(F sep /F∞ ). (8.11.2) If D

T ←−−−−−−−−→  ←−  −− D −−→ Φ −−−−−  −−−−−−− y − − → ←− A

T∗   Φ  y A∗

∗ b ad is a duality diagram with T, T ∗ ∈ Db ((ad R[GF ] Mod)R−f t ) and A, A ∈ D ((R[GF ] Mod)R−cof t ), then the functors FΓ and FΓ from 8.3-4 define a duality diagram over R D

FΓ (T ) ←−−−−−−−−→ FΓ (T ∗ )ι ←−     −− D −−→ −−−−−   Φ Φ −−−−−−− y y − − → ←− FΓ (A) FΓ (A∗ )ι

(8.11.2.1)

with FΓ (T ), FΓ (T ∗ ) ∈ Db ((ad Mod)R−f t ) and FΓ (A), FΓ (A∗ ) ∈ Db ((ad Mod)R−cof t ). Define R[G ] R[G ] F

F

RΓIw (F∞ /F, T ) = RΓcont (GF , FΓ (T )). As in the global case, we have ∼

• • Ccont (GF , FΓ (A)) −→ lim −→ Ccont (GFα , A)

α ∼ i i −→ lim Hcont (GFα , A) −→ Hcont (GF∞ , A) −→ α ∼ i H i (RΓIw (F∞ /F, T )) −→ ← lim Hcont (GFα , T ), α−

i Hcont (GF , FΓ (A)) i HIw (F∞ /F, T ) :=

∼

where Fα /F runs through all finite subextensions of F∞ /F . Applying Theorem 5.2.6 to (8.11.2.1) we obtain a duality diagram D

RΓIw (F∞ /F, T ) ←−−−−−−−−→ RΓIw (F∞ /F, T ∗ (1))ι [2] ←−     −− D −−→ −−−−−   Φ Φ − − − − − y y − −→ −− − ← RΓcont (GF∞ , A) RΓcont (GF∞ , A∗ (1))ι [2]

(8.11.2.2)

b in D(co)f t (R Mod). On the level of cohomology, (8.11.2.2) yields isomorphisms ∼

2−i i HIw (F∞ /F, T )ι −→ D(Hcont (GF∞ , A∗ (1)))

(8.11.2.3)

and a spectral sequence 2−j j i+j E2i,j = ExtiR (HIw (F∞ /F, T ∗ (1))ι , ωR ) = ExtiR (D(Hcont (GF∞ , A)), ωR ) =⇒ HIw (F∞ /F, T ).

(8.11.2.4)

For simplicity, we shall use in the rest of Sect. 8.11 the following notation: E i (−) = ExtiR (−, ωR ),

i i HIw (−) = HIw (F∞ /F, −).

Note that the isomorphisms (8.11.2.3) are obtained from Tate’s local duality isomorphisms over Fα ∼

2−i i Hcont (GFα , T ) −→ D(Hcont (GFα , A∗ (1)))

by taking the projective limit. 201

(8.11.3) Lemma. If T is concentrated in degree zero, then: i (i) (∀i 6= 1, 2) HIw (T ) = 0. ∼ 2 (ii) HIw (T ) −→ D(H 0 (GF∞ , A∗ (1))ι ) is an R-module of finite type of dimension ≤ d = dim(R). 2 (iii) (∀q < r) E q (HIw (T )) = 0. i Proof. (i) The group HIw (T ) vanishes for i < 0 (resp., i > 2) as τ0 T = 0 and 0 cdp (GF ) = 2). Finally, HIw (T ) = 0 by Proposition 8.3.5(iii). The statement (ii) is just (8.11.2.3) for i = 2, 2 combined with the fact that D(H 0 (GF∞ , A∗ (1))) is a quotient of D(A∗ (1)) = T (−1). As dim(HIw (T )) ≤ d, local duality over R and lemma 2.4.7(iii) show that ∼

d+r−q 2 2 D(E q (HIw (T ))) −→ H{m} (HIw (T )) = 0

for q < r. (8.11.4) Proposition. If both T and T ∗ (1) are concentrated in degree zero, then: (i) There is an exact sequence 0 −→

2 E 1 (HIw (T ∗ (1)))ι

−→

1 1 HIw (T ) −→ E 0 (HIw (T ∗ (1)))ι

−→

−→

2 E 2 (HIw (T ∗ (1)))ι

−→

2 1 HIw (T ) −→ E 1 (HIw (T ∗ (1)))ι

−→

−→

2 E 3 (HIw (T ∗ (1)))ι

−→

and isomorphisms

0

∼

1 2 E q (HIw (T ∗ (1))) −→ E q+2 (HIw (T ∗ (1)))

(q ≥ 2).

2 HIw (T )

(ii) If r > 1, then is pseudo-null over R. 1 1 1 (iii) If r > 1 and R is regular, then HIw (T ) injects into E 0 (HIw (T ∗ (1)))ι = HomR (HIw (T ∗ (1)), H 0 (ωR )) and is torsion-free over R. q Proof. (i) This follows from the spectral sequence (8.11.2.4) and the vanishing of HIw (Z) (Z = T, T ∗ (1)) for q 6= 1, 2. The statement (ii) follows from Lemma 8.11.3(ii). As regards (iii), regularity of R (hence of R) implies that E 1 (M ) = 0 for any pseudo-null R-module M ; we conclude by (ii) and the fact that H 0 (ωR ) is torsion-free over R. ∼

(8.11.5) Proposition. If R = O is a discrete valuation ring (finite over Zp ), Γ −→ Zrp (r ≥ 1), T is supported in degree zero and is torsion-free over O, then: (i) There are isomorphisms of R-modules  0, q 6= r, r + 1    ∼ 2 q=r E q (HIw (T ∗ (1)))ι −→ H 0 (GF∞ , T ),    0 H (GF∞ , A)/B, q = r + 1, where B is the maximal O-divisible submodule of H 0 (GF∞ , A). 1 (ii) The torsion submodule of HIw (T ) is isomorphic to ( 0 H (GF∞ , T ), ∼ 1 HIw (T )R−tors −→ 0,

r=1 r > 1.

1 (iii) HIw (T ) contains no non-zero pseudo-null R-submodules.

Proof. The proof of Lemma 9.1.6 below apllies in the present situation and yields (i). As TO−tors = 0, T ∗ (1) is also concentrated in degree zero, hence the exact sequence from Proposition 8.11.4(i) gives ( 0 H (GF∞ , T ), r=1 ∼ 1 1 2 ∗ ι ∼ HIw (T )R−tors −→ E (HIw (T (1))) −→ 0, r > 1, 202

2 using (i). Finally, as HIw (T ∗ (1)) is R-torsion, (iii) follows from 9.1.3(vi).

(8.11.6) Universal norms. We define
i i i N∞ Hcont (GF , T ) = Im HIw (T )Γ −→ Hcont (GF , T ) = \
cor i i = Im Hcont (GFα , T )−−→Hcont (GF , T ) , α

where Fα are as in 8.11.1. By (8.11.2.3), there is an isomorphism
∼ res 2−i 2−i i D(Hcont (GF , T )/N∞ ) −→ Ker Hcont (GF , A∗ (1))−−→Hcont (GF∞ , A∗ (1)) .

(8.11.6.1)

If T is concentrated in degree zero, so is A∗ (1), hence ∼

1 1 D(Hcont (GF , T )/N∞ ) −→ Hcont (Γ, H 0 (GF∞ , A∗ (1))),

(8.11.6.2)

by the Hochschild-Serre spectral sequence (note that A∗ (1) is a discrete GF -module). ∼

b ad r (8.11.7) Proposition. Assume that T ∈ DR−f t (R[GF ] Mod) and Γ −→ Zp . Then: (i) There is a (homological) spectral sequence −j −i−j 2 Ei,j = Hi,cont (Γ, HIw (T )) =⇒ Hcont (GF , T ), 2 in which all terms Ei,j are R-modules of finite type. ∼ (ii) If τ≤0 T −→ T and S ⊂ R is a multiplicative set such that D(H 0 (GF , D(H 0 (T ))(1)))S = 0, then, for all i, 2 (∀j < −2) (Ei,j )S = 0 1 1 and the localization at S of the canonical map HIw (T )Γ −→ Hcont (GF , T ) is an isomorphism of RS -modules. 0 0 In particular, if codimR (suppR (D(H (GF , D(H (T ))(1))))) ≥ 1, then 1 codimR (suppR (Hcont (GF , T )/N∞ )) ≥ 1. ∼

j Proof. Proposition 8.4.8.1(ii) gives (i). As τ≤0 T −→ T and cdp (GF ) = 2, we have HIw (T ) = 0 for j > 2. The remaining statements follow by the same argument as in the proof of Proposition 8.10.4(ii). ∼

(8.11.8) Corollary. Let R = O, Γ −→ Zrp and T be as in Proposition 8.11.5. If H 0 (GF , T ∗ (1)) = 0, then 1 the group Hcont (GF , T )/N∞ is finite. Proof. This is a special case of Proposition 8.11.7, but it can be proved more directly as follows: the assumption H 0 (GF , T ∗ (1)) = 0 is equivalent to the finiteness of H 0 (GF , A∗ (1)) = H 0 (Γ, H 0 (GF∞ , A∗ (1))) (where A∗ = D(T )). Lemma 8.10.5 then implies that H 1 (Γ, H 0 (GF∞ , A∗ (1))) is finite, too. We conclude by (8.11.6.2). (8.11.9) Over each finite subextension Fα /F of F∞ /F , the pairing ev2 : T ⊗R T ∗ (1) −→ ωR (1) induces cup products (5.2.2.1) j i h , iα : Hcont (GFα , T ) ⊗R Hcont (GFα , T ∗ (1)) −→ H i+j−2 (ωR ).

Similarly, the pairing 203

ev2 : FΓ (T ) ⊗R FΓ (T ∗ (1))ι −→ ωR (1) (cf. 8.4.6.3) induces products on the “Iwasawa cohomology”

h, i:

   ι j i ∗ lim H (G , T ) ⊗ lim H (G , T (1)) −→ H i+j−2 (ωR ) = Fα Fα cont cont R ← ← α− α−
= H i+j−2 (ωR ) ⊗R R = lim H i+j−2 (ωR ) ⊗R R[Gal(Fα /F )] . ← − α

(8.11.10) Proposition. In the situation of 8.11.9,  h(xα ), (yα )i = 

X

 hxα , σyα iα ⊗ [σ]

σ∈Gal(Fα /F )

α

j (where σ acts on Hcont (GFα , T ∗ (1)) by conjugation).

Proof. As the pairing h , i is R-bilinear, it suffices to check that the coefficient at [1] in the projection of the L.H.S. to H i+j−2 (ωR ) ⊗R R[Gal(Fα /F )] is equal to hxα , yα iα . This, in turn, is a consequence of Lemma 8.1.6.5 and the fact that the local invariant maps invv commute with corestriction (cf. 9.2.2 below). (8.12) In the absence of (P ) (8.12.1) The discussion in 6.9 applies to R and FΓ (X), FΓ (Y )ι .

204

9. Classical Iwasawa theory In this chapter we apply the duality results from Chapter 8 to classical Iwasawa theory, obtaining new results on ideal class groups in Zrp -extensions of number fields (Sect. 9.4-5). In Sect. 9.6 we compare groups arising in our theory to classical Selmer groups and in Sect. 9.7 we show that well-behaved perfect complexes naturally appear in this context. These results were used in the work of Mazur and Rubin [M-R 2] on “organizing modules” in Iwasawa theory of elliptic curves. Let K∞ /K be as in 8.5.1 (hence we also assume that (P ) from 5.1 holds). (9.1) Generalities (9.1.1) We consider the following special case of 8.5: R = O is the ring of integers in a finite extension F ∼ ∼ of Qp and Γ −→ Zrp (r ≥ 1); we fix a prime element $ ∈ O. In this case I −→ F/O and the ring R = R[[Γ]] is equal to the usual Iwasawa algebra ∼

Λ = O[[Γ]] −→ O[[X1 , . . . , Xr ]], ∼

which is a regular ring of dimension dim(Λ) = r + 1. This implies that ωΛ −→ Λ (by 2.7(iii), since regular local rings are Gorenstein). (9.1.2) In the category (Λ Mod)/(pseudo − null), every Λ-module of finite type M is isomorphic to Λa ⊕

b M

Λ/fi Λ

(fi ∈ Λ − {0}).

i=1

The element charΛ (M ) = char(M ) = f1 · · · fb (mod Λ∗ ) ∈ Frac(Λ)∗ /Λ∗ (“the characteristic power series of M ”) depends only on the isomorphism class of M in (Λ Mod)/(pseudo − null). (9.1.3) For every Λ-module of finite type M and i ≥ 0, put E i (M ) = ExtiΛ (M, Λ) (E 1 (M ) is the “Iwasawa adjoint” of M ). These Λ-modules have the following properties. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

E i (M ) is a Λ-module of finite type. codimΛ (supp(E i (M ))) ≥ i. E i (M ) = 0 for i < dim(Λ) − dim(M ). E 0 (M ) is a reflexive Λ-module. The canonical map E 1 (M ) −→ E 1 (Mtors ) is an isomorphism in (Λ Mod)/(pseudo − null). E 1 (Mtors ) has no non-zero pseudo-null submodules. E 1 (Mtors ) is isomorphic to Mtors in (Λ Mod)/(pseudo − null). If M is a Cohen-Macaulay Λ-module, then E i (M ) = 0 for i 6= dim(Λ) − dim(M ). ∼ If M = Λ/(x1 , . . . , xa )Λ is the quotient of Λ by a regular sequence of length a, then E a (M ) −→ M .

For (i), take a finitely generated free resolution of M ; (ii), (iii) and (viii) follow from Lemma 2.4.7 (ii)-(iv) and Local Duality 2.5; (vi) and (vii) are proved in ([PR 1], Ch. I, Prop. 8), while (ix) follows from the Koszul resolution of M . As the kernel (resp., cokernel) of the canonical map ε : M −→ E 0 (E 0 (M )) is torsion (resp., pseudo-null), the exact sequence of Ext’s together with (vi) and (vii) prove (iv). Finally, the exact sequence of Ext’s together with (ii) show that, for each x ∈ Λ − {0}, E 1 (M/Mtors )/x is pseudo-null; it follows that E 1 (M/Mtors ) itself is pseudo-null, proving (v). (9.1.4) Let T be a free O-module of finite type equipped with a continuous O-linear action of GK,S . Then the O[GK,S ]-modules 205

T ∗ = HomO (T, O)

A∗ = HomO (T, F/O) = T ∗ ⊗O F/O A = HomO (T ∗ , F/O) = T ⊗O F/O are related by the duality diagram D

T ←−−−−−−−−→  ←−  −− D −−→ Φ −−−−−  −−−−−−− y − − → ←− A

T∗   Φ  y A∗

over O. According to Theorem 8.5.6, there is a spectral sequence 3−j i+j E2i,j = E i (Hc,Iw (K∞ /K, T ∗(1)))ι = E i (DΛ (H j (KS /K∞ , A))) =⇒ HIw (K∞ /K, T ).

(9.1.4.1)

Above, DΛ (M ) coincides with the Pontrjagin dual of M , for every Λ-module of finite or co-finite type. The Λ-action on DΛ (M ) = HomΛ (M, DΛ (Λ)) = HomO,cont (M, F/O) is given by (λ · f )(m) = f (λm). Similarly, there is a spectral sequence 0

3−j i+j E2i,j = E i (HIw (K∞ /K, T ∗ (1)))ι = E i (DΛ (Hcj (KS /K∞ , A))) =⇒ Hc,Iw (K∞ /K, T ).

(9.1.4.2)

Together with 9.1.3(ii), these spectral sequences imply that j corkΛ H j (KS /K∞ , A) = rkΛ HIw (K∞ /K, T ) j corkΛ Hcj (KS /K∞ , A) = rkΛ Hc,Iw (K∞ /K, T )

(9.1.5) Lemma. (i) 0 E2i,j = 0 for j 6= 1, 2. (ii) The spectral sequence 0 Er induces isomorphisms of Λ-modules ∼

2 1 E 0 (HIw (K∞ /K, T ∗(1))) −→ Hc,Iw (K∞ /K, T )ι ∼

2 1 E i (HIw (K∞ /K, T ∗(1))) −→ E i−2 (HIw (K∞ /K, T ∗ (1)))

(i > 3)

and an exact sequence 2 2 1 0 −→ E 1 (HIw (K∞ /K, T ∗(1))) −→ Hc,Iw (K∞ /K, T )ι −→ E 0 (HIw (K∞ /K, T ∗(1))) −→ 2 3 1 −→ E 2 (HIw (K∞ /K, T ∗(1))) −→ Hc,Iw (K∞ /K, T )ι −→ E 1 (HIw (K∞ /K, T ∗(1))) −→ 2 −→ E 3 (HIw (K∞ /K, T ∗(1))) −→ 0.

(iii) (cf. [Gre 2], Prop. 4) DΛ (H 2 (KS /K∞ , A∗ (1))) is a reflexive Λ-module. S Proof. (i) As cdp (GK,S ) = 2, we have 0 E2i,j = 0 for j 6= 1, 2, 3. Put T0 = α H 0 (Gal(KS /Kα ), T ) ⊆ T ; then 0 0 HIw (K∞ /K, T ) = HIw (K∞ /K, T0 ). The transition maps in the projective system H 0 (Gal(KS /Kα ), T0 ) are 0 given, ultimately, by the multiplication by [Kβ : Kα ]. Hence HIw (K∞ /K, T0 ) = 0 and 0 E2i,3 = 0. (ii) This follows from (i). ∼ 1 2 (iii) By (ii), DΛ (H 2 (KS /K∞ , A∗ (1))) −→ Hc,Iw (K∞ /K, T )ι is isomorphic to E 0 (HIw (K∞ /K, T ∗(1))), which is reflexive by 9.1.3(iv). 206

(9.1.6) Lemma. There is a canonical isomorphism of Λ-modules

E2i,0

 0,    ∼ −→ T GK∞ ,    GK A ∞ /B,

i 6= r, r + 1 i=r i = r + 1,

where B = T GK∞ ⊗O F/O is the maximal O-divisible submodule of AGK∞ . Proof. Let M = DΛ (AGK∞ ). We distinguish three cases: (i) AGK∞ = H 0 (Gal(KS /K∞ ), A) is finite. (ii) AGK∞ = A. (iii) The general case. (i) As M is a Λ-module of finite length, we have E i (M ) = 0 for i 6= dim(Λ) = r + 1, by 9.1.3(iii). Local duality implies that ∼

0 E r+1 (M ) −→ DΛ (H{m} (M )) = DΛ (M ) = AGK∞ .

(ii) As an O-module, M = DΛ (A) = T ∗ is free of finite rank. Thus E i (M ) = 0 for i 6= dim(Λ) − 1 = r, by 9.1.3(viii). The exact sequence of Ext’s attached to pn

0 −→ M −→M −→ M/pn M −→ 0 together with case (i) give ∼

∼

E r (M )/pn E r (M ) −→ E r+1 (M/pn M ) −→ A[pn ], hence ∼

r n r E r (M ) −→ lim ←− E (M )/p E (M ) = Tp (A) = T. n

GK∞

(iii) Note that B is a Λ-submodule of A

. The exact sequence of Ext’s attached to

0 −→ DΛ (AGK∞ /B) −→ M −→ DΛ (B) −→ 0 together with cases (i) and (ii) give isomorphisms ∼

∼

E r+1 (M ) −→ E r+1 (DΛ (AGK∞ /B)) −→ AGK∞ /B ∼

∼

E r (M ) ←− E r (DΛ (B)) −→ Tp (B) = T GK∞ ∼

E i (M ) −→ 0

(i 6= r, r + 1).

(9.1.7) Corollary. There is a canonical isomorphism of Λ-modules ( 1 HIw (K∞ /K, T )tors

∼

−→

E21,0

∼

−→

T GK∞ ,

r=1

0,

r > 1.

(9.2) Cohomology of Zp (1) In this section, O = Zp . (9.2.1) Every finite discrete GK,S -module M determines an ´etale sheaf Met on Spec(OK,S ). If the order of M is a unit in OK,S , then the spectral sequence E2i,j = lim H i (Gal(L/K), H j (Spec(OL,S )et , Met )) =⇒ H i+j (Spec(OK,S )et , Met ), −→ L in which L runs through finite subextensions of KS /K, degenerates into isomorphisms ∼

E2i,0 = H i (GK,S , M ) −→ H i (Spec(OK,S )et , Met ). 207

More precisely, there is a canonical isomorphism ∼

RΓ(Spec(OK,S )et , Met ) −→ RΓcont (GK,S , M ) in D+ (Ab). The exact sequence of sheaves pn

0 −→ µpn −→ Gm −→Gm −→ 0 on Spec(OK,S )et and the standard description of the Brauer group Br(K) yield the following exact sequences (cf. [Sch 1]): ∼

µpn (K) −→H 0 (GK,S , µpn ) ∗ 0 −→ OK,S ⊗ Z/pn Z −→H 1 (GK,S , µpn ) −→ Pic(OK,S )[pn ] −→ 0 M Σ 0 −→ Pic(OK,S ) ⊗ Z/pn Z −→H 2 (GK,S , µpn ) −→ Z/pn Z−→Z/pn Z −→ 0. v∈Sf

Passing to the inductive (resp., projective) limit with respect to n, we obtain ∼

H 0 (GK,S , Qp /Zp (1)) −→ µp∞ (K) ∗ 0 −→ OK,S ⊗ Qp /Zp −→H 1 (GK,S , Qp /Zp (1)) −→ Pic(OK,S )[p∞ ] −→ 0 M Σ 0 −→H 2 (GK,S , Qp /Zp (1)) −→ Qp /Zp −→Qp /Zp −→ 0

(9.2.1.1)

v∈Sf i

H (GK,S , Qp /Zp (1)) = 0

(i > 2).

resp., ∼

∗ 1 OK,S ⊗ Zp −→Hcont (GK,S , Zp (1))

M

2 0 −→ Pic(OK,S )[p∞ ] −→Hcont (GK,S , Zp (1)) −→

Σ

Zp −→Zp −→ 0

v∈Sf i Hcont (GK,S , Zp (1)) = 0

(9.2.1.2)

(i 6= 1, 2).

In particular, 1 2 rkZp Hc,cont (GK,S , Zp ) = rkZp Hcont (GK,S , Zp (1)) = |Sf | − 1 2 1 rkZp Hc,cont (GK,S , Zp ) = rkZp Hcont (GK,S , Zp (1)) = r1 + r2 + |Sf | − 1 q 3−q rkZp Hc,cont (GK,S , Zp ) = rkZp Hcont (GK,S , Zp (1)) = 0

(q 6= 1, 2),

hence X X q 0 (−1)q rkZp Hc,cont (GK,S , Zp ) = r1 + r2 = rkZp Hcont (Gv , Zp ), q

v|∞

in line with Theorem 5.3.6. S (9.2.2) Write K∞ = Kα as a union of finite extensions of K. Let Sα be the set of all primes of Kα above S. If Kα ⊂ Kβ and wβ ∈ Sβ lies above vα ∈ Sα , then the corresponding local Brauer groups are related by Br((Kα )vα )  inv y vα Q/Z

res

−→ Br((Kβ )wβ )  invw y β nαβ

−→

cor

Br((Kβ )wβ )  invw y β

id

Q/Z,

Br((Kα )vα ) ←−  inv y vα

Q/Z

Q/Z

where nαβ = [(Kβ )wβ : (Kα )vα ]. Put 208

←−

∗ Eα0 = OK ⊗ Zp , α ,Sα

0 E∞

=

lim Eα0 , −→ α

A0∞

A0α = Pic(OKα ,Sα )[p∞ ] 0 = lim −→ Aα ,

0 X∞ = lim A0α . ← α−

α

0 It is well-known that X∞ and DΛ (A0∞ ) are torsion Λ-modules of finite type ([Gre 1], proof of Thm. 1).

Applying (9.2.1.1-2) to each Kα and using the above description of the transition maps for the local Brauer groups, we obtain the following exact sequences (and isomorphisms): H i (KS /K∞ , Qp /Zp (1)) = 0

(i > 2)

∼

0

H (KS /K∞ , Qp /Zp (1)) −→ µp∞ (K∞ ) 0 −→

0 E∞

⊗ Qp /Zp −→H 1 (KS /K∞ , Qp /Zp (1)) −→ A0∞ −→ 0 M ∼ H 2 (KS /K∞ , Qp /Zp (1)) −→ DΛ (Λ).

(9.2.2.1)

v∈Sf Γv =0

i HIw (K∞ /K, Zp (1)) = 0 1 HIw (K∞ /K, Zp (1))

∼

−→

0 2 0 −→ X∞ −→HIw (K∞ /K, Zp (1)) −→

(i 6= 1, 2) 0 lim ← − Eα α

M

(9.2.2.2)

Λv −→ Zp −→ 0,

v∈Sf

where Γv ⊂ Γ is the decomposition group of any prime v∞ of K∞ above v, and Λv = Zp [[Γ/Γv ]]. In particular, the Λ-module ∼

DΛ (H 2 (KS /K∞ , Qp /Zp (1))) −→

M

∼

2 2 Λ −→ HIw (K∞ /K, Zp (1))/HIw (K∞ /K, Zp (1))tors

(9.2.2.3)

v∈Sf Γv =0

is free and 0 2 0 −→ X∞ −→ HIw (K∞ /K, Zp (1))tors −→

M

Λv −→ Zp −→ 0

(9.2.2.4)

v∈Sf Γv 6=0

is an exact sequence of torsion Λ-modules. (9.3) Pseudo-null submodules (9.3.1) Proposition. (cf. [Gre 2], Prop. 5) If, in the notation of 9.1, H 2 (KS /K∞ , A) = 0, then DΛ (H 1 (KS /K∞ , A)) has no non-zero pseudo-null submodules. Proof. Lemma 9.1.5(ii) applied to T ∗ (1) instead of T (combined with the fact that E 0 (−) is torsion-free) yields isomorphisms ∼

∼

2 2 2 DΛ (H 1 (KS /K∞ , A))tors −→ Hc,Iw (K∞ /K, T ∗ (1))ιtors −→ E 1 (HIw (K∞ /K, T )) = E 1 (HIw (K∞ /K, T )tors ),

where the last equality follows from 2 rkΛ HIw (K∞ /K, T ) = rkΛ DΛ (H 2 (KS /K∞ , A)) = 0.

We conclude by 9.1.3(vi). 209

(9.3.2) Corollary. ([Ng], Thm. 3.1) Let M∞ be the maximal pro-p-abelian extension of K∞ , unramified outside the primes above S. If H 2 (KS /K∞ , Qp /Zp ) = 0 (i.e. if K∞ satisfies the “weak Leopoldt conjecture”), then Gal(M∞ /K∞ ) has no non-zero pseudo-null submodules. Proof. We have ∼

DΛ (Gal(M∞ /K∞ )) −→ H 1 (KS /K∞ , Qp /Zp )ι (the involution ι appears, because of different sign conventions for the Γ- and Λ- action on DΛ (−)). Apply Proposition 9.3.1 with T = Zp . (9.3.3) Proposition. DΛ (A0∞ ) has no non-zero pseudo-null submodules. Proof. By (9.2.2.1), DΛ (A0∞ ) is contained in ∼

∼

2 2 DΛ (H 1 (KS /K∞ , Qp /Zp (1)))tors −→ Hc,Iw (K∞ /K, Zp )ιtors −→ E 1 (HIw (K∞ /K, Zp (1))),

where the last isomorphism follows from the spectral sequence (9.1.4.2) for T = Zp . As the quotient 2 2 HIw (K∞ /K, Zp (1))/HIw (K∞ /K, Zp (1))tors is free over Λ (by (9.2.2.3)), it follows that 2 2 E 1 (HIw (K∞ /K, Zp (1))) = E 1 (HIw (K∞ /K, Zp (1))tors );

we again conclude by 9.1.3(vi). (9.3.4) Greenberg ([Gre 6], Thm. 2) recently proved the following generalization of Proposition 9.3.1 (and of Lemma 9.1.5(iii)): denote, using the notation of 9.2.2,  X (K∞ , A) = Ker H j (KS /K∞ , A) −→ j

j Hloc,v (K∞ , A) = − lim → α

j HIw ((K∞ /K)v , T )

= lim ←− α

M

M

 j Hloc,v (K∞ , A)

v∈Sf

H j (Gvα , A) = H j (Gv , FΓ (A))

vα |v

M

j j Hcont (Gvα , T ) = Hcont (Gv , FΓ (T )).

vα |v

Then the Λ-module DΛ (X2 (K∞ , A)) is reflexive; furthermore, if X2 (K∞ , A) = 0, then DΛ (H 1 (KS /K∞ , A)) has no non-zero pseudo-null submodules. We present in 9.3.5-7 below an alternative proof of Greenberg’s result. (9.3.5) Lemma. (i) The sequence 0 −→ X2 (K∞ , A) −→ H 2 (KS /K∞ , A) −→

M

2 Hloc,v (K∞ , A) −→ 0

v∈Sf

is exact. 2 (ii) For each v ∈ Sf , the Pontrjagin dual of Hloc,v (K∞ , A) is a free Λ-module of rank ( uv =

0,

Γv 6= 0

rkO H 0 (Gv , T ∗ (1)),

Γv = 0.

(iii) The composite map (induced by the isomorphism from the proof of Lemma 9.1.5(iii)) ∼

2 2 HIw (K∞ /K, T ) −→ E 0 (E 0 (HIw (K∞ /K, T ))) −→ E 0 (DΛ (H 2 (KS /K∞ , A))) −→ M ∼ 2 −→ E 0 (DΛ (Hloc,v (K∞ , A))) −→ Λu v∈Sf Γv =0

210

P (where u = v∈Sf uv ) is surjective. (iv) The sequence obtained by applying E 0 ◦ DΛ to the exact sequence from (i) is exact. Proof. (i),(ii) The Poitou-Tate exact sequence 5.1.6 yields, in the limit over all finite subextensions Kα /K of K∞ /K, an exact sequence M

0 −→ X2 (K∞ , A) −→ H 2 (KS /K∞ , A) −→


ι 2 0 Hloc,v (K∞ , A) −→ DΛ HIw (K∞ /K, T ∗(1)) −→ 0;

v∈Sf 0 the Λ-module HIw (K∞ /K, T ∗(1)) vanishes, by (the proof of) Lemma 9.1.5(i). The same argument also 2 0 shows that the Λ-module DΛ (Hloc,v (K∞ , A)) = HIw ((K∞ /K)v , T ∗ (1))ι vanishes if Γv = 6 0. If Γv = 0, then
0 ∗ 0 ∗ HIw ((K∞ /K)v , T (1)) = FΓ H (Gv , T (1)) is a free Λ-module of rank uv . (iii) After applying ι ◦ DΛ , it is enough to show that the map

M

M M
ι
ι ◦ DΛ ◦ E 0 FΓ (H 0 (Gv , T ∗ (1))) = FΓ ΦO (H 0 (Gv , T ∗ (1))) = FΓ (H 0 (Gv , A∗ (1))div ) ,→

v∈Sf Γv =0

,→

M

v∈Sf Γv =0

v∈Sf Γv =0

0 Hloc,v (K∞ , A∗ (1)) −→ Hc1 (KS /K∞ , A∗ (1))

v∈Sf Γv =0

(where we have used Lemma 8.4.6.4 for the first equality) is injective; this follows from the exact sequence 0 −→ H 0 (KS /K∞ , A∗ (1)) −→

M

0 Hloc,v (K∞ , A∗ (1)) −→ Hc1 (KS /K∞ , A∗ (1))

v∈Sf

and the fact that there exists v ∈ Sf for which Γv 6= 0. (iv) We must show that the map M M


2 2 E 0 DΛ (H 2 (KS /K∞ , A)) −→ E 0 DΛ (Hloc,v (K∞ , A)) = E 0 DΛ (Hloc,v (K∞ , A)) v∈Sf

v∈Sf Γv =0

is surjective; this follows from (iii). (9.3.6) Proposition ([Gre 6], Thm. 2). The Λ-module DΛ (X2 (K∞ , A)) is reflexive (in particular, X2 (K∞ , A) = 0 ⇐⇒ corkΛ X2 (K∞ , A) = 0). Proof. By Lemma 9.3.5, there is an exact sequence 0 −→ Λu −→ DΛ (H 2 (KS /K∞ , A)) −→ DΛ (X2 (K∞ , A)) −→ 0;

(9.3.6.1)

moreover, the sequence obtained by applying E 0 to (9.3.6.1) 0 −→ E 0 (DΛ (X2 (K∞ , A))) −→ E 0 (DΛ (H 2 (KS /K∞ , A))) −→ E 0 (Λu ) −→ 0

(9.3.6.2)

∼

is also exact. As E 0 (Λu ) −→ Λu is a free Λ-module, it follows that the double dual of (9.3.6.1) 0 −→ E 0 (E 0 (Λu )) −→ E 0 (E 0 (DΛ (H 2 (KS /K∞ , A)))) −→ E 0 (E 0 (DΛ (X2 (K∞ , A)))) −→ 0 is also exact. As both Λu and DΛ (H 2 (KS /K∞ , A)) are reflexive (by Lemma 9.1.5(iii)), we deduce from the Snake Lemma that the third module DΛ (X2 (K∞ , A)) is also reflexive. 211

(9.3.7) Proposition. If X2 (K∞ , A) = 0, then: ∼ 2 2 (i) HIw (K∞ /K, T )/HIw (K∞ /K, T )tors −→ Λu is a free Λ-module of
rank∼ u.
2 (ii) There is an isomorphism of Λ-modules DΛ (H 1 (KS /K∞ , A)) tors −→ E 1 HIw (K∞ /K, T )tors . (iii) ([Gre 6], Thm. 2) The Λ-module DΛ (H 1 (KS /K∞ , A)) has no non-zero pseudo-null submodules. Proof. (i) The assumption X2 (K∞ , A) = 0 implies, thanks to Lemma 9.3.5(i)-(ii), that X 2 2 rkΛ HIw (K∞ /K, T ) = corkΛ H 2 (KS /K∞ , A) = corkΛ Hloc,v (K∞ , A) = u. v∈Sf 2 According to Lemma 9.3.5(iii), there exists a surjection HIw (K∞ /K, T )  Λu ; its kernel must be equal to 2 HIw (K∞ /K, T )tors . (ii) It follows from Lemma 9.1.5(ii) that


2 2 DΛ (H 1 (KS /K∞ , A)) tors = Hc,Iw (K∞ /K, T )ιtors = E 1 (HIw (K∞ /K, T )). 2 2 On the other hand, (i) implies that E 1 (HIw (K∞ /K, T )) = E 1 (HIw (K∞ /K, T )tors ). (iii) This follows from (ii), by 9.1.3(vi). 0 (9.4) Relating A0∞ and X∞

0 (9.4.1) It is generally expected that X∞ and DΛ (A0∞ ) are isomorphic in (Λ Mod)/(pseudo − null); in particular, their charateristic power series should coincide. 0 It is known that DΛ (A0∞ ) is isomorphic to E 1 (X∞ ) in (Λ Mod)/(pseudo − null) (resp., in (Λ Mod)), provided r = 1 ([Iw], Thm. 11) (resp., |Sf | = 1 ([McCa 2], Thm. 8)).

In this section we define a canonical homomorphism of Λ-modules 0 α0 : X∞ −→ E 1 (DΛ (A0∞ )),

show that Coker(α0 ) is very close to being pseudo-null and that 0 charΛ (DΛ (A0∞ )) charΛ (X∞ ). n (9.4.2) Denote by F ∗ HIw (K∞ /K, Zp (1)) the filtration induced by the spectral sequence (9.1.4.1) for T = Zp (1): i+j E2i,j = E i (DΛ (H j (KS /K∞ , Qp /Zp (1)))) =⇒ HIw (K∞ /K, Zp (1)).

As codimΛ (supp(E2i,j )) ≥ i and

( E2i,2

=

0,

i 6= 0

Λ − free,

i=0

2 2 by (9.2.2.1), we have F 1 HIw (K∞ /K, Zp (1)) = HIw (K∞ /K, Zp (1))tors .

The terms E2i,0 can be determined from Lemma 9.1.6: writing µp∞ (K∞ ) = µpm with m ∈ N ∪ {∞} and ( 0, m=∞ ε= 1, m < ∞, then ( E2i,0

=

0,

i 6= r + ε

Zp /pm Zp (1),

i=r+ε

∞

(with the convention Zp /p Zp = Zp ). The spectral sequence Er induces a map 212

2 δ : HIw (K∞ /K, Zp (1))tors −→ E21,1

with pseudo-null kernel and cokernel. More precisely, the previous discussion and the exact sequence d1,1

δ

3,0 2 2 E22,0 −→ HIw (K∞ /K, Zp (1))tors −→E21,1 −−→E 2

imply the following result (cf. [McCa 2], Thm. 13). (9.4.3) Lemma. (i) If r + ε 6= 2, 3, then δ is an isomorphism. (ii) If r + ε = 2 (resp., r + ε = 3), then there is an exact sequence δ

2 Zp /pm Zp (1) −→ HIw (K∞ /K, Zp (1))tors −→E21,1 −→ 0

resp.,

δ

2 0 −→ HIw (K∞ /K, Zp (1))tors −→E21,1 −→ Zp /pm Zp (1).

(9.4.4) We shall use the following notation: 0 Y = DΛ (E∞ ⊗ Qp /Zp ) Sur = {v ∈ Sf | v is unramified in K∞ /K} M Zur = Λv v∈Sur Γv 6=0



 M   Z = Ker  Λv −→ Zp  . v∈Sf Γv 6=0

The canonical projection Z −→ Zur is surjective; denote its kernel by Zram . (9.4.5) The map δ is a bridge between two exact sequences: 0

−→

i

0 X∞ .. .

−→

E 1 (DΛ (A0∞ ))

←−

0

↓α

E 2 (Y ) ←−

i0

2 HIw (K∞ /K, Zp (1))tors   yδ

−→

E21,1

←−

j

Z

−→

0 (9.4.5.1)

E 1 (Y ) ←−

0

The top (resp., bottom) row is equal to (9.2.2.4) (resp., to E • ◦ DΛ applied to (9.2.2.1)). We define 0 α0 : X∞ −→ E 1 (DΛ (A0∞ ))

to be the composite map α0 = j ◦ δ ◦ i. Conjecturally, both Ker(α0 ) and Coker(α0 ) are pseudo-null. The morphism δ is invertible in the category (Λ Mod)/(pseudo − null), hence defines a morphism δ −1

2 γ : E 1 (Y ) −→ E21,1 −−→HIw (K∞ /K, Zp (1))tors −→ Z

in the latter category. The statement that Ker(α0 ) is pseudo-null (resp., Coker(α0 ) is pseudo-null) is equivalent to Ker(γ) = 0 (resp., Coker(γ) = 0) in (Λ Mod)/(pseudo − null). (9.4.6) Definition. A prime ideal p ∈ Spec(Λ) with ht(p) = 1 is exceptional if there is v ∈ Sf − Sur such ∼ that Γv = hγv i −→ Zp and γv − 1 ∈ p. (9.4.7) Proposition. If p ∈ Spec(Λ) with ht(p) = 1 is not exceptional, then Coker(α0 )p = 0. Proof. We must show that Coker(γ)p = 0. The assumption on p implies that (Zram )p = 0, which means that it is enough to show that the composite map 213

γ

γur : E 1 (Y )−→Z −→ Zur has Coker(γur ) = 0 in (Λ Mod)/(pseudo − null). For each v ∈ Sf there is a semi-local version of the spectral sequence Er , namely v

E2i,j = E i (DΛ (

M

i+j H j ((K∞ )v∞ , Qp /Zp (1)))) =⇒ HIw ((K∞ /K)v , Zp (1)),

(9.4.7.1)

v∞ |v

where we denote M v∞ |v ∼

H j ((K∞ )v∞ , −) = − lim → α

M

H j ((Kα )vα , −).

vα |v

r(v)

As Γv −→ Zp with 0 ≤ r(v) ≤ r, the ring Λv is a quotient of Λ by a regular sequence of length r(v). As in (9.2.2.1) and 9.4.2, we have ( 0, i 6= 0 or Γv 6= 0 v i,2 ∼ E2 −→ Λ, i = 0, Γv = 0. The invariant maps invvα for vα |v together with a choice of v∞ |v induce an isomorphism ∼

2 HIw ((K∞ /K)v , Zp (1)) −→ Λv ,

which implies that ( F

1

2 HIw ((K∞ /K)v , Zp (1))

=

2 HIw ((K∞ /K)v , Zp (1)),

Γv 6= 0

0,

Γv = 0.

As in the global case, we have µp∞ ((K∞ )v∞ ) = µpm(v) for some m(v) ∈ N ∪ {∞}. Put ( 0, m(v) = ∞ ε(v) = 1, m(v) < ∞; then ( v i,0 E2

=

0,

i 6= r(v) + ε(v)

Zp /pm(v) Zp [[Γ/Γv ]],

i = r(v) + ε(v).

The exact sequence v

v 1,1

d

δ

v 2 2 v 1,1 v 3,0 E22,0 −→ F 1 HIw ((K∞ /K)v , Zp (1))−−→ E2 −−→ E2

implies that, in the case r(v) = 1, δv is an isomorphism in (Λ Mod)/(pseudo − null). Fix v ∈ Sur . For each Kα , the valuations vα |v define surjective maps M

(vα )

H 1 ((Kα )vα , Qp /Zp (1))−−→

vα |v

M

Qp /Zp ,

vα |v

which induce in the limit, after dualization, injective maps   M βv : Λv ,→ DΛ  H 1 ((Kα )vα , Qp /Zp (1)) . vα |v

In the global situation, the composite map 214

0 E∞ ⊗ Qp /Zp −→

M M

(vα )

H 1 ((Kα )vα , Qp /Zp (1))−−→

v∈Sur vα |v

M M

Qp /Zp

v∈Sur vα |v

is also surjective, giving rise to an injective map   M M M (βv ) β: Λv −−→ DΛ  H 1 ((Kα )vα , Qp /Zp (1)) −→ Y. v∈Sur

v∈Sur

vα |v

The induced maps on E 1 sit in the following commutative diagram: 2 HIw (K∞ /K, Zp (1))tors   yδ

−→

E21,1 x  

−→

E 1 (Y )

−−→

L

E 1 (β)

v∈Sur Γv 6=0

∼

2 F 1 HIw ((K∞ /K)v , Zp (1)) −→  (δ ) y v L v 1,1 v∈Sur E2 Γv 6=0   1 yE (βv ) L 1 v∈Sur E (Λv )

L

v∈Sur Γv 6=0

Λv = Zur

Γv 6=0

For each v ∈ Sur , the condition Γv 6= 0 implies that r(v) = 1; thus δv is an isomorphism. We know that Ker(δ) and Coker(δ) are pseudo-null. This is also true for Coker(E 1 (β)), by the injectivity of β, hence ∼ also for Coker(E 1 (βv )). As E 1 (Λv ) −→ Λv (by 9.1.3(ix)), it follows that the composite map E 1 (βv ) ◦ (δv ) has pseudo-null kernel and cokernel. Putting all this together, we see that the composite morphism in (Λ Mod)/(pseudo − null) M δ −1 ∼ 2 2 E 1 (Y ) −→ E21,1 −−→HIw (K∞ /K, Zp (1))tors −→ F 1 HIw (K∞ /K, Zp (1)) −→ Zur , v∈Sur Γv 6=0

which is equal to γur , has Coker(γur ) = 0. This proves the claim. ∼

r(v)

(9.4.8) Corollary. If, for every v ∈ Sf ramified in K∞ /K, we have Γv −→ Zp map 0 α0 : X∞ −→ E 1 (DΛ (A0∞ ))

with r(v) ≥ 2, then the

has pseudo-null cokernel. Proof. Under these assumptions, Zram is pseudo-null and there are no exceptional p ∈ Spec(Λ). (9.4.9) Ranks (9.4.9.1) Definition. Let A be a noetherian domain with fraction field F , and M an A-module of finite type. The rank of M is rkA (M ) = dimF (M ⊗A F ). If rkA (M ) = 0 and p ∈ Spec(A) has height ht(p) = 1, put ep (M ) = `Ap (Mp ) < ∞. ∼

∼

(9.4.9.2) Let Γ = Γ0 × Γ0 , where Γ0 −→ Zp and Γ0 −→ Zr−1 (r ≥ 1). Fix a topological generator γ0 ∈ Γ0 ; p n ∼ 0 0 0 then Λ −→ Λ [[γ0 − 1]], where Λ = Zp [[Γ ]]. For each n ≥ 0, put ωn = γ0p , νn = ωn /ωn−1 (where ω−1 = 1). Then each (νn ) is a prime ideal in Λ with rkΛ0 (Λ/νn Λ) = ϕ(pn ). If p ∈ Spec(Λ) and ht(p) = 1, then 215

( n

ep (Zp [[Γ/p Γ0 ]]) = ep (Λ/ωn Λ) =

1,

p = (νi ),

0,

otherwise.

0≤i≤n

(9.4.9.3) Lemma. Let M be a Λ-module of finite type. For each n ≥ 0 we have ∼

(Mtors )(νn ) −→

kn M

Λ(νn ) /(νnm(n,i) )

(m(n, i) ≥ 1).

i=1

Then, for each n ≥ 0, (i) 0 ≤ kn ≤ e(νn ) (Mtors ). Q kj (ii) divides charΛ (Mtors ). j≥ 0 (νj ) Pn (iii) rkΛ0 (M/ωn M ) = pn rkΛ (M ) + j=0 kj ϕ(pj ). (iv) rkΛ0 (M/ωn M ) = pn rkΛ (M ) + O(1) for n −→ ∞. Q e Pkj Proof. (i), (ii) charΛ (Mtors ) is divisible by νj j , where ej = eνj (Mtors ) = i=1 m(j, i) ≥ kj ≥ 0. (iii) Fix n ≥ 0. In (Λ Mod)/(pseudo − null), M is isomorphic to Λa ⊕ M1 ⊕ M2 , where a = rkΛ (M ), M1 , M2 are Λ-torsion, M1 =

kj n M M

m(j,i)

Λ/νj

Λ

j=0 i=1

and supp(M2 ) ∩ {ν0 , . . . , νn } = ∅. Then

rkΛ0 (M1 /ωn M1 ) =

n X

kj rkΛ0 (Λ/νj Λ) =

j=0

n X

kj ϕ(pj ),

j=0

rkΛ0 (Λa /ωn Λa ) = pn a.

rkΛ0 (M2 /ωn M2 ) = 0,

The equality (iii) follows. Finally, (iv) is an immediate consequence of (iii). ∼

(9.4.9.4) Proposition. Let M be a Λ-module of finite type, where Λ = Zp [[Γ]], Γ −→ Zrp , r ≥ 1. Assume that we are given a finite collection {Γv ⊆ Γ | v ∈ S 0 } of non-zero closed subgroups of Γ, an integer a ≥ 0 and a real number C ≥ 0 such that X rkZp (MΓα ) − a[Γ : Γα ] ≤ C |Γα \Γ/Γv | v∈S 0

holds for every open subgroup Γα of Γ. Then rkΛ (M ) = a. Proof. Induction on r. If r = 1, then Γα = pn Γ, [Γ : Γα ] = pn and |Γα \Γ/Γv | = O(1) (for each v ∈ S 0 ), hence rkZp (Mpn Γ ) = pn a + O(1); the result then follows from Lemma 9.4.9.3(iv). Assume that r > 1 and the statement has been proved for r − 1. Choose a decomposition Γ = Γ0 × Γ0 as in 9.4.9.2 such that we have, for all v ∈ S 0 , Γv 6⊂ Γ0 . Fix n ≥ 0 and put M 0 = Mpn Γ0 , viewed as a Λ0 -module. For every open subgroup Γ0β of Γ0 we have rkZp (M 0 0 ) − apn [Γ0 : Γ0 ] = rkZp (Mpn Γ ×Γ0 ) − a[Γ : (pn Γ0 × Γ0 )] ≤ β β Γβ 0 β X X ≤C |((pn Γ0 \Γ0 ) × (Γ0β \Γ0 ))/Γv | ≤ pn C |Γ0β \Γ0 /Γ0v |, v∈S 0

v∈S 0 ∼

where Γ0v = Im(Γv ,→ Γ −→ Γ/Γ0 −→ Γ0 ) 6= 0. The induction hypothesis implies that rkΛ0 (M/ωn M ) = rkΛ0 (M 0 ) = apn , hence rkΛ (M ) = a, by Lemma 9.4.9.3(iv). 216

(9.4.9.5) Proposition. Let Λ = Zp [[Γ]] = Λ0 [[γ0 − 1]] be as in 9.4.9.2. Assume that M is a Λ-module of finite type, b ≥ 0 an integer, {Γv ⊆ Γ | v ∈ S 0 } a finite set of closed subgroups of Γ and {Xv | v ∈ S 0 } a collection of (right) Γv -sets such that (?)

X

rkZp (MΓα ) = b [Γ : Γα ] +

|(Xv × (Γα \Γ))/Γv | − 1

v∈S 0

for each open subgroup Γα of Γ. Put a=b+

X

|Xv |.

v∈S 0 Γv =0

Then (i) For each n ≥ 0, n

rkΛ0 (M/ωn M ) = p a +

(

X

n

|(Xv × (p Γ0 \Γ0 ))/Γv | +

v∈S 0 06=Γv ⊆Γ0

−1,

r=1

0,

r > 1.

(ii) rkΛ (M ) = a. (iii) If |Xv | = 1 for each v ∈ S 0 , then rkΛ0 (M/ωn M ) = pn a +

n X

kj ϕ(pj )

(∀n ≥ 0),

j=0

where

( kn − kn+1 = |{v ∈ S 0 | Γv ⊆ Γ0 , [Γ0 : Γv ] = pn }| +

and charΛ (Mtors ) is divisible by

Q

−1,

r = 1, n = 0

0,

otherwise,

kn n≥0 (νn ) .

Proof. Induction on r. If r = 1, then Γ = Γ0 , Λ0 = Zp , pn Γ0 = Γα is open in Γ, [Γ : Γα ] = pn and M/ωn M = MΓα , hence (i) is just (?). For v ∈ S 0 we have ( |Xv |, Γv = 0 |(Xv × (pn Γ\Γ))/Γv | lim = n−→∞ pn 0, Γv = 6 0; applying Lemma 9.4.9.3(iv) gives (ii). The statement (iii) follows from (i) and Lemma 9.4.9.3(iii). Assume that r > 1 and the statement holds for r − 1. Fix n ≥ 0 and put M 0 = M/ωn M = Mpn Γ0 , viewed as a Λ0 -module. The assumption (?) implies that we have, for every open subgroup Γ0β of Γ0 , X rkZp (MΓ0 0 ) = rkZp (Mpn Γ0 ×Γ0β ) = apn [Γ0 : Γ0β ] + aβ (v) − 1 β

v∈S 0 Γv 6=0

with aβ (v) = |(Xv × (pn Γ0 \Γ0 ) × (Γ0β \Γ0 ))/Γv |. Fix v ∈ S 0 such that Γv 6= 0. If Γv ⊆ Γ0 , then [Γ0 : Γv ] < ∞ and aβ (v) = [Γ0 : Γ0β ] · |(Xv × (pn Γ0 \Γ0 ))/Γv |. ∼

If Γv 6⊆ Γ0 , then Γ0v := Im(Γv ,→ Γ −→ Γ/Γ0 −→ Γ0 ) 6= 0 and aβ (v) ≤ pn |Xv | · |Γ0β \Γ0 /Γv |. 217

Applying Proposition 9.4.9.4 to M 0 and Λ0 , we obtain (i). The statements (ii) and (iii) follow from (i) by the same argument as in the case r = 1. ∼

∼

(9.4.9.6) In the remainder of Sect. 9.4.9, let O be as in 9.1.1, Γ −→ Zrp (r ≥ 1) and Λ = O[[Γ]] −→ O[[X1 , . . . , Xr ]] (Xi = γi − 1). For each open subgroup Γα ⊂ Γ, set Jα = Ker (Λ −→ O[Γ/Γα ]) ; then Λ/Jα = O[Γ/Γα ],

rkO (Λ/Jα ) = (Γ : Γα ).

(9.4.9.7) Proposition. Let Γα ⊂ Γ be an open subgroup. For each Λ-module of finite type M , define χα (M ) :=

r X

(−1)i rkO TorΛ i (M, Λ/Jα ).

i=0

(i) The integer χα (M ) is well-defined. (ii) If 0 −→ M 0 −→ M −→ M 00 −→ 0 is an exact sequence of Λ-modules of finite type, then χα (M ) = χα (M 0 ) + χα (M 00 ). (iii) If f ∈ Λ − {0}, then χα (Λ/f Λ) = 0. (iv) χα (Λ) = (Γ : Γα ). (v) χα (M ) = rkΛ (M ) (Γ : Γα ), for each Λ-module of finite type M . Proof. There exists a set of topological generators γ1 , . . . , γr of Γ and integers n1 , . . . , nr ≥ 0 such that Γα is n1 nr topologically generated by γ1p , . . . , γrp . This implies that the ideal Jα is generated by the regular sequence L

ni

x = (x1 , . . . , xr ), where xi = γip − 1; thus M ⊗Λ Λ/Jα is represented by the Koszul complex KΛ• (M, x)[r]. r−i The statements (i) and (ii) follow from this description of TorΛ (KΛ• (M, x)), while (iii) is i (M, Λ/Jα ) = H a consequence of (ii), applied to the exact sequence f

0 −→ Λ−→Λ −→ Λ/f Λ −→ 0. The formula χα (Λ) = rkO (Λ/Jα ) = (Γ : Γα ) is immediate. In order to prove (v), note that there exists a filtration M = M0 ⊃ M1 ⊃ · · · ⊃ Mt = 0 by Λ-submodules such that each graded quotient Mj /Mj+1 is isomorphic to Λ or Λ/fj Λ (fj ∈ Λ − {0}). Applying (ii)-(iv), we deduce that χα (M ) =

t−1 X

χα (Mj /Mj+1 ) = (Γ : Γα )

j=0

t−1 X

rkΛ (Mj /Mj+1 ) = (Γ : Γα ) rkΛ (M ).

j=0

(9.4.9.8) Proposition. Let M be a Λ-module of finite type. (i) If rkΛ (M ) = 0, then rkO (MΓα ) lim = 0. Γα −→0 (Γ : Γα ) (ii) For each i ≥ 1, lim

Γα −→0

rkO TorΛ i (M, Λ/Jα ) = 0. (Γ : Γα )

(iii) lim

Γα −→0

rkO (MΓα ) = rkΛ (M ). (Γ : Γα ) 218

Above, Γα runs through all open subgroups of Γ. Proof. We apply d´evissage in the following form: if 0 −→ M 0 −→ M −→ M 00 −→ 0 is an exact sequence such that (i) (resp., (ii)) holds for M 0 and M 00 , then it also holds for M . (i) If M is O-torsion, then (i) holds for trivial reasons. By d´evissage, we can replace M by M/MO−tors , hence assume that M has no O-torsion. Fix a uniformizing element π ∈ O and set k = O/πO, Λ = Λ/πΛ = k[[Γ]], M = M/πM . The assumption MO−tors = 0 implies that rkΛ (M ) = 0; as rkO (MΓα ) ≤ dimk (M Γα ), it is enough to show that dimk (M Γα ) ? = 0. Γα −→0 (Γ : Γα ) lim

There exists a filtration M = N0 ⊃ N1 ⊃ · · · ⊃ Nt = 0 by Λ-submodules such that each graded quotient Nj /Nj+1 is isomorphic to Λ/gj Λ (gj ∈ Λ − {0}). By d´evissage, it is sufficient to consider the case M = Λ/gΛ (g ∈ Λ − {0}). We have Λ = k[[X1 , . . . , Xr ]] (Xi = γi − 1). After renumbering the topological generators γi of Γ, we can assume that

C := dimk Λ/(g, X2 , . . . , Xr ) = dimk M /(X2 , . . . , Xr )M < ∞. An induction argument then shows that

(∀ni ≥ 1) dimk M /(X1n1 , . . . , Xrnr )M ≤ dimk M /(X2n2 , . . . , Xrnr )M ≤ C(n2 −1) · · · (nr −1) < Cn2 · · · nr . αi

If Γα is topologically generated by γip lim

Γα −→0

α1

αr

(i = 1, . . . , r), then Jα Λ = (X1p , . . . , Xrp ) and

dimk (M Γα ) dimk (M /Jα M ) C = < α1 , (Γ : Γα ) pα1 +···+αr p

which proves (i). (ii) Fix i ≥ 1. Filter M as in the proof of Proposition 9.4.9.7(v); by d´evissage, we reduce the proof of (ii) to the case M = Λ or M = Λ/f Λ (f ∈ Λ − {0}). If M = Λ or (M = Λ/f Λ and i ≥ 2), then TorΛ i (M, Λ/Jα ) = 0. In the remaining case M = Λ/f Λ and i = 1, the exact sequence f

0 −→ TorΛ 1 (M, Λ/Jα ) −→ Λ/Jα −→Λ/Jα −→ MΓα −→ 0 implies that rkO TorΛ 1 (M, Λ/Jα ) = rkO (MΓα ), hence (ii) follows from (i). (iii) As MΓα = M ⊗Λ Λ/Jα = TorΛ 0 (M, Λ/Jα ), the statement follows from (ii) and Proposition 9.4.9.7(v). (9.4.9.9) Note that the statements of Proposition 9.4.9.4 and Proposition 9.4.9.5(ii) follow directly from Proposition 9.4.9.8(iii). (9.4.10) We now return to the extension K∞ /K and the Λ-modules 0 Y = DΛ (E∞ ⊗ Qp /Zp )   M   Z = Ker  Λv −→ Zp  . v∈Sf Γv 6=0

For a number field L, we use the standard notation r1 (L) (resp., r2 (L)) for the number of real (resp., complex) primes of L. 219

(9.4.11) Proposition. (i) For every open subgroup Γα ⊆ Γ, the canonical map 0 iα : Eα0 ⊗ Qp /Zp −→ (E∞ ⊗ Qp /Zp )Γα

has finite kernel and cokernel. (ii) rkΛ (Y ) = r1 (K) + r2 (K) + |{v ∈ Sf | Γv = 0}|. (iii) We have the divisibility of characteristic power series charΛ (Z) charΛ (Ytors ). Proof. (i) The following diagram is commutative and has exact rows and columns: 0   y H 1 (Γα , µp∞ (K∞ ))   y 0

−→

Eα0 ⊗ Qp /Zp  i yα

−→

H 1 (KS /Kα , Qp /Zp (1))   y

−→

A0α   y

0

−→

0 (E∞ ⊗ Qp /Zp )Γα

−→

H 1 (KS /K∞ , Qp /Zp (1))Γα   y

−→

(A0∞ )Γα

−→

0

H 2 (Γα , µp∞ (K∞ )) By the Snake Lemma (and the finiteness of A0α ), it is enough to show that H j (Γα , µp∞ (K∞ )) is finite for j = 1, 2. This is clear if µp∞ 6⊂ K∞ . If µp∞ ⊂ K∞ , then the cyclotomic character χα : Γα ,→ Γ −→ AutZp (µp∞ ) = Z∗p ∼

has infinite image Im(χα ) −→ Zp . The Hochschild-Serre spectral sequence E2i,j = H i (Γα /Γ0α , H j (Γ0α , µp∞ )) =⇒ H i+j (Γα , µp∞ ) for Γ0α = Ker(χα ) ⊂ Γα degenerates into exact sequences 0

0 −→ H j−1 (Γ0α , µp∞ )Γα /Γ0α −→ H j (Γα , µp∞ ) −→ H j (Γ0α , µp∞ )Γα /Γα −→ 0, in which H j (Γ0α , µp∞ ) = (µp∞ )⊕(

r−1 j

),

with Γα /Γ0α acting via χα . This implies that the groups H j (Γα , µp∞ ) −→ (µp∞ (Kα ))⊕( ∼

)

r−1 j

are finite for all j ≥ 0. (ii) The statement (i) together with Dirichlet’s theorem on units imply that X ∗ rkZp (YΓα ) = rkZ (OK ) = (r (K) + r (K))[Γ : Γ ] + |Γα \Γ/Γv | − 1 1 2 α ,S α α

(9.4.11.1)

v∈Sf

holds for every open subgroup Γα of Γ (note that ri (Kα ) = [Kα : K] ri (K), thanks to the assumption (P )); apply Proposition 9.4.9.5(ii). 220

∼

(iii) Put S 0 = {v ∈ Sf | Γv −→ Zp }; for v ∈ S 0 let Γsat v := Γ ∩ (Γv ⊗Zp Qp ) ⊂ Γ ⊗Zp Qp 0 0 00 sat be the saturation of Γv in Γ. Put S 00 = {Γsat v | v ∈ S } and let π : S −→ S be the map π(v) = Γv . 00 −1 n(v) Fix Γ0 ∈ S ; for every v ∈ π (Γ0 ) we have Γv = p Γ0 for suitable n(v) ≥ 0. Then ( 1, if r > 1 Y Y charΛ (Z) = ωn(v) (Γ0 ) · ∼ 1 if Γ = hγi −→ Zp , Γ0 ∈S 00 v∈π −1 (Γ0 ) γ−1 , n

where ωn (Γ0 ) = γ0p − 1, for a fixed topological generator γ0 of Γ0 . The desired divisibility charΛ (Z) charΛ (Ytors ) then follows from Proposition 9.4.9.5(iii) and the formula (9.4.11.1). 0 (9.4.12) Proposition. The characteristic power series charΛ (X∞ ) is divisible by charΛ (E 1 (DΛ (A0∞ ))) = charΛ (DΛ (A0∞ )).

Proof. The map δ in the diagram (9.4.5.1) is an isomorphism in (Λ Mod)/(pseudo − null), hence 0 charΛ (X∞ ) charΛ (Z) = charΛ (E 1 (Y )) charΛ (E 1 (DΛ (A0∞ ))) = charΛ (Ytors ) charΛ (DΛ (A0∞ )).

However, charΛ (Z) divides charΛ (Ytors ), by Proposition 9.4.11(iii). (9.5) Relating A∞ and X∞ (9.5.1) For every finite subextension Kα /K of K∞ /K, put ∗ Eα = OK ⊗ Zp , α

Aα = Pic(OKα )[p∞ ]

and let E∞ = lim Eα , −→ α

A∞ = lim Aα , −→ α

X∞ = lim Aα . ← α−

In the case when r = 1, Iwasawa ([Iw], Thm. 11) showed that there is a surjective morphism of Λ-modules A∞ −→ DΛ (E 1 (X∞ )), with pseudo-null kernel. In this section we construct a canonical morphism in (Λ Mod)/(pseudo − null) α : X∞ −→ E 1 (DΛ (A∞ )) and show that Coker(α) is close to zero in (Λ Mod)/(pseudo − null).

e 1 (GK,S , Qp /Zp (1); ∆(Qp /Zp (1))) (9.5.2) From now on, let Sf = {v|p}. We first relate Pic(OK )[p∞ ] to H f for appropriate local conditions ∆(Qp /Zp (1)). For each v|p the valuation v defines a morphism of complexes v⊗id

• τ≤1 Ccont (Gv , Z/pn Z(1)) −→ H 1 (Gv , Z/pn Z(1))[−1] = Kv∗ ⊗ Z/pn Z[−1]−−→Z/pn Z[−1];

put v⊗id

• Uv+ (Z/pn Z(1)) = Cone (τ≤1 Ccont (Gv , Z/pn Z(1))−−→Z/pn Z[−1]) [−1].

This complex is equipped with a canonical morphism 221

• + n n i+ v : Uv (Z/p Z(1)) −→ Ccont (Gv , Z/p Z(1));

put Uv− (Z/pn Z(1)) = Cone(−i+ v ). • The Pontrjagin dual of Uv− (Z/pn Z(1)) is quasi-isomorphic to Ccont (Gv /Iv , Z/pn Z)[2], hence to the complex

fv −1

0

[Z/pn Z−−→Z/pn Z] = [Z/pn Z−→Z/pn Z] = Z/pn Z[2] ⊕ Z/pn Z[1] (in degrees −2, −1). It follows that Uv− (Z/pn Z(1)) is quasi-isomorphic to Z/pn Z[−2] ⊕ Z/pn Z[−1]. Passing to the inductive (resp., projective) limit with respect to n, we obtain local conditions ∆(Qp /Zp (1)) (resp., ∆(Zp (1))). The exact triangles gf (GK,S , X(1)) −→ RΓcont (GK,S , X(1)) −→ RΓ

M

(X[−2] ⊕ X[−1])

(X = Zp , Qp /Zp )

v|p

together with (9.2.1.1-2) give ∼ e f0 (GK,S , Qp /Zp (1)) −→ H µp∞ (K) ∗ e f1 (GK,S , Qp /Zp (1)) −→ Pic(OK )[p∞ ] −→ 0 0 −→ OK ⊗ Qp /Zp −→H

(9.5.2.1)

∼ e f3 (GK,S , Qp /Zp (1)) −→ H Qp /Zp

e fi (GK,S , Qp /Zp (1)) = 0 H

(i 6= 0, 1, 3),

resp., ∼ ∗ e f1 (GK,S , Zp (1)) −→ H OK ⊗ Zp ∼ e f2 (GK,S , Zp (1)) −→ H Pic(OK )[p∞ ]

(9.5.2.2)

∼ e f3 (GK,S , Zp (1)) −→ H Zp

e fi (GK,S , Zp (1)) = 0 H

(i 6= 1, 2, 3).

(9.5.3) The corresponding local conditions in the limit over K∞ /K are equal to   M v ⊗id α • Uv+ (FΓ (Qp /Zp (1))) = Cone τ≤1 Ccont (Gv , FΓ (Qp /Zp (1)))−−→ lim Qp /Zp [−1] [−1] −→ α vα |v

resp.,  v ⊗id

α • Uv+ (FΓ (Zp (1))) = Cone τ≤1 Ccont (Gv , FΓ (Zp (1)))−−→ lim ←−

α

For Kα ⊂ Kβ , the transition maps   M M Qp /Zp −→ Qp /Zp  vα |v

resp.,

wβ |vα

M vα |v

M

 Zp [−1] [−1].

vα |v

 

M

 Zp −→ Zp 

wβ |vα

are given by the ramification indices e(wβ |vα ) (resp., the inertia degrees f (wβ |vα )). It follows that 222

H

1

(Uv− (FΓ (Qp /Zp (1))))

= lim −→ α

H 1 (Uv− (FΓ (Zp (1)))) = lim ←− α

M

( ∼

Qp /Zp −→

vα |v

M

( ∼

Zp −→

vα |v

0,

I(Γv ) 6= 0

DΛ (Λv ),

I(Γv ) = 0

0,

[Γv : I(Γv )] = ∞

Λv ,

[Γv : I(Γv )] < ∞.

Combined with the functoriality of the local Brauer groups (cf. 9.2.2), this gives isomorphisms in D(Λ Mod)  DΛ (Λ)[−1] ⊕ DΛ (Λ)[−2], Γv = 0    ∼ Uv− (FΓ (Qp /Zp (1))) −→ DΛ (Λv )[−1], Γv 6= 0 = I(Γv ) (9.5.3.1)    0, I(Γv ) 6= 0 resp., ( Λv [−2] ⊕ Λv [−1], [Γv : I(Γv )] < ∞ ∼ − Uv (FΓ (Zp (1))) −→ (9.5.3.2) Λv [−2], [Γv : I(Γv )] = ∞. (9.5.4) For each v|p there are canonical isomorphisms in D(Λ Mod) ∼

DΛ (Λv ) −→ Λv [−r(v)]

(9.5.4.1)

∼

ΦΛ (Λv ) −→ DΛ (Λv )[r(v)], ∼

r(v)

where Γv −→ Zp

. Put Sex = {v|p : r(v) = 1, v is ramified in K∞ /K}.

(9.5.5) Lemma. (i) If v is unramified in K∞ /K, then there is a canonical isomorphism in D(Λ Mod) ∼

DΛ (Uv− (FΓ (Zp (1)))) −→ DΛ (Uv− (FΓ (Qp /Zp (1)))). (ii) If v is ramified in K∞ /K and r(v) > 1, then there are isomorphisms in D((Λ Mod)/(pseudo − null)) ∼

∼

DΛ (Uv− (FΓ (Zp (1)))) −→ DΛ (Uv− (FΓ (Qp /Zp (1)))) −→ 0. (iii) If v ∈ Sex , then there are isomorphisms in D(Λ Mod) ∼

∼

DΛ (Uv− (FΓ (Zp (1)))) −→ Λv ⊕ Λv [1],

DΛ (Uv− (FΓ (Qp /Zp (1)))) −→ 0.

Proof. This follows from (9.5.3.1-2), (9.5.4.1) and the fact that Λv is pseudo-null if r(v) > 1. gf (KS /K∞ , Qp /Zp (1)) (resp., RΓ gf,Iw (K∞ /K, Zp (1))) represented by (9.5.6) The Selmer complexes RΓ • • Cf (GK,S , FΓ (Qp /Zp (1)); ∆) (resp., by Cf (GK,S , FΓ (Zp (1)); ∆)) - for the local conditions defined in 9.5.3 have cohomology equal to e i (KS /K∞ , Qp /Zp (1)) = lim H ei H f −→ f (GKα ,Sα , Qp /Zp (1)) α

e i (K∞ /K, Zp (1)) = lim H ei H f,Iw ←− f (GKα ,Sα , Zp (1)). α

It follows from (9.5.2.1-2) that we have ∼ e 0 (KS /K∞ , Qp /Zp (1)) −→ H µp∞ (K∞ ) f

e 1 (KS /K∞ , Qp /Zp (1)) −→ A∞ −→ 0 0 −→ E∞ −→H f e i (KS /K∞ , Qp /Zp (1)) H f 223

=0

(9.5.6.1) (i > 1),

resp., ∼ e 1 (K∞ /K, Zp (1)) −→ H lim f,Iw ←− Eα α

∼ e 2 (K∞ /K, Zp (1)) −→ H X∞ f,Iw

(9.5.6.2)

∼ e 3 (K∞ /K, Zp (1)) −→ H Zp f,Iw

e i (K∞ /K, Zp (1)) = 0 H f,Iw

(i 6= 1, 2, 3).

(9.5.7) The exact triangles in D(Λ Mod) gf (KS /K∞ , Qp /Zp (1)) −→ RΓcont (GK,S , FΓ (Qp /Zp (1))) −→ RΓ

M

Uv− (FΓ (Qp /Zp (1)))

v|p

gf,Iw (K∞ /K, Zp (1)) −→ RΓcont (GK,S , FΓ (Zp (1))) −→ RΓ

M

Uv− (FΓ (Zp (1)))

v|p

together with the canonical isomorphisms ∼

∼

DΛ (RΓcont (GK,S , FΓ (Qp /Zp (1)))) −→ DΛ (ΦΛ (RΓcont (GK,S , FΓ (Zp (1))))) −→ ∼

−→ DΛ (RΓcont (GK,S , FΓ (Zp (1)))) and Lemma 9.5.5 yield an exact triangle in D((Λ Mod)/(pseudo − null)) M

gf (KS /K∞ , Qp /Zp (1))) −→ DΛ (RΓ gf,Iw (K∞ /K, Zp (1))) −→ DΛ (RΓ

(Λv [2] ⊕ Λv [1]).

v∈Sex

Applying DΛ we obtain another exact triangle in D((Λ Mod)/(pseudo − null)) gf,Iw (K∞ /K, Zp (1)) −→ W −→ RΓ

M

(Λv [−1] ⊕ Λv [−2]),

(9.5.7.1)

v∈Sex

in which gf (KS /K∞ , Qp /Zp (1)))). W = DΛ (DΛ (RΓ (9.5.8) The hyper-cohomology spectral sequence e j (KS /K∞ , Qp /Zp (1)))) =⇒ H i+j (W ) E2i,j = E i (DΛ (H f satisfies E2i,j = 0

codimΛ (supp(E2i,j )) ≥ i.

(j 6= 0, 1),

In particular, we have isomorphisms in (Λ Mod)/(pseudo − null) H j (W ) = 0 2

∼

H (W ) −→

(j 6= 0, 1, 2) E21,1 .

Combining E i (9.5.6.1) with the cohomology sequence of the triangle (9.5.7.1), we obtain a diagram in (Λ Mod)/(pseudo − null) 224

0   y E 1 (DΛ (E∞ ⊗ Qp /Zp ))   y

L v∈Sex

Λv

−→

X∞

H 2 (W )   y

−→

−→

L v∈Sex

Λv

−→

Zp −→ 0

(9.5.8.1)

E 1 (DΛ (A∞ ))   y 0 with exact row and column. This diagram defines a morphism in (Λ Mod)/(pseudo − null) α : X∞ −→ H 2 (W ) −→ E 1 (DΛ (A∞ )). (9.5.9) Proposition. If p ∈ Spec(Λ) with ht(p) = 1 is not exceptional (in the sense of 9.4.6), then Coker(α)p = 0. Proof. The assumption on p implies that M

! Λv

= 0.

v∈Sex

p

The statement follows by localizing the diagram (9.5.8.1) at p. ∼

r(v)

(9.5.10) Corollary. If, for every v ∈ Sf ramified in K∞ /K, we have Γv −→ Zp map α : X∞ −→ E 1 (DΛ (A∞ ))

with r(v) ≥ 2, then the

is an epimorphism in (Λ Mod)/(pseudo − null). Proof. Under these assumptions there are no exceptional p ∈ Spec(Λ). (9.6) Comparison with classical Selmer groups (9.6.1) Let O, T, T ∗ , A, A∗ be as in 9.1.4; put V = T ⊗O F , V ∗ = T ∗ ⊗O F . Assume that we are given, for each v ∈ Σ = {v|p}, an F [Gv ]-submodule Vv+ ⊂ V . Put Tv+ = T ∩ Vv+ ,

+ + A+ v = Vv /Tv ⊂ V /T = A,

Xv− = X/Xv+

(X = T, V, A; v ∈ Σ)

and, for each v|p, ∓ V ∗ (1)± v = HomF (Vv , F )(1),

∓ T ∗ (1)± v = HomO (Tv , O)(1),

∗ ± ∗ ± A∗ (1)± v = V (1)v /T (1)v .

These data induce Greenberg’s local conditions for X = T, V, A (and also for X = T ∗ (1), V ∗ (1), A∗ (1)) with ( • Ccont (Gv , Xv+ ), (v ∈ Σ) + Uv (X) = • Ccont (Gv /Iv , X Iv ), (v ∈ Σ0 ), 225

gf (X) and their cohomology where Σ0 = {v ∈ Sf ; v - p}, hence the corresponding Selmer complexes RΓ i e groups Hf (X). Greenberg [Gre 2,3] defined his Selmer groups (resp., strict Selmer groups) as ! M M 1 1 − 1 SX (K) = Ker Hcont (GK,S , X) −→ Hcont (Iv , Xv ) ⊕ Hcont (Iv , X) v∈Σ0

v∈Σ str SX (K)

= Ker

1 Hcont (GK,S , X)

−→

M

1 Hcont (Gv , Xv− )

⊕

M

! 1 Hcont (Iv , X)

.

v∈Σ0

v∈Σ

(9.6.2) These groups satisfy the following properties: there is an exact sequence M str 1 0 −→ SX (K) −→ SX (K) −→ Hcont (Gv /Iv , (Xv− )Iv ), v|p

isomorphisms ∼

∼

STstr (K) ⊗O F −→ SVstr (K)

ST (K) ⊗O F −→ SV (K), and canonical injective maps

str STstr (K) ⊗O F/O ,→ SA (K)

ST (K) ⊗O F/O ,→ SA (K), with finite cokernels.

(9.6.3) Lemma. For each X = T, V, A there is an exact sequence e f0 (X) −→ X GK −→ 0 −→ H

M

str e f1 (X) −→ SX (Xv− )Gv −→ H (K) −→ 0.

v|p

Proof. This follows from the exact triangle gf (X) −→ RΓcont (GK,S , X) −→ RΓ

M

Uv− (X)

v∈Sf 1 • and the fact that Uv− (X) (defined in 6.1.3) is quasi-isomorphic to Ccont (Gv /Iv , Hcont (Iv , X))[−1] (resp., • − Ccont (Gv , Xv )) if v - p (resp., v|p).

(9.6.4) Corollary. If V GK = 0, then str e f1 (A)) (= dimF (H e f1 (V ))) = corkO (SA corkO (H (K)) +

M

dimF ((Vv− )Gv ).

v|p

e f1 (A)) corkO (H

≥ corkO (SA (K)).

Proof. Combine the exact sequence of Lemma 9.6.3 with the statements from 9.6.2 and the equality
1 dimF Hcont (Gv /Iv , (Vv− )Iv ) = dimF ((Vv− )Gv ). ∼

(9.6.5) Let K ⊂ K∞ ⊂ KS , with Γ = Gal(K∞ /K) −→ Zrp (r ≥ 1). We define SA (K∞ ) = lim −→ SA (Kα ), α

str str SA (K∞ ) = lim −→ SA (Kα ), α

where Kα , as usual, runs through all finite subextensions of K∞ /K. The exact sequence of Lemma 9.6.3 yields, in the limit, an exact sequence MM G v∞ e 1 (KS /K∞ , A) −→ S str (K∞ ) −→ 0. AGK∞ −→ (A− −→ H (9.6.5.1) v) f A v|p v∞ |v

226

For each v|p there is an isomorphism of Λv -modules   M G v∞  ∼ G v∞ DΛv  (A− −→ D((A− ) ⊗O Λv ; v ) v ) v∞ |v

thus e 1 (KS /K∞ , A) = corkΛ S str (K∞ ) + corkΛ H f A

M

dimF ((Vv− )Gv ).

(9.6.5.2)

v|p Γv =0

∼

r(v)

If we write, as usual, Γv −→ Zp

, then there is an isomorphism in (Λ Mod)/(pseudo − null) ( 0, r(v) ≥ 2 ∼ − G v∞ D((Av ) ) ⊗O Λv −→ G v∞ D((A− r(v) = 1. v )O−div ) ⊗O Λv ,

(9.6.6) Proposition. The canonical surjective map e 1 (KS /K∞ , A) −→ S str (K∞ ) β:H f A has the following properties. (i) If (Vv− )Gv∞ = 0 for all primes v|p, then DΛ (Ker(β)) ⊗ Q = 0. (ii) Assume that no v|p splits completely in K∞ /K, and that (Vv− )Gv∞ = 0 for all primes v|p satisfying r(v) = 1. Then DΛ (Ker(β)) is Λ-pseudo-null. Gv G v∞ (iii) If (A− = 0 for all primes v|p, then (A− = 0 for all v|p and β is an isomorphism. v ) v) Proof. The statements (i) and (ii) follow from the discussion in 9.6.5. As regards (iii), for each v|p, N := G v∞ Gv D((A− ) is a Λv -module of finite type satisfying NΓv = D((A− ) = 0; thus N = 0 by Nakayama’s v ) v ) − G v∞ Lemma, hence (Av ) = D(N ) = 0. The exact sequence (9.6.5.1) then implies that Ker(β) = 0. (9.6.7) Abelian varieties. In the rest of 9.6 we let O = Zp , F = Qp . Let B be an abelian variety over K with good reduction outside Sf . Then T = Tp (B) is a representation of GK,S and the Weil pairing identifies b where B b denotes the dual abelian variety; hence A = B[p∞ ] and A∗ (1) = B[p b ∞ ]. Let T ∗ (1) with Tp (B), S be any finite set of primes of K containing all primes of bad reduction of B and all primes dividing p∞. Put, as before, Σ = {v|p}, Σ0 = Sf − Σ. (9.6.7.1) The classical Selmer groups for the pn -descent   M Sel(B/K, pn ) = Ker H 1 (GK,S , B[pn ]) −→ H 1 (Gv , B[pn ])/Im(δv,n ) v∈Sf

are defined by the local conditions
Im δv,n : B(Kv ) ⊗ Z/pn Z ,→ H 1 (Gv , B[pn ]) . The corresponding discrete and compact Selmer groups  n  1 Sel(B/K, p∞ ) = lim −→ Sel(B/K, p ) = Ker H (GK,S , A) −→ n

 n  1 Sp (B/K) = ← lim − Sel(B/K, p ) = Ker H (GK,S , T ) −→ n

227

M v∈Sf

M v∈Sf

 H 1 (Gv , A)/Lv (A)  H 1 (Gv , T )/Lv (T ) ,

where we have put
Lv (A) = − lim Im(δv,n ) = Im B(Kv ) ⊗ Qp /Zp ,→ H 1 (Gv , A) → n
1 b p −→ Hcont Lv (T ) = ← lim Im(δ ) = Im B(K ) ⊗Z (G , T ) , v,n v v n− depend only on the GK,S -modules A = B[p∞ ] and T = Tp (B), respectively. More precisely, they coincide with the Bloch-Kato Selmer groups Sel(B/K, p∞ ) = Hf1 (K, A),

Sp (B/K) = Hf1 (K, T )

([B-K], 3.11). If v - p, then 1 Lv (T ) = Hcont (Gv , T ).

Lv (A) = 0,

(9.6.7.1.1)

(9.6.7.2) We shall consider only the following “elementary” case: Σ = {v|p} = Σo ∪ Σt , where (Ord) (∀v ∈ Σo ) (T or) (∀v ∈ Σt )

B has good ordinary reduction at v. B has completely toric reduction at v.

Under this assumption, for each v|p there is a canonical sub-Qp [Gv ]-module Vv+ ,→ V = T ⊗Zp Qp as in 9.6.1, with Vv+ arising from the kernel of the reduction map at v (resp., from a p-adic uniformization by a torus) in the case (Ord) (resp., (T or)). In either case, Vv− = V /Vv+ is an unramified Gv -module. For v ∈ Σt , the geometric Frobenius element fv acts on Vv− by an element of finite order, while for v ∈ Σo all eigenvalues of fv acting on Vv− are v-Weil numbers of weight −1 (see 12.4.8.1 below). b is isogeneous to B, which implies that, for each v ∈ Σo (resp., v ∈ Σt ), The dual abelian variety B b B also has good ordinary (resp., completely toric) reduction at v. This means that the same construction ∗ ∗ b defines sub-Qp [Gv ]-modules V ∗ (1)+ v ,→ V (1) = T (1) ⊗Zp Qp = Tp (B) ⊗Zp Qp (v ∈ Σ), which coincide with the abstract modules defined in 9.6.1. b defines an injective morphism of Zp [GK,S ]-modules T = Tp (B) ,→ T ∗ (1) = Any polarization λ : B −→ B b with finite cokernel. For each v|p, the sub-Zp [Gv ]-module T + ⊂ T is mapped into T ∗ (1)+ , again with Tp (B) v v finite cokernel. In other words, the Weil pairing attached to λ defines a skew-symmetric bilinear form V ⊗Qp V −→ Qp (1), which induces isomorphisms of Qp [GK,S ]-modules ∼

V −→ V ∗ (1) = HomQp (V, Qp )(1) resp., of Qp [Gv ]-modules ∼

Vv± −→ (Vv∓ )∗ (1) = HomQp (Vv∓ , Qp )(1). The following results are well-known (cf. [Co-Gr], [Gre 5]); we record them for the sake of completeness. (9.6.7.3) Lemma. (i) There are exact sequences 0 −→ STstr (K) −→ Sp (B/K) −→

M

1 Hcont (Gv , Tv− )tors ⊕

0 −→ Sel(B/K, p ) −→

str SA (K)

−→

M

1 1 Hcont (Gv , T )/Hur (Gv , T )

v∈Σ0

v|p

∞

M

Im H

1

(Gv , A+ v)

M
1 −→ H 1 (Gv , A) /div ⊕ Hur (Gv , A). v∈Σ0

v|p

228

1 1 1 (ii) For each v ∈ Σ0 , the groups Hcont (Gv , T )/Hur (Gv , T ) and Hur (Gv , A) are finite, of common order equal Tamv (T,(p)) to p . (iii) For each v|p, ∼

1 Hcont (Gv , Tv− )tors −→ H 0 (Gv , A− v )/div

1 + 1 D Im H (Gv , Av ) −→ H (Gv , A) /div ⊆ H 0 (Gv , A∗ (1)− v )/div.

(iv) The groups

Sp (B/K)/STstr (K),

str SA (K)/Sel(B/K, p∞)

are finite. Proof. For v|p,
1 Lv (A) = Im H 1 (Gv , A+ v ) −→ H (Gv , A) div b and applying Tate’s local (cf. [Gre 5], Prop. 2.2 and pp. 69–70 in the case dim(B) = 1). Replacing B by B duality, we obtain
1 1 Lv (T ) = Ker Hcont (Gv , T ) −→ Hcont (Gv , Tv− )/tors . Combining these expressions with (9.6.7.1.1), we obtain (i). ∼ (ii) The self-dual Qp [Gv ]-module V −→ V ∗ (1) is known to satisfy the weight-monodromy conjecture (see, 1 1 e.g., [Ja 2], Sect. 5,7), which implies that V Gv = 0, hence Hur (Gv , V ) = 0 (=⇒ Hur (Gv , A) is finite). Applying the duality isomorphism ∼

1 1 1 D(Hcont (Gv , T )/Hur (Gv , T )) −→ Hur (Gv , A∗ (1)),

the statement then follows from 7.6.9 and 7.6.10.11. (iii) The isomorphism in the first row is standard. By Tate’s local duality, the L.H.S. of the second row is isomorphic to
1 1 1 ∗ − 0 ∗ − Im Hcont (Gv , T ∗ (1)) −→ Hcont (Gv , T ∗ (1)− v ) tors ⊆ Hcont (Gv , T (1)v )tors = H (Gv , A (1)v )/div. (iv) This is a consequence of (i)-(iii). (9.6.7.4) For K∞ /K as in 9.6.5, put ∞ Sel(B/K∞) = lim −→ Sel(B/Kα , p ),

Sp (B/K∞ ) = lim ←− Sp (B/Kα );

α

α

then str Sel(B/K∞ , p∞ ) ⊆ SA (K∞ ),

lim STstr (Kα ) ⊆ Sp (B/K∞ ). ← α−

(9.6.7.5) Proposition. Assume that each v|p is ramified in K∞ /K. Then: (i) There exist exact sequences of Λ-modules of finite type str 0 −→ lim ←− ST (Kα ) −→ Sp (B/K∞ ) −→ α

M

M

1 D(Hur (Gv , A∗ (1))) ⊗Zp Λ

v∈Σ0 Γv =0

1 str D(Hur (Gv , A)) ⊗Zp Λ −→ DΛ (SA (K∞ )) −→ DΛ (Sel(B/K∞ , p∞ )) −→ 0.

v∈Σ0 Γv =0

(ii) Put

c = max{Tamv (T, (p)) | v ∈ Σ0 , v splits completely in K∞ /K}. 229

(9.6.7.4.1)

Then

  str str pc · (SA (K∞ )/Sel(B/K∞, p∞ )) = pc · Sp (B/K∞ )/ lim S (K ) = 0. α ←− T α

0

In particular, if Tamv (T, (p)) = 0 for all v ∈ Σ that split completely in K∞ /K, then str Sel(B/K∞ , p∞ ) = SA (K∞ ),

str Sp (B/K∞ ) = ← lim − ST (Kα ). α

Proof. (i) Fix v|p, a finite sub-extension Kα /K of K∞ /K and a prime vα |v of Kα . Our assumption implies that, possibly after replacing Kα by a finite extension contained in K∞ , there exists a Zp -extension S ∼ Kα = F ⊂ F1 ⊂ · · · ⊂ F∞ = n Fn ⊂ K∞ (Gal(Fn /F ) −→ Z/pn Z), which is totally ramified at vα . As the ∗ − Gv -modules A− v , A (1)v are unramified, the corestriction maps in the projective systems
(H 0 (Gvn , A∗ (1)− v ) n≥1

H 0 (Gvn , A− v ))n≥1 ,

(where vn denotes the unique prime of Fn above vα ) are given by multiplication by p, hence the corresponding projective limits vanish. It follows that, if we apply Lemma 9.6.7.3(i),(iii) over each finite sub-extension Kα /K of K∞ /K and pass to the limit, the terms corresponding to v|p will disappear and we shall be left (using Tate’s local duality) with the exact sequences

M

str 0 −→ ← lim − ST (Kα ) −→ Sp (B/K∞ ) −→ α

Mv (A) −→

str DΛ (SA (K∞ ))

M

Mv (A∗ (1))

v∈Σ0

−→ DΛ (Sel(B/K∞ , p∞ )) −→ 0.

v∈Σ0

Here we have used the notation  Mv (X) := DΛ lim −→ α

M

 1 Hur (Gvα , X) ,

vα |v

0

for each v ∈ Σ and any p-primary torsion discrete Gv -module X. Fix v ∈ Σ0 ; as v is unramified in K∞ /K, ∼ we have either Γv −→ Zp , or Γv = 0. ∼ If Γv −→ Zp , then the tower of local fields (Kα )vα exhausts the maximal pro-p-unramified extension of any fixed (Kα0 )vα0 ; this implies that Mv (X) = 0 (for any X as above). If Γv = 0, then v splits completely in K∞ /K, hence 1 Mv (X) = D(Hur (Gvα , X)) ⊗Zp Λ.

The statement (i) is proved. 1 (ii) This follows from (i), as the groups Hur (Gv , X) (X = A, A∗ (1)) have common order, equal to pTamv (T,(p)) 0 (for each v ∈ Σ ). (9.6.7.6) Lemma. (i) If v ∈ Σo , then H 0 (Gv , Vv− ) = 0 and the group H 0 (Gv , A− v ) is finite. (ii) If v ∈ Σt , let Tv be the torus over Kv attached to B and denote by t(v) (0 ≤ t(v) ≤ dim(B)) the dimension of the maximal Kv -split subtorus of Tv . Then ∼

H 0 (Gv , Vv− ) −→ Q⊕t(v) , p (iii) We have

∼

⊕t(v) H 0 (Gv , A− . v ) −→ (Qp /Zp )

e 1 (T ) = corkZp Sel(B/K, p∞) + rkZp H f

X

t(v).

v∈Σt

Proof. (i) All eigenvalues of fv acting on Vv− have absolute values (N v)−1/2 , hence there are no fv -invariants. (ii) This follows from the fact that Tv− is isomorphic, as an Gv /Iv -module, to X∗ (Tv ) ⊗Z Zp , where X∗ (Tv ) is the cocharacter group of the torus Tv . Finally, (iii) follows from (i)-(ii) and the exact sequence from Lemma 9.6.3 (as V GK = 0). 230

(9.6.7.7) Corollary. If dim(B) = 1, i.e. if B = E is an elliptic curve, then e 1 (T ) = corkZp Sel(E/K, p∞ ) + |{v ∈ Σt | E has split multiplicative reduction at v}|. rkZp H f (9.6.7.8) Proposition. For K∞ /K as in 9.6.5, str corkΛ SA (K∞ ) = corkΛ Sel(B/K∞, p∞ ).

Proof. In view of the proof of Proposition 9.6.7.5(i), it is enough to show that, for each prime v | p which is unramified in K∞ /K, the Λ-module M [ Nv = lim H 0 (Gvα , A∗ (1)− (K∞ = Kα , [Kα : K] < ∞) v )/div ← − α,cor vα |v

satisfies rkΛ (Nv ) = 0. If Γv = 0, then Nv = Λ ⊗Zp (H 0 (Gv , A∗ (1)− v )/div) is killed by some power of p. If ∼ Γv 6= 0, then Γv −→ Zp and Nv is a Zp [Γ/Γv ]-module of finite type, hence rkΛ (Nv ) = 0. (9.7) Duality and perfectness The notation from 9.6.1 is in force. (9.7.1) For each intermediate field K ⊂ L ⊂ K∞ , put ΓL = Gal(K∞ /L), ΓL = Gal(L/K) = Γ/ΓL , ΛL = O[[ΓL ]]. Greenberg’s local conditions attached to the data from 9.6.1 define “Selmer complexes” gf (KS /L, Y ) ∈ Db (ΛL Mod), RΓ cof t

gf,Iw (L/K, Z) ∈ Db (ΛL Mod) RΓ ft

(Y = A, A∗ (1); Z = T, T ∗ (1)),

e i (KS /L, Y ) which do not, in fact, depend on S (by Proposition 7.8.8), and whose cohomology groups H f i i e e (L0 , Y ) (resp., (resp., H (L/K, Z)) are equal, respectively, to the inductive (resp., projective) limit of H f,Iw f e i (L0 , Z)), where L0 /K runs through all finite subextensions of L/K (by Proposition 8.8.6). of H f We also put gf,Iw (L/K, V ) = RΓ gf,Iw (L/K, T ) ⊗ΛL (ΛL ⊗ Q) ∈ Db (ΛL ⊗Q Mod) RΓ ft (and similarly for V ∗ (1)). These Selmer complexes have the following properties. (9.7.2) Proposition. (i) There is an isomorphism   ∼ g gf,Iw (L/K, T ) −→ DΛL RΓ RΓf (KS /L, A∗ (1))ι [3]. gf,Iw (L/K, Z) ∈ D[0,3] (Λ Mod) (ii) RΓ (Z = T, T ∗ (1)). L parf e 0 (L/K, Z) = 0 (iii) If [L : K] = ∞, then H (Z = T, T ∗ (1)). f,Iw

e 0 (K, A∗ (1)) = 0, then H e 0 (KS /L, A∗ (1)) = 0 and RΓ gf,Iw (L/K, T ) ∈ D[0,2] (ΛL Mod). (iv) If H f f parf e 0 (KS /L, A∗ (1)) is finite, then RΓ gf,Iw (L/K, V ) ∈ D[0,2] (ΛL ⊗Q Mod). (v) If H f

parf

gf,Iw (L/K, Z) is represented by the complex Proof. (i) See 8.9.6.1. (ii) By definition, RΓ Cone (X1 ⊕ X2 ⊕ X20 −→ X3 ) [−1], where • X1 = Ccont (GK,S , FΓL (Z)),

X20

=

M

X2 =

M v|p

Iv

Ccont (Gv /Iv , FΓL (Z )), •

v∈Σ0

X3 =

• Ccont (Gv , FΓL (Zv+ )),

M v∈Sf

231

• Ccont (Gv , FΓL (Z)).

As each O-module Z, Zv+ , Z Iv is free, the corresponding ΛL -modules FΓL (Z), FΓL (Zv+ ), FΓL (Z Iv ) are also free. Applying Proposition 4.2.9, we obtain that [0,2]

X1 , X2 , X3 ∈ Dparf (ΛL Mod),

[0,1]

X20 ∈ Dparf (ΛL Mod)

(as cdp (GK,S ) = cdp (Gv ) = 2, cdp (Gv /Iv ) = 1), which proves the claim. e 0 (L0 , Z) ⊆ H 0 (L0 , Z) vanishes, by (iii) This follows from the fact that the projective limit of the groups H f Proposition 8.3.5(iii) (cf. the proof of Lemma 9.1.5(i)). e 0 (KS /L, A∗ (1)) = 0, which in turns implies the vanishing of (iv) It follows from Proposition 8.10.14 that H f  ι ∼ 3 e f,Iw e f0 (KS /L, A∗ (1)) = 0. H (L/K, T ) −→ D H As noted in 4.2.8, this is sufficient to prove the claim. The same argument proves (v), as  ι ∼ e 3 (L/K, V ) −→ e 0 (KS /L, A∗ (1)) ⊗ Q = 0 H D H f,Iw f in this case. (9.7.3) Proposition. Let K ⊂ L0 ⊂ L ⊂ K∞ be arbitrary intermediate fields. Then: (i) There is a canonical isomorphism in Dfb t (ΛL0 Mod) L

∼ g gf,Iw (L/K, Z) ⊗Λ ΛL0 −→ RΓ RΓf,Iw (L0 /K, Z) L

(Z = T, T ∗ (1)).

(ii) There are natural pairings in Dfb t (ΛL Mod) L

gf,Iw (L/K, T ) ⊗ΛL RΓ gf,Iw (L/K, T ∗(1))ι −→ ΛL [−3], RΓ compatible with the isomorphisms from (i). Denote by

 ι gf,Iw (L/K, T ) −→ DΛL RΓ gf,Iw (L/K, T ∗ (1)) [−3] αT : RΓ

the corresponding adjoint map. (iii) The map

 ι gf,Iw (L/K, V ) −→ DΛ ⊗Q RΓ gf,Iw (L/K, V ∗ (1)) [−3] αV = αT ⊗ Q : RΓ L

is an isomorphism in Dfb t (ΛL ⊗Q Mod). (iv) The following conditions are equivalent: αT is an isomorphism in Dfb t (ΛL Mod) ⇐⇒ (∀v ∈ Σ0 )

Tamv (T, ($)) (= Tamv (T ∗ (1), ($))) = 0.

0 Proof. (i) Applying Proposition 8.10.10 to K∞ = L, L0 , we obtain canonical isomorphisms L

∼ gf,Iw (L/K, Z) ⊗ΛL ΛL0 −→ RΓ

  L L L ∼ ∼ g RΓf,Iw (K∞ /K, Z) ⊗Λ ΛL ⊗ΛL ΛL0 −→ RΓf,Iw (K∞ /K, Z) ⊗Λ ΛL0 −→ ∼

gf,Iw (L0 /K, Z). −→ RΓ (ii) These are the duality pairings (8.9.6.3.1) (cf. Proposition 8.10.10). (iii) This is a special case of Theorem 8.9.11. (iv) Proposition 8.9.7.6(iv),(v) and Theorem 8.9.12 apply to the extension L/K. In particular, αT is an isomorphism in Dfb t (ΛL Mod) ⇐⇒ (∀v ∈ Σ0 ) 0

⇐⇒ (∀v ∈ Σ )

0

Errv (DΛL , FΓL (T )) = 0 ⇐⇒

Errv (D, T ) = 0 ⇐⇒ (∀v ∈ Σ ) Tamv (T, ($)) = 0.

e 0 (KS /L, A) H f ∗

gf,Iw (L/K, V ) ∈ D[1,3] (ΛL ⊗Q Mod). (9.7.4) Proposition. (i) If is finite, then RΓ parf e 0 (KS /L, Y ) (Y = A, A (1)) are finite, then RΓ gf,Iw (L/K, Z) ∈ D[1,2] (ΛL ⊗Q Mod) (Z = V, V ∗ (1)). (ii) If H f parf gf,Iw (L/K, V ∗ (1)) ∈ Proof. (i) According to Proposition 9.7.2(v) (applied to A and V ∗ (1)), we have RΓ [0,2] Dparf (ΛL ⊗Q Mod). Applying DΛL [−3] and using the duality isomorphism of Proposition 9.7.3(iii), we obtain the claim. The statement (ii) follows from (i) and Proposition 9.7.2(v). 232

(9.7.5) Proposition. Assume that (∀v ∈ Σ0 ) Tamv (T, ($)) = 0. Then: e 0 (K, A) = 0, then H e 0 (KS /L, A) = 0 and RΓ gf,Iw (L/K, T ) ∈ D[1,3] (ΛL Mod). (i) If H f f parf e 0 (K, Y ) = 0 (Y = A, A∗ (1)), then H e 0 (KS /L, Y ) = 0 and RΓ gf,Iw (L/K, Z) ∈ D[1,2] (ΛL Mod) (ii) If H f f parf ∗ (Z = T, T (1)). Proof. The proof of 9.7.4 applies, using Proposition 9.7.2(iv) and 9.7.3(iv) instead of 9.7.2(v) and 9.7.3(iii). (9.7.6) Proposition (Self-dual case). Assume that there exists an isomorphism of F [GK,S ]-modules ∼ j : V −→ V ∗ (1) which is skew-symmetric (i.e. j ∗ (1) = −j) and satisfies j(Vv+ ) = V ∗ (1)+ v (v ∈ Σ). Then: (i) j induces an isomorphism in Dfb t (ΛL ⊗Q Mod) ∼

gf,Iw (L/K, V ) −→ RΓ gf,Iw (L/K, V ∗ (1)). j∗ : RΓ (ii) The induced pairing L

L

id⊗j∗

gf,Iw (L/K, V ) ⊗ΛL RΓ gf,Iw (L/K, V )ι −−−−→RΓ gf,Iw (L/K, V ) ⊗ΛL RΓ gf,Iw (L/K, V ∗ (1))ι −→ ΛL [−3] ∪ : RΓ is skew-hermitian, i.e. satisfies ∪ι ◦ s12 = −ν ι ◦ ∪, in the notation of Corollary 6.6.7. gf,Iw (L/K, V ) ∈ D[1,2] (Λ ⊗Q Mod). e 0 (KS /L, A) is finite, then RΓ (iii) If H L f parf e 0 (KS /L, A) is finite and corkΛL (H e 1 (KS /L, A)) = 0, then (iv) If H f

f

(∀i 6= 2) the ΛL ⊗ Q-module

i e f,Iw H (L/K, V ) = 0,

 ι ∼ e 2 (L/K, V ) −→ e 1 (KS /L, A) ⊗ Q H DΛL H f,Iw f

gf,Iw (L/K, V ) can be represented by a complex is torsion and RΓ u

Cone(M −−→M )[−2], where M is a free ΛL ⊗ Q-module of finite type and u an injective endomorphism of M . Proof. The statement (i) is trivial and (ii) follows from Corollary 6.6.7 (which applies thanks to 7.7.2). + ∗ e0 (iii) As A∗ (1) (resp., A∗ (1)+ v , v ∈ Σ) differs from A (resp., from Av ) by a finite group, Hf (KS /L, A (1)) is also finite, hence (iii) follows from Proposition 9.7.4(ii). e i (L/K, V ) vanish for i 6= 1, 2, and H 1 is (iv) By (iii), the ΛL ⊗ Q-modules of finite type H i := H f,Iw torsion-free. On the other hand, the duality isomorphisms 9.7.2(i) and 9.7.3(iii) imply that e f1 (KS /L, A)) = 0, rkΛL ⊗Q (H 1 ) = rkΛL ⊗Q (H 2 ) = corkΛL (H gf,Iw (L/K, V ) can be represented hence H 1 = 0 vanishes and H 2 is a torsion ΛL ⊗Q-module. It follows that RΓ u by a complex [P 1 −−→P 2 ] (in degrees 1, 2), where P 1 , P 2 are projective (hence free) ΛL ⊗ Q-modules of the same rank (hence isomorphic to each other) and Ker(u) = 0. (9.7.7) Proposition (Integral self-dual case). In the situation of 9.7.6, assume that j(T ) = T ∗ (1) (hence j(Tv+ ) = T ∗ (1)+ v for all v ∈ Σ). Then: (i) j induces an isomorphism in Dfb t (ΛL Mod) ∼

gf,Iw (L/K, T ) −→ RΓ gf,Iw (L/K, T ∗(1)). j∗ : RΓ (ii) The induced pairing L

L

id⊗j∗

gf,Iw (L/K, T ) ⊗ΛL RΓ gf,Iw (L/K, T )ι−−−−→RΓ gf,Iw (L/K, T ) ⊗ΛL RΓ gf,Iw (L/K, T ∗ (1))ι −→ ΛL [−3] ∪ : RΓ 233

is skew-hermitian. gf,Iw (L/K, T ) ∈ D[1,2] (Λ Mod). e 0 (K, A) = 0 and (∀v ∈ Σ0 ) Tamv (T, ($)) = 0, then RΓ (iii) If H L f parf e 1 (KS /L, A)) = 0, then (iv) If, under the assumptions of (iii), corkΛL (H f

(∀i 6= 2) the ΛL -module

i e f,Iw H (L/K, T ) = 0,

 ι ∼ 2 e f,Iw e f1 (KS /L, A) H (L/K, T ) −→ DΛL H

gf,Iw (L/K, T ) can be represented by a complex is torsion and RΓ u

Cone(M −−→M )[−2], where M is a free ΛL -module of finite type and u an injective endomorphism of M . Proof. The proof of Proposition 9.7.6 applies, using 9.7.5 instead of 9.7.4. (9.7.8) More general local conditions. In Sect. 9.6 and 9.7.1-7, we considered only Greenberg’s local conditions for Σ = {v | p}. It is often useful to consider Greenberg’s local conditions attached to an arbitrary intermediate set {v | p} ⊂ Σ ⊂ Sf : define, for each v ∈ Σ(p) := Σ − {v | p}, Xv+ = 0,

(∀X = T, V, A)

(∀X = T ∗ (1), V ∗ (1), A∗ (1))

Xv− = Xv

Xv+ = Xv ,

Xv− = 0.

We incorporate Σ into the notation by writing gf,Σ (L, X), RΓ

gf,Σ (KS /L, Y ), RΓ

gf,Iw,Σ (L/K, Z) RΓ

for the Selmer complexes attached to such local conditions (and we drop Σ from the notation if Σ = {v | p}). (9.7.9) Proposition. Assume that, for each finite extension K 0 of K contained in K∞ , each v ∈ Σ(p) and each prime v 0 | v of K 0 , we have H 0 (Gv0 , V ) = 0

(Gv0 = Gal(K v /Kv0 0 )).

(∗)

Then, for each intermediate field K ⊂ L ⊂ K∞ : (i) If [L : K] < ∞, then the canonical maps gf,Σ (L, V ) −→ RΓ gf (L, V ), RΓ

gf (L, V ∗ (1)) −→ RΓ gf,Σ (L, V ∗ (1)) RΓ

are isomorphisms in Dfb t (F Mod). (ii) If p ∈ Spec(ΛL ⊗ Q) is contained in the augmentation ideal of ΛL ⊗ Q, then the canonical maps gf,Iw,Σ (L/K, V )p −→ RΓ gf,Iw (L/K, V )p , RΓ

gf,Iw (L/K, V ∗ (1))p −→ RΓ gf,Iw,Σ (L/K, V ∗ (1))p RΓ

are isomorphisms in Dfb t (ΛL ⊗Q Mod). (iii) For each j ∈ Z and each minimal prime ideal q ∈ Spec(ΛL ), the ranks   ∗ ι ej e 3−j rk(ΛL )q H = f,Iw,Σ (L/K, T )q = rk(ΛL )q Hf,Iw,Σ (L/K, T (1)) q     e j (KS /L, A)) = rk(Λ ) DΛL (H e 3−j (KS /L, A∗ (1)))ι = rk(ΛL )q DΛL (H L q f,Σ f,Σ q

do not depend on Σ (and vanish for j 6= 1, 2). Proof. (i) There are exact triangles in Dfb t (F Mod) 234

q

M M

gf,Σ (L, V ) −→ RΓ gf (L, V ) −→ RΓ

v∈Σ(p)

gf (L, V ∗ (1)) −→ RΓ gf,Σ (L, V ∗ (1)) −→ RΓ

RΓur (Gw , V )

w|v

 ι M DF  RΓur (Gw , V ) [−2],

M

w|v

v∈Σ(p)

1 where Gw = Gal(K v /Lw ) and RΓur (Gw , V ) = RΓcont (Gw /Iw , V Iw ). As dim Hur (Gw , V ) = dim H 0 (Gw , V ), ∼ 0 the assumption (*) for K = L implies that RΓur (Gw , V ) −→ 0. (ii) There are exact triangles in Dfb t (ΛL ⊗Q Mod)

M

gf,Iw,Σ (L/K, V ) −→ RΓ gf,Iw (L/K, V ) −→ RΓ

RΓur (Gv , FΓ (V ))

v∈Σ(p)

gf,Iw (L/K, V ∗ (1)) −→ RΓ gf,Iw,Σ (L/K, V ∗ (1)) −→ RΓ

M

ι

DΛL ⊗Q (RΓur (Gv , FΓ (V ))) [−2].

v∈Σ(p) ∼

Write ΓL = Gal(L/K) as a product ΓL = ΓL,0 × ∆L , where |∆L | < ∞ and ΓL,0 −→ Zrp . Applying (the proof of) Proposition 8.9.7.7(ii) with R = O, Γ0 = ΓL,0 , ∆ = ∆L , Wv = T and all minimal prime ideals q0 ∈ Spec(∆L ), and using the assumption (*) with K 0 = LΓL,0 , we obtain (∀v ∈ Σ(p) )

RΓur (Gv , FΓ (V ))p = 0.

(iii) This follows from (ii), 8.9.6.1 and (8.9.6.4.2). (9.7.10) Let B be an abelian variety defined over K, which has good reduction outside Sf , and such that {v | p} = Σo ∪ Σt in the notation of 9.6.7.2. The assumption of Proposition 9.7.9 are then satisfied for ∼ T = Tp (B) (V = Vp (B) −→ V ∗ (1)), since Vv satisfies the monodromy-weight conjecture for each v - p (as remarked in the proof of Lemma 9.6.7.3(ii)). (9.7.11) In the situation 9.7.7(iv), it was proved in ([M-R 2], Prop. 6.5) that the duality isomorphism αT can be represented by a skew-hermitian pairing on the module M , provided that p 6= 2 and M is “minimal” in a suitable sense.

235

10. Generalized Cassels-Tate pairings Let K and S be as in 5.1. In this chapter we reformulate the duality results from Chapters 6 and 8 in terms of generalized Cassels-Tate pairings. The general mechanism is very simple: in the situation of 6.2.5(B) with n = 0, the discussion in 6.3.5 gives homomorphisms (under suitable finiteness assumptions on the local conditions) e fi (X1 )R−tors −→ Ext1R (H e j (X2 ), ωR ) H f

(i + j = 4)

with controlled (1) kernels and cokernels. Observing that, for every R-module M of finite type, the group Ext1R (M, ωR ) is very close (1) to HomR (MR−tors , H 0 (ωR ) ⊗R (Frac(R)/R)), we obtain bilinear forms e i (X1 )R−tors ⊗R H e j (X2 ) H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f f R−tors

(i + j = 4)

in (R Mod)/(pseudo − null) (sometimes even in (R Mod)) with controlled kernels. These pairings - at least for i = j = 2 - are natural generalizations of the Cassels-Tate (and Flach [Fl 1]) pairings in our context. In the self-dual case, we obtain skew-symmetric (or skew-hermitian) pairings, from which we deduce various parity results (Sect. 10.6-7). (10.1) A topological analogue (10.1.1) Assume that X is a connected compact oriented topological manifold of dimension D. The cohomology groups H i (X, Z) are finitely generated abelian groups; the exact sequences 0 −→ H i (X, Z) ⊗ Z/nZ −→ H i (X, Z/nZ) −→ H i+1 (X, Z)[n] −→ 0 give, in the limit, 0 −→ H i (X, Z) ⊗ Q/Z −→ H i (X, Q/Z) −→ H i+1 (X, Z)tors −→ 0 and ∼

b := lim H i (X, Z/nZ) −→ H i (X, Z) ⊗ Z. b H i (X, Z) ←− n

∼

For m|n, Poincar´e duality and the orientation H D (X, A) −→ A give perfect pairings H i (X, Z/mZ) ×   y

H D−i (X, Z/mZ) x  

−→ Z/mZ   y

H i (X, Z/nZ)

H D−i (X, Z/nZ)

−→

×

Z/nZ,

which induce in the limit Pontrjagin duality b −→ Q/Z. H i (X, Q/Z) × H D−i (X, Z) The orthogonal complement of the maximal divisible subgroup H i (X, Q/Z)div = H i (X, Z) ⊗ Q/Z ⊂ H i (X, Q/Z) is the torsion subgroup D−i b H D−i (X, Z) (X, Z)Z−tors . Z−tors = H

As a consequence, we obtain perfect pairings of finite abelian groups (1)

At least if R has no embedded primes 236

(10.1.1.1)

h , ii+1,D−i : H i+1 (X, Z)tors × H D−i (X, Z)tors −→ Q/Z. More precisely, Poincar´e duality identifies (10.1.1.1) with the Pontrjagin dual of b ←− H D−i (X, Z) b ←− H D−i (X, Z) 0 ←− (H D−i (X, Z)/tors) ⊗ Z tors ←− 0. (10.1.2) The pairing ha, bii+1,D−i can be described in terms of cocycles as follows. Represent a (resp., b) by a (singular) cocycle α ∈ Z i+1 (X, Z) (resp., β ∈ Z D−i (X, Z)). There exist n ≥ 1 and α0 ∈ C i (X, Z) (resp., β 0 ∈ C D−i−1 (X, Z)) such that nα = dα0 , nβ = dβ 0 . Then α0 ∪ β, β 0 ∪ α ∈ C D (X, Z)

d(α0 ∪ β) = n(α ∪ β), d(β 0 ∪ α) = n(β ∪ α) ∈ C D (X, nZ). This means that 1 0 (α ∪ β) (mod C D (X, Z)), n

1 0 (β ∪ α) (mod C D (X, Z)) n

are elements of Z D (X, n1 Z/Z); their cohomology classes in H D (X,

1 ∼ 1 Z/Z) −→ Z/Z ⊂ Q/Z n n

are equal to ha, bii+1,D−i and hb, aiD−i,i+1 , respectively. (10.1.3) As d(β 0 ∪ α0 ) = n(β ∪ α0 ) + (−1)D−i−1 n(β 0 ∪ α) = (−1)i(D−i) n(α0 ∪ β) + (−1)D−i−1 n(β 0 ∪ α) is a coboundary in C D (X, Z), we have (−1)i(D−i) ha, bii+1,D−i + (−1)D−i−1 hb, aiD−i,i+1 = 0, hence hb, aiD−i,i+1 = (−1)(D−i)(i+1) ha, bii+1,D−i = (−1)D(i+1) ha, bii+1,D−i .

(10.1.3.1)

In particular, if D = 2n − 1, then the pairing h , in,n : H n (X, Z)tors × H n (X, Z)tors −→ Q/Z satisfies hb, ain,n = (−1)n ha, bin,n . (10.1.4) The pairings h , ii,j can be described in a more abstract way in terms of a composition of two cup products in the derived category. For every complex A• of abelian groups put −incl

H!i (A• ) = H i (RΓ! (A• )),

RΓ! (A• ) = A• ⊗Z [Z−−→Q],

where the complex [Z −→ Q] is in degrees 0, 1. This defines an exact functor RΓ! : D∗ (Z Mod) −→ D∗ (Z Mod)

(∗ = ∅, +, −, b).

The cohomology sequence attached to the exact sequence of complexes 237

0 −→ (A• ⊗ Q)[−1] −→ RΓ! (A• ) −→ A• −→ 0 gives 0 −→ H i−1 (A• ) ⊗ Q/Z −→ H!i (A• ) −→ H i (A• )tors −→ 0. For A, B ∈ D− (Z Mod), the same construction as in 2.10.7-8 defines a canonical product L

L

RΓ! (A)⊗Z RΓ! (B) −→ RΓ! (A⊗Z B). The induced cup products L

H!i (A) ⊗Z H!j (B) −→ H!i+j (A⊗Z B) factor through L

H i (A)tors ⊗Z H j (B)tors −→ H!i+j (A⊗Z B). (10.1.5) Applying this construction to A = B = RΓ(X, Z) and the truncated cup product L

∼

RΓ(X, Z)⊗Z RΓ(X, Z) −→ τ≥D RΓ(X, Z) −→ Z[−D], we obtain products L

∼

RΓ! (RΓ(X, Z))⊗Z RΓ! (RΓ(X, Z)) −→ RΓ! (Z[−D]) −→ Q/Z[−D − 1]

(10.1.5.1)

and ∪i+1,D−i : H i+1 (X, Z)tors ⊗Z H D−i (X, Z)tors −→ H!D+1 (Z[−D]) = Q/Z. Alternatively, there are canonical isomorphisms in D(Z Mod) ∼

∼

RΓ! (RΓ(X, Z)) −→ RΓ(X, [Z −→ Q]) −→ RΓ(X, Q/Z[−1]), and the product (10.1.5.1) is induced by L

∼

Q/Z[−1]⊗Z Q/Z[−1] −→ Q/Z[−1] and L

∼

RΓ(X, Q/Z[−1])⊗Z RΓ(X, Q/Z[−1]) −→ τ≥D+1 RΓ(X, Q/Z[−1]) −→ Q/Z[−D − 1]. (10.1.6) Lemma. The pairings h , ii+1,D−i , ∪i+1,D−i : H i+1 (X, Z)tors ⊗Z H D−i (X, Z)tors −→ Q/Z from 10.1.2 (resp., 10.1.5) are related by ∪ = (−1)D−1 h , i . Proof. In the notation of 10.1.2, α e = α ⊗ 1 + (−1)i+1 α0 ⊗

1 , n

1 βe = β ⊗ 1 + (−1)D−i β 0 ⊗ n

are cocycles of degrees i + 1 (resp., D − i) in RΓ! (C • (X, Z)) lifting α (resp., β). Using the notation from Lemma 2.10.7(i), the cup product a ∪i+1,D−i b is represented by 238

e = v(α ⊗ β ⊗ 1 ⊗ 1 + (−1)D−i α ⊗ β 0 ⊗ 1 ⊗ 1 + (−1)D+1 α0 ⊗ β ⊗ 1 ⊗ 1+ v ◦ s23 (e α ⊗ β) n n 1 1 1 i 0 0 D−i 0 +(−1) α ⊗ β ⊗ ⊗ ) = (α ∪ β) ⊗ 1 + (−1) (α ∪ β ) ⊗ . n n n As α ∪ β ∈ Z D+1 (X, Z) = dC D (X, Z), it follows that a ∪i+1,D−i b is represented by 1 (−1)D−i (α ∪ β 0 ) (mod C D (X, Z)), n hence a ∪i+1,D−i b = (−1)D−i (−1)(i+1)(D−i−1) hb, aiD−i,i+1 = = (−1)D−i (−1)(i+1)(D−i−1) (−1)(i+1)(D−i) ha, bii+1,D−i = (−1)D−1 ha, bii+1,D−i . (10.2) Abstract Cassels-Tate pairings In this section we construct generalized Cassels-Tate pairings, mimicking the discussion in 10.1.4. Throughout 10.2, we assume that R has no embedded primes ( ⇐⇒ R satisfies (S1 )). • (10.2.1) Assume that J = ωR [n] for some n ∈ Z (hence DJ = Dn ). Let X1 , X2 be bounded complexes of admissible R[GK,S ]-modules with cohomology of finite type over R, and

π : X1 ⊗R X2 −→ J(1) a morphism of complexes of R[GK,S ]-modules. Finally, assume that we are given orthogonal local conditions ∆(X1 ) ⊥π,hS ∆(X2 ) such that the complexes US+ (Xi ) (i = 1, 2) have cohomology of finite type over R. e• (Xi ) = C e• (GK,S , Xi ; ∆(Xi )) (i = 1, 2) (10.2.2) Under the assumptions of 10.2.1, the Selmer complexes C f f also have cohomology of finite type over R. Recall from 6.3.1 the cup products e • (X1 ) ⊗R C e • (X2 ) −→ J[−3] = ω • [n − 3] ∪π,r,h : C f f R

(r ∈ R)

and their adjoints • ef• (X1 ) −→ Hom•R (C ef• (X2 ), ωR γπ,r,hS = adj(∪π,r,h ) : C [n − 3]).

The construction from 2.10.14 applied to ∪π,r,h defines cup products (independent of r ∈ R) e fi (X1 ) e j (X2 ) ∪π,hS ,i,j : H ⊗R H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f R−tors R−tors

(i + j = 4 − n)

in (R Mod)/(pseudo − null) (or even in (R Mod), if R is Cohen-Macaulay). Furthermore, we obtain from 2.10.15-17 isomorphisms in (R Mod)/(pseudo − null) ∼ ∼ e j (X2 ) e j (X2 ), ωR ) −→ HomR (H , H 0 (ωR ) ⊗R (Frac(R)/R)) −→ Ext1R (H f f R−tors   ∼ gf (X2 )) −→ H i DJ[−3] (RΓ , (i + j = 4 − n) R−tors

the composition of which with 239

e i (X1 ) e j (X2 ) adj(∪π,hS ,i,j ) : H −→ HomR (H , H 0 (ωR )⊗R (Frac(R)/R)) f f R−tors R−tors

(i+j = 4−n)

coincides, up to a sign, with the restriction of the map   e i (X1 ) −→ H i DJ[−3] (RΓ gf (X2 )) . (γπ,r,hS )∗ : H f (10.2.3) Theorem. Under the assumptions of 10.2.1, let i + j = 4 − n and Err = Err(∆(X1 ), ∆(X2 ), π). Assume that π is a perfect duality in the sense of 6.2.6. Then, in the category (R Mod)/(pseudo − null), (i) Ker(adj(∪π,hS ,i,j )) is isomorphic to a subquotient of H i−1 (Err). (ii) If H i−1 (Err) is R-torsion, then Coker(adj(∪π,hS ,i,j )) (resp., Ker(adj(∪π,hS ,i,j ))) is isomorphic to a subobject (resp., a quotient) of H i (Err) (resp., of H i−1 (Err)). ∼ In particular, if H i−1 (Err) = H i (Err) −→ 0 in (R Mod)/(pseudo−null), then adj(∪π,hS ,i,j ) is an isomorphism (again in (R Mod)/(pseudo − null)). Proof. This follows from Proposition 2.10.17 applied to ∪π,r,h and Theorem 6.3.4. (10.2.4) Proposition. Assume that, in addition to 10.2.1, the local conditions ∆(Xi ) (i = 1, 2) admit transposition data 6.5.3.1-5. Then the cup products attached to π and π ◦ s12 : X2 ⊗R X1 −→ X1 ⊗R X2 −→ J(1) are related by x ∪π,hS ,i,j y = (−1)ij y ∪π◦s12 ,h0S ,j,i x

(i + j = 4 − n).

Proof. This follows from Corollary 6.5.5 and (2.10.14.1). (10.2.5) Proposition (Self-dual case). Assume that, in 10.2.1, Xi = X, ∆(Xi ) = ∆(X) (i = 1, 2), π 0 := π ◦ s12 = c · π with c = ±1 and ∆(X) admit transposition operators as in Proposition 6.6.2. Then x ∪π,hS ,i,j y = c(−1)ij y ∪π,hS ,j,i x

(i + j = 4 − n).

In particular, if n = 0, then the bilinear form e f2 (X) e f2 (X) ∪π,hS ,2,2 : H ⊗R H −→ H 0 (ωR ) ⊗R (Frac(R)/R) R−tors R−tors is symmetric (resp., skew-symmetric) if c = +1 (resp., c = −1). Proof. Combine Proposition 6.6.2 and Proposition 10.2.4. (10.2.6) In the case n = 0, the formula y ∪π,hS ,j,i x = c(−1)i x ∪π,hS ,i,j y

(i + j = 4)

in Proposition 10.2.5 should be compared to (10.1.3.1) for D = 3: hb, aij,i = (−1)i ha, bii,j

(i + j = 4).

The extra factor c comes from the fact that X is identified with its dual DJ (X)(1) = D(X)(1) by using the bilinear form π of parity c. (10.2.7) Hermitian case (10.2.7.1) Assume that R is equipped with an involution ι : R −→ R, and that ν : J ι −→ J is as in 6.6.4; denote by ν∗ : H 0 (ωR )ι −→ H 0 (ωR ) • ι the map induced by ν[−n] : J ι [−n] = (ωR ) −→ J[−n]. Assume that we are in the situation of 6.6.5, i.e.

240

(X1 , ∆(X1 )) = (X, ∆(X)),

ι

ι

(X2 , ∆(X2 )) = (X ι , ∆(X) ),

∆(X) ⊥π,hS ∆(X)

and π ◦ s12 = c · (ν ◦ π ι ),

c = ±1.

It is sometimes more convenient to view the cup products from 10.2.2 e i (X) e j (X)ι ∪π,hS ,i,j : H ⊗R H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f f R−tors R−tors

(i + j = 4 − n)

as hermitian pairings e i (X) e j (X) h , iπ,hS ,i,j : H ×H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f f R−tors R−tors

(i + j = 4 − n)

satisfying hαx, βyiπ,hS ,i,j = α ι(β) hx, yiπ,hS ,i,j

(α, β ∈ R).

(10.2.7.2) Proposition. Assume that X is as in 10.2.7.1 and admits transposition operators as in Proposition 6.6.6. Then   hx, yiπ,hS ,i,j = c(−1)ij (ν∗ ⊗ ι) hy, xiπ,hS ,j,i (i + j = 4 − n). In particular, if n = 0, then the hermitian form e f2 (X) e f2 (X) h , i = h , iπ,hS ,2,2 : H ×H −→ H 0 (ωR ) ⊗R (Frac(R)/R) R−tors R−tors satisfies hαx, βyi = α ι(β) hx, yi,

hx, yi = c · (ν∗ ⊗ ι)(hy, xi)

(α, β ∈ R).

Proof. Combine Proposition 6.6.6 and (2.10.14.1). (10.2.8) Duality of error terms. For v - p, let X, Y be bounded complexes of admissible R[Gv ]-modules • • with cohomology of finite type over R and π : X ⊗R Y −→ ωR (1) = σ≥0 ωR (1) a morphism of complexes of R[Gv ]-modules. Assume that p ∈ Spec(R) satisfies dim(Rp ) = depth(Rp ) = 1 and πp : Xp ⊗Rp Yp −→ • (ωR )p (1) is a perfect duality over Rp . By Corollary 7.6.8(i), there are isomorphisms in Df t (Rp Mod) h i fv −1 ∼ ur 0 1 0 1 Errv (∆ur v (X), ∆v (Y ), π)p −→ H{p} (Hcont (Iv , X)p )−−→H{p} (Hcont (Iv , X)p ) h i fv −1 ∼ ur 0 1 0 1 Errv (∆ur v (Y ), ∆v (X), π ◦ s12 )p −→ H{p} (Hcont (Iv , Y )p )−−→H{p} (Hcont (Iv , Y )p ) , where the complexes on the R.H.S. are in degrees 1, 2. The duality theory for Iv described in 7.5.8 comes from the cup product Qis

• • • • • • • II Ccont (Iv , X) ⊗R Ccont (Iv , Y ) −→ Ccont (Iv , ωR (1)) −→ τ≥1 Ccont (Iv , ωR (1))−→ ωR [−1].

(10.2.8.1)

Localizing (10.2.8.1) at p, the construction in 2.10.7-9 gives rise to products ∼

j • 0 i 0 1 ∪ij : H{p} (Hcont (Iv , X)p ) ⊗Rp H{p} (Hcont (Iv , Y )p ) −→ H{p} ((ωR )p ) −→ IRp

which induce fv -equivariant isomorphisms of Rp -modules of finite length 241

(i + j = 2),

∼

j 0 i 0 H{p} (Hcont (Iv , X)p ) −→ DRp (H{p} (Hcont (Iv , Y )p ))

(i + j = 2),

(10.2.8.2)

by a localized verson of the duality (7.5.8.2) and Proposition 2.10.12. Using the fv -equivariance of (10.2.8.2) for i = j = 1, we obtain isomorphisms ∼

ur 3−q ur H q (Errv (∆ur (Errv (∆ur v (X), ∆v (Y ), π)p ) −→ DRp (H v (Y ), ∆v (X), π ◦ s12 )p ))

(q = 1, 2). (10.2.8.3)

Comparing the lengths (over Rp ) of the both sides in (10.2.8.3), we obtain Tamv (X, p) = Tamv (Y, p). (10.3) Greenberg’s local conditions In this section we investigate the abstract pairings from Sect. 10.2 in the context of Greenberg’s local conditions (including Iwasawa theory). Throughout 10.3, we assume that R has no embedded primes. • (10.3.1) Everything in 10.2 works under the assumptions of 7.8.2: let J = ωR = σ≥0 J, X1 = X, X2 = Y and π : X ⊗R Y −→ J(1) be as in 6.7.5(B) (in particular, all complexes X, Y , Xv+ , Yv+ are bounded) and Xv+ ⊥π Yv+ for all v ∈ Σ. Under these assumptions, the local conditions ∆(Z) (Z = X, Y ) defined in 7.8.2 satisfy

∆(X) ⊥π,0 ∆(Y ) and admit transposition operators satisfying 6.5.3.1-5, by 7.8.3. ur 0 If π is a perfect duality, then the cohomology of the error terms Errv (∆ur v (X), ∆v (Y ), π) for v ∈ Σ are given in (R Mod)/(pseudo − null) by 7.8.4.5. In particular,

M

∼

H i (Errv (∆v (X), ∆v (Y ), π)) −→ 0

(∀i 6= 1, 2)

v∈Σ0

in (R Mod)/(pseudo − null). If π is a perfect duality, then the error terms Errv (∆v (X), ∆v (Y ), π) for v ∈ Σ are given by Proposition ∼ 6.7.6(iv). In particular, if Xv+ ⊥⊥π Yv+ , then Errv (∆v (X), ∆v (Y ), π) −→ 0 in Db (R Mod). (10.3.2) Specializing the results of 10.2.2-4, the cup products • ef• (X) ⊗R C ef• (Y ) −→ J[−3] = ωR ∪π,r,0 : C [−3]

(r ∈ R)

give rise to pairings e i (X) e j (Y ) ∪π,0,i,j : H ⊗R H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f f R−tors R−tors

(i + j = 4)

in (R Mod)/(pseudo − null) (or even in (R Mod), if R is Cohen-Macaulay), satisfying x ∪π,0,i,j y = (−1)ij y ∪π◦s12 ,0,j,i x

(i + j = 4).

(10.3.2.1)

The kernels and cokernels of the adjoint maps e fi (X) e j (Y ) adj(∪π,0,i,j ) : H −→ HomR (H , H 0 (ωR ) ⊗R (Frac(R)/R)) f R−tors R−tors

(i + j = 4)

are as in Theorem 10.2.3. If K 0 /K is a finite Galois subextension of KS /K in which all primes v ∈ Σ0 are unramified, then 242

(∀g ∈ Gal(K 0 /K))

Adf (g)(x) ∪π,0,i,j Adf (g)(y) = x ∪π,0,i,j y,

(10.3.2.2)

by Proposition 8.8.9. In the self-dual case, i.e. if Y = X, Yv+ = Xv+ (v ∈ Σ) and π ◦ s12 = c · π with c = ±1, then Proposition 10.2.5 implies that ∪π,0,2,2 is symmetric (resp., skew-symmetric) if c = +1 (resp., c = −1). (10.3.3) Iwasawa theory ∼ (10.3.3.1) Let K∞ /K be as in 8.8.1, with Γ = Gal(K∞ /K) = Γ0 × ∆, where Γ0 −→ Zrp (r ≥ 1) and ∆ is a finite abelian group. We assume that the condition (U ) from 8.8.1 is satisfied, i.e. each prime v ∈ Σ0 is unramified in K∞ /K (this is automatic if Γ = Γ0 ). The ring R = R[[Γ]] is equipped with the canonical R-linear involution ι : R −→ R induced by γ −→ γ −1 (γ ∈ Γ). As R = R0 [∆] with ∼ R0 = R[[Γ0 ]] −→ R[[X1 , . . . , Xr ]], it follows from Lemma 2.10.13.3(iii) that Frac(R) = Frac(R0 )[∆]. As in 8.4.6.2, we fix, for each S = R, R, a complex ωS• = σ≥0 ωS• of injective S-modules representing ωS . There exists a quasi-isomorphism (unique up to homotopy) • • ϕ : ωR ⊗R R −→ ωR .

Fix ϕ; then there exists a morphism of complexes of R-modules • ι • ν : (ωR ) −→ ωR

(again unique up to homotopy) making the following diagram commutative: ι

ϕι

• ωR ⊗R R   yid⊗ι

−−→

• ωR ⊗R R

−−→

ϕ

• ι (ωR )  ν y • ωR .

• • ι Fix such a morphism ν; then the map ν ι : ωR −→ (ωR ) is a homotopy inverse of ν. The induced maps on cohomology

ι

• ι • ν∗ : H ∗ (ωR ) = H ∗ (ωR ) ⊗R R −→ H ∗ (ωR ) ⊗R R = H ∗ (ωR )

are equal to id ⊗ ι. By abuse of language, we shall denote ν∗ , as well as
ι ν∗ ⊗ ι : H 0 (ωR ) ⊗R (Frac(R)/R) −→ H 0 (ωR ) ⊗R (Frac(R)/R), by ι. (10.3.3.2) We need a slightly stronger version of the assumptions 6.7.5(B): we require that, in each degree i ∈ Z, the components X i , Y i , (Xv+ )i , (Yv+ )i (v ∈ Σ) are of finite type over R. The recipe 8.9.2 then defines Greenberg’s local conditions for FΓ (Z) = (Z ⊗R R) < −1 >,

ι

FΓ (Z)ι = (Z ⊗R R ) < −1 >

(Z = X, Y )

over R, together with a pairing F (π)

ϕ(1)

• • π : FΓ (X) ⊗R FΓ (Y )ι −−−−→ωR (1) ⊗R R−−→ωR (1),

under which ι + FΓ (X)+ v ⊥π (FΓ (Y ) )v

(v ∈ Σ).

As in 6.6.5, we have ∆(FΓ (Y )ι ) = ∆(FΓ (Y ))ι ,

e • (FΓ (Y )ι ) = C e • (FΓ (Y ))ι . C f f

If π is a perfect duality, so is π, by Corollary 8.4.6.5; if, in addition, Xv+ ⊥⊥π Xv+ (v ∈ Σ), then 243

ι + FΓ (X)+ v ⊥⊥π (FΓ (Y ) )v .

(10.3.3.3) As R has no embedded primes, the ring R has the same property. This implies that the theory from 10.2 (as specialized in 10.3.1-2) applies to FΓ (X), FΓ (Y )ι and π: the cup products ι • e• (FΓ (X)) ⊗ C e• ∪π,r,0 : C f R f (FΓ (Y )) −→ ωR [−3]

(r ∈ R)

induce pairings e i (K∞ /K, X) e j (K∞ /K, Y )ι ∪π,0,i,j : H ⊗ H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f,Iw R−tors R f,Iw R−tors

(i + j = 4)

in (R Mod)/(pseudo − null) (or even in (R Mod), if R - hence R - is Cohen-Macaulay). Similarly, s

π

12 • π ◦ s12 : Y ⊗R X −−→X ⊗R Y −−→ωR (1)

gives rise to ι F (π◦s12 )

ϕ(1)

• • π ◦ s12 : FΓ (Y ) ⊗R FΓ (X) −−−−−−→ωR (1) ⊗R R−−→ωR (1)

and - using the notation from 6.6.4 - cup products • ef• (FΓ (Y ))ι ⊗ C e• (∪π◦s12 ,r,0 )ι : C R f (FΓ (X)) −→ ωR [−3]

(r ∈ R)

and ∪π◦s12 ,0,j,i


ι


ι i 0 e j (K∞ /K, Y )ι e f,Iw :H ⊗ H (K /K, X) −→ H (ω ) ⊗ (Frac(R)/R) . ∞ f,Iw R R R R−tors R−tors

As in 10.2.7.1, it is sometimes convenient to view ∪π,0,i,j and ∪π◦s12 ,0,j,i


ι

(i + j = 4) as hermitian pairings

e i (K∞ /K, X) e j (K∞ /K, Y ) h , iπ,0,i,j : H ×H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f,Iw f,Iw R−tors R−tors


ι ι i e j (K∞ /K, Y ) e f,Iw h , iπ◦s12 ,0,j,i : H ×H (K∞ /K, X)R−tors −→ H 0 (ωR ) ⊗R (Frac(R)/R) f,Iw R−tors (10.3.3.3.1) satisfying hαx, βyiπ,0,i,j = α ι(β) hx, yiπ,0,i,j ι

(α, β ∈ R; i + j = 4)

ι

hβy, αxiπ◦s12 ,0,j,i = ι(β) α hy, xiπ◦s12 ,0,j,i

In order to avoid any confusion, we stress that the action of ι(β) α in the second formula is with respect to the R-module structure (−)ι on the R.H.S. (10.3.3.4) Lemma. The following diagram is commutative: π:

FΓ (X) ⊗R FΓ (Y )  s y 12 ι

ι

F (π)

−−−−−−→ F (π◦s12 )ι

ι

(π ◦ s12 ) : FΓ (Y ) ⊗R FΓ (X) −−−−−−→

• ωR (1) ⊗R R x  id⊗ι • ωR (1) ⊗R R

ι

ϕ(1)

−−−−→ ϕι (1)

−−−−→

• ωR (1) x ν(1)  • ι (ωR ) (1).

Proof. Commutativity of the left (resp., right) square follows from the commutative diagram ι

−−−−−−→

R ⊗R R

−−−−−−→

R ⊗R R  s12 y ι

id⊗ι

(id⊗ι)ι

(resp., from 10.3.3.1). 244

R x ι  ι

R

(10.3.3.5) Proposition. Under the assumptions of 10.3.3.2, the hermitian pairings (10.3.3.3.1) are related by   ι hαx, βyiπ,0,i,j = (−1)ij ι hβy, αxiπ◦s12 ,0,j,i (i + j = 4). Proof. As in the proof of Corollary 6.5.5, the existence of transposition operators for FΓ (X) and FΓ (Y )ι , together with Lemma 10.3.3.4, imply that the following diagram is commutative up to homotopy (for any r ∈ R): ∪π,r,0 :

ι e • (FΓ (X)) ⊗ C e• C f R f (FΓ (Y ))  s y 12

−−→

• ωR [−3] x ν[−3] 

e • (FΓ (Y ))ι ⊗ C e• (∪π◦s12 ,1−r,0 )ι : C f R f (FΓ (X)) −−→

• ι (ωR ) [−3].

The result follows by applying (2.10.14.1). (10.3.3.6) Alternatively, one can reformulate the formula in Proposition 10.3.3.5 as x ∪π,0,i,j y = (−1)ij ι y ∪π◦s12 ,0,j,i


ι
x

(i + j = 4),

where i e f,Iw x∈H (K∞ /K, X)R−tors ,

ι

e j (K∞ /K, Y ) y∈H f,Iw R−tors

(i + j = 4).

(10.3.4) Self-dual case (10.3.4.1) Back to the situation of 10.3.1, assume that Y = X, ∆(Y ) = ∆(X) and π : X ⊗R X −→ J(1) satisfies π ◦ s12 = c · π,

c = ±1.

(10.3.4.2) Proposition. (i) The pairings e fi (X) e j (X) ∪π,0,i,j : H ⊗R H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f R−tors R−tors satisfy

x ∪π,0,i,j y = c(−1)ij y ∪π,0,j,i x

(i + j = 4)

(i + j = 4).

In particular, ∪π,0,2,2 is symmetric (resp., skew-symmetric) if c = +1 (resp., c = −1). (ii) If X, Xv+ (v ∈ Σ) satisfy the assumptions of 10.3.3.2, then the pairings e i (K∞ /K, X) e j (K∞ /K, X)ι ∪π,0,i,j : H ⊗ H −→ H 0 (ωR )⊗R (Frac(R)/R) f,Iw R−tors R f,Iw R−tors satisfy

x ∪π,0,i,j y = c(−1)ij ι y ∪π◦s12 ,0,j,i


ι
x

(i + j = 4).

In particular, ∪π,0,2,2 is symmetric hermitian (resp., skew-hermitian) if c = +1 (resp., c = −1). Proof. (i) This is a special case of Proposition 10.2.5. (ii) The statement follows from 10.3.3.6, since π ◦ s12 = c · π,

∪π◦s12 ,0,j,i = c · ∪π,0,j,i .

Alternatively, one can apply Proposition 10.2.7.2, as π ◦ s12 = ν ◦ (π ◦ s12 )ι = c · (ν ◦ π ι ). (10.3.5) Iwasawa theory - dihedral case Let X, Y and π be as in 10.3.3.2. 245

(i + j = 4)

(10.3.5.1) Assume that K∞ /K extends to a dihedral Galois extension K∞ /K + , i.e. [K : K + ] = 2 and Γ+ = Gal(K∞ /K + ) is a semi-direct product Γ+ = Γ o {1, τ } for some τ ∈ Γ+ − Γ such that τ 2 = 1,

τ γτ −1 = γ −1

(γ ∈ Γ)

(which is then true for every τ ∈ Γ+ − Γ). We also assume that Greenberg’s local conditions for X, Y are defined over K + , in the following sense: (10.3.5.1.1) There is a finite set of primes S + of K + such that S = {v : (∃v + ∈ S + ) v|v + } (hence KS /K + is a Galois extension). (10.3.5.1.2) Both X and Y are (bounded) complexes of (admissible) R[Gal(KS /K + )]-modules and the morphism π is Gal(KS /K + )-equivariant. (10.3.5.1.3) For a suitable subset Σ+ ⊂ Sf+ , Σ0 = {v ∈ Sf : (∃v + ∈ Sf+ − Σ+ ) v|v + }.

Σ = {v ∈ Sf : (∃v + ∈ Σ+ ) v|v + },

(10.3.5.1.4) (∀Z = X, Y ) (∀v + ∈ Σ+ ) there is a (bounded) complex of (admissible) R[Gv+ ]-modules Zv++ and a morphism of complexes of R[Gv+ ]-modules jv++ (Z) : Zv++ −→ Z such that, for each v ∈ Σ, v|v + , the restriction of jv++ (Z) to Gv ⊂ Gv+ coincides with jv+ (Z). (10.3.5.1.5) Each prime v ∈ Σ0 is unramified in K/K + . (10.3.5.2) In order to simplify the notation, put G+ = Gal(KS /K + ), G = GK,S = Gal(KS /K); denote by ρ : G+ −→ Γ+ the canonical projection and fix τ ∈ ρ−1 (τ ). The tautological character ρ

χΓ : G−→Γ ,→ R

∗

satisfies χΓ (τ 2 ) = 1,

χΓ (τ gτ −1 ) = χΓ (g −1 )

(g ∈ G).

The R-algebra R[[Γ+ ]] = R ⊕ Rτ = R ⊕ τ R

(10.3.5.2.1)

is equipped with an involution ι, induced by γ + 7→ (γ + )−1 (γ + ∈ Γ+ ), which extends ι on R. (10.3.5.3) For Z = X, Y , Shapiro’s Lemma gives a quasi-isomorphism +

• • sh : Ccont (G+ , IndG G (FΓ (Z))) −→ Ccont (G, FΓ (Z))

(and similarly for FΓ (Z)ι ). Recall that FΓ (Z) = (Z ⊗R R) < −1 >,

ι

FΓ (Z)ι = (Z ⊗R R ) < −1 > .

Denote by FΓ+ (Z) the following complex in (ad Mod)R−f t : R[G+ ] FΓ+ (Z) = Z ⊗R R[[Γ+ ]], with r ∈ R and g + ∈ G+ acting by r(z ⊗ a) = z ⊗ ra,

g + (z ⊗ a) = g + z ⊗ aρ(g + )−1 ,

respectively. Note that FΓ+ (Z)ι is canonically isomorphic to 246

(z ∈ Z, a ∈ R[[Γ+ ]])

Z ⊗R R[[Γ+ ]], where r ∈ R and g + ∈ G+ act by g + (z ⊗ a) = g + z ⊗ aρ(g + )−1

r(z ⊗ a) = z ⊗ ι(r)a,

(z ∈ Z, a ∈ R[[Γ+ ]]).

The decomposition (10.3.5.2.1) defines canonical isomorphisms of R[G]-modules ∼

∼

FΓ+ (Z) −→ (Z ⊗R R) < −1 > ⊕(Z ⊗R R)τ < −1 >−→ (Z ⊗R R) < −1 > ⊕(Z ⊗R R) < 1 > τ ι ι ι ι ∼ ∼ FΓ+ (Z)ι −→ (Z ⊗R R ) < −1 > ⊕(Z ⊗R R )τ < −1 >−→ (Z ⊗R R ) < −1 > ⊕(Z ⊗R R ) < 1 > τ. (10.3.5.4) Lemma. (i) For Z = X, Y , the formula f 7→ f (1) + (τ −1 ⊗ 1)f (τ )(1 ⊗ τ ) defines isomorphisms of complexes in (ad Mod)R−f t R[G+ ] +

∼

+

G IndG G ((Z ⊗R R) < −1 >) (= IndG (FΓ (Z))) −→ FΓ+ (Z) ι

+

∼

+

G ι ι IndG G ((Z ⊗R R ) < −1 >) (= IndG (FΓ (Z) )) −→ FΓ+ (Z) ∼

+

+ IndG G ((X ⊗R Y ) ⊗R R) −→ (X ⊗R Y ) ⊗R R[Γ /Γ],

which make the following diagram commutative: +

+

ι

id⊗(id⊗ι)

IndG G ((X ⊗R Y ) ⊗R R)  o y

prod

(X ⊗R Y ) ⊗R R[Γ+ /Γ].

G IndG G ((X ⊗R R) < −1 >) ⊗R IndG ((Y ⊗R R ) < −1 >) −−−−−−→  o y

FΓ+ (X) ⊗R FΓ+ (Y )ι

−−−−−−→

+

Above, the map prod is given by prod : (x ⊗ (a1 + a2 τ )) ⊗ (y ⊗ (b1 + b2 τ )) 7→ (x ⊗ y) ⊗ (a1 ι(b1 ) + a2 ι(b2 )τ )

(aj , bj ∈ R).

(ii) The formulas uX : x ⊗ (a1 + a2 τ ) 7→ x ⊗ (ι(a2 ) + ι(a1 )τ ) vY : y ⊗ (b1 + b2 τ ) 7→ y ⊗ (ι(b2 ) + ι(b1 )τ ) define isomorphisms of complexes of R[G+ ]-modules ∼

∼

uX : FΓ+ (X) −→ FΓ+ (X)ι ,

vY : FΓ+ (Y )ι −→ FΓ+ (Y )

satisfying vZ = u−1 Z (Z = X, Y ) and making the following diagram commutative: FΓ+ (X) ⊗R FΓ+ (Y )ι  u ⊗v y X Y

prod

−−−−→ prodι

FΓ+ (X)ι ⊗R FΓ+ (Y ) −−−−→

(X ⊗R Y ) ⊗R R[Γ+ /Γ] x id  X⊗Y ⊗(ι⊗τ ) ι

(X ⊗R Y ) ⊗R R [Γ+ /Γ].

+

+ Proof. (i) Let f ∈ IndG G ((Z ⊗R R) < −1 >); then f : G −→ Z ⊗R R is a function satisfying

f (gg + ) = (g ⊗ χΓ (g)−1 )f (g + )

(g ∈ G, g + ∈ G+ ),

on which g1+ ∈ G+ acts by (g1+ ∗ f )(g + ) = f (g + g1+ ). In particular, 247

(τ ∗ f )(1) = f (τ ),

(g ∗ f )(1) = (g ⊗ χΓ (g)−1 )f (1),

(τ ∗ f )(τ ) = (τ 2 ⊗ 1)f (1),

(g ∗ f )(τ ) = f ((τ gτ −1 )τ ) = (τ gτ −1 ⊗ χΓ (g))f (τ ).

(g ∈ G)

Putting j(f ) = f (1) + (τ −1 ⊗ 1)f (τ )(1 ⊗ τ ) = f1 + f2 τ, where fi ∈ Z ⊗R R are equal to f2 = (τ −1 ⊗ 1)f (τ ),

f1 = f (1), then

j(τ ∗ f ) = f (τ ) + (τ −1 ⊗ 1)(τ 2 ⊗ 1)f (1)(1 ⊗ τ ) = (τ ⊗ 1)(f2 + f1 τ ) = τ ∗ j(f ) j(g ∗ f ) = (g ⊗ χΓ (g)−1 )f1 + (g ⊗ χΓ (g))f2 τ = g ∗ j(f )

(g ∈ G).

This shows that the (bijective) map +

j : IndG G ((Z ⊗R R) < −1 >) −→ FΓ+ (Z) is a homomorphism of R[G+ ]-modules, hence an isomorphism. The same argument also works for +

ι

∼

ι j ι : IndG G ((Z ⊗R R ) < −1 >) −→ FΓ+ (Z) .

The corresponding statement for (X ⊗R Y ) ⊗R R is just one of the isomorphisms from 8.1.3. As regards the commutativity of the diagram involving prod, let +

ι

+

fX ∈ IndG G ((X ⊗R R) < −1 >),

fY ∈ IndG G ((Y ⊗R R ) < −1 >).

Writing j(fX ) = u1 + u2 τ, j ι (fY ) = v1 + v2 τ,

u1 = fX (1), u2 = (τ −1 ⊗ 1)fX (τ ) ∈ X ⊗R R ι v1 = fY (1), v2 = (τ −1 ⊗ 1)fY (τ ) ∈ Y ⊗R R ,

we must express the values w1 = f (1),

w2 = (τ −1 ⊗ τ −1 ⊗ 1)f (τ ) ∈ (X ⊗R Y ) ⊗R R

of the function f (g + ) = fX (g + ) ⊗ (id ⊗ ι)fY (g + )

(g + ∈ G+ )

in terms of ui , vi . As w1 = u1 ⊗ (id ⊗ ι)v1 ,

w2 = u2 ⊗ (id ⊗ ι)v2 ,

the formula for prod which makes the diagram commutative follows. (ii) Clearly uZ ◦ vZ = id, vZ ◦ uZ = id. The remaining statement follows from the commutativity of the following diagram: prod :

(x ⊗ (a1 + a2 τ )) ⊗ (y ⊗ (b1 + b2 τ ))  u ⊗v y X Y

prodι : (x ⊗ (ι(a2 ) + ι(a1 )τ )) ⊗ (y ⊗ (ι(b2 ) + ι(b1 )τ )) 248

7→

(x ⊗ y) ⊗ (a1 ι(b1 ) + a2 ι(b2 )τ ) x id⊗(ι⊗τ ) 

7→

(x ⊗ y) ⊗ (b2 ι(a2 ) + b1 ι(a1 )τ ).

(10.3.5.5) In this section we show that the assumptions 10.3.5.1.1-5 allow us to replace Selmer complexes e • (FΓ (Z)) (Z = X, Y ) by certain generalized Selmer complexes C e • (G+ , FΓ+ (Z)). C f f + + As a first step we define, for each v ∈ S , local conditions ∆v+ (FΓ+ (Z)) (Z = X, Y ) by ( + i+ v + (FΓ (Z))

:

• • Uv++ (FΓ+ (Z)) = Ccont (Gv+ , FΓ+ (Zv++ )) −→ Ccont (Gv+ , FΓ+ (Z)),

(v + ∈ Σ+ )

• • Uv++ (FΓ+ (Z)) = Cur (Gv+ , FΓ+ (Z)) −→ Ccont (Gv+ , FΓ+ (Z)),

(v + ∈ Sf+ − Σ+ )

and the corresponding generalized Selmer complex for G+ by ef• (G+ , FΓ+ (Z)) = C





resS + −(i++ )

M M v f  •  • Cone Ccont (G+ , FΓ+ (Z)) ⊕ Uv++ (FΓ+ (Z))−−−−−−−−→ Ccont (Gv+ , FΓ+ (Z)) [−1] v + ∈Sf+

v + ∈Sf+

(and similarly for FΓ+ (Z)ι ). Lemma 10.3.5.4 together with a variant of the theory from 8.6-8.8 (which applies thanks to the assumptions 10.3.5.1.1-5) give functorial quasi-isomorphisms of complexes of R-modules shf ∼ e• G+ + e• + e• sh+ f : Cf (G , FΓ+ (Z)) ←− Cf (G , IndG (FΓ (Z)))−−→Cf (FΓ+ (Z))

(Z = X, Y ).

• It follows from Lemma 10.3.5.4(i) that the morphisms of complexes π : X ⊗R Y −→ ωR (1) and

ι

F (π)

ϕ(1)

• • π : FΓ (X) ⊗R FΓ (Y ) −−−−→ωR (1) ⊗R R−−→ωR (1)

induce products ϕ(1)

• • π + : FΓ+ (X) ⊗R FΓ+ (Y )ι −→ ωR (1) ⊗R R[Γ+ /Γ]−−→ωR (1) ⊗R R[Γ+ /Γ]

and   • ι • • • ∪π+ : Ccont (Gv+ , FΓ+ (X)) ⊗R Ccont (Gv+ , FΓ+ (Y )) −→ ωR ⊗R R[Γ+ /Γ] [−2]

(v + ∈ Sf+ ),

with respect to which Uv++ (FΓ+ (X)) ⊥π+ ,0 Uv++ (FΓ+ (Y )ι )

(v + ∈ Sf+ ).

Applying the construction from 6.3.1 and using 8.5.3, we obtain cup products (r ∈ R) ι



ef• (G+ , FΓ+ (X)) ⊗ C e• + ∪π+ ,r,0 : C R f (G , FΓ+ (Y )) −→



resS +

M shc f  •  • • • −→ τ≥3 Cone Ccont (G+ , ωR (1) ⊗R R[Γ+ /Γ])−−−−→ Ccont (Gv+ , ωR (1) ⊗R R[Γ+ /Γ]) [−1]−−→ v + ∈Sf+

• • −→ τ≥3 Cc,cont (GK,S , ωR (1)) −→ ωR [−3] •

such that the diagram ι e• (G+ , FΓ+ (X)) ⊗ C e• + C f R f (G , FΓ+ (Y ))  sh+ ⊗sh+ y f f ι

∪π+ ,r,0

−−−−→

• ωR [−3]

k ∪π,r,0

• e• (FΓ (X)) ⊗ C e• C −−−−→ ωR [−3] f R f (FΓ (Y )) is commutative up to homotopy. This implies that the pairings ∪π,0,i,j from 10.3.3.3 can also be defined using ∪π+ ,r,0 instead of ∪π,r,0 .

249

(10.3.5.6) Lemma. The following diagram is commutative up to homotopy: ι e• (G+ , FΓ+ (X)) ⊗ C e• + C f R f (G , FΓ+ (Y ))  (u ) ⊗(v ) y X∗ Y ∗

∪π+ ,r,0

−−−−−−→ )ι

(∪

π + ,r,0 e• (G+ , FΓ+ (X))ι ⊗ C e• + C f R f (G , FΓ+ (Y )) −−−−−−→

• ωR [−3] x ν[−3]  • ι (ωR ) [−3].

Proof. Using the previous discussion, the statement follows from Lemma 10.3.5.4(ii) and Proposition 6.4.2. Note that the action of τ on R[Γ+ /Γ] in the right vertical arrow in 10.3.5.4(ii) corresponds to the homotopy • action of Ad(τ ) on τ≥3 Cc,cont (GK,S , J(1)), which makes the following diagram commutative up to homotopy: • τ≥3 Cc,cont (GK,S , J(1))  Ad(τ ) y • τ≥3 Cc,cont (GK,S , J(1))

Qis

−→

J[−3] k

Qis

−→

J[−3].

(10.3.5.7) Corollary. The pairings ι

i e f,Iw e j (K∞ /K, X) ∪π,0,i,j : H (K∞ /K, X)R−tors ⊗R H −→ H 0 (ωR )⊗R (Frac(R)/R) f,Iw R−tors

satisfy

x ∪π,0,i,j y = ι (uX (x) (∪π,0,i,j )ι vY (y))

(i + j = 4)

(i + j = 4).

Proof. This follows from Lemma 10.3.5.6 and (2.10.14.1). (10.3.5.8) Proposition (Self-dual dihedral case). If, under the assumptions 10.3.5.1.1-5, Y = X, • ∆(Y ) = ∆(X) and π : X ⊗R X −→ ωR (1) satisfies π ◦ s12 = c · π, then the formula

(x, y)i,j := x ∪π,0,i,j uX (y)

c = ±1, (i + j = 4)

defines R-bilinear pairings 0 e i (K∞ /K, X) e j (K∞ /K, X) ( , )i,j : H ⊗ H f,Iw R−tors −→ H (ωR ) ⊗R (Frac(R)/R) R−tors R f,Iw

(i + j = 4)

in (R Mod)/(pseudo − null) (or even in (R Mod), if R is Cohen-Macaulay) satisfying (x, y)i,j = c(−1)ij (y, x)j,i

(i + j = 4).

In particular, ( , )2,2 is a symmetric (resp., skew-symmetric) bilinear form if c = +1 (resp., c = −1). Proof. Combining Corollary 10.3.5.7 with Proposition 10.3.4.2, we obtain (x, y)i,j = x ∪π,0,i,j uX (y) = ι (uX (x) (∪π,0,i,j )ι y) = c(−1)ij y ∪π,0,j,i uX (x) = c(−1)ij (y, x)j,i . (10.3.5.9) It is very likely that the conclusions of Corollary 10.3.5.7 and Proposition 10.3.5.8 still hold if the assumption 10.3.5.1.5 is weakened (it was included above in order to apply 8.7.6). (10.3.5.10) There is an alternative way of constructing R-bilinear generalized Cassels-Tate pairings on e ∗ (K∞ /K, −) H in the dihedral situation of 10.3.5.1. Mimicking the definition of an invariant bilinear f,Iw R−tors form on irreducible two-dimensional representations of a dihedral group, we define a (R-bilinear) pairing λ : FΓ+ (X) ⊗R FΓ+ (Y ) −→ (X ⊗R Y ) ⊗R R 250

by the formula λ(x ⊗ (a1 + τ a2 ) ⊗ y ⊗ (b1 + τ b2 )) = x ⊗ y ⊗ (a1 ι(b2 ) + ι(a2 )b1 ).

(10.3.5.10.1)

It is easy to check that this pairing is, indeed, R-bilinear, symmetric λ ◦ s12 = (s12 ⊗ id) ◦ λ

(10.3.5.10.2)

and G+ -invariant (∀g + ∈ G+ )

(g + ⊗ id) ◦ λ = λ ◦ (g + ⊗ g + ).

If we denote by π 0 the composite pairing λ

ϕ(1)

π⊗id

• • π 0 : FΓ+ (X) ⊗R FΓ+ (Y )−−→(X ⊗R Y ) ⊗R R−−−−→ωR (1) ⊗R R−−→ωR (1),

then we have, for each v + ∈ Σ+ , FΓ+ (Xv++ ) ⊥π0 FΓ+ (Yv++ ). If π is a perfect duality, so is π 0 , by Corollary 8.4.6.5. If, in addition, Xv++ ⊥⊥π Yv++ , then FΓ+ (Xv++ ) ⊥⊥π0 FΓ+ (Yv++ ) (again by Corollary 8.4.6.5). The cup products (r ∈ R) • e • (G+ , FΓ+ (X)) ⊗ C e• + ∪π0 ,r,0 : C f R f (G , FΓ+ (Y )) −→ ωR [−3]

together with the quasi-isomorphisms sh+ f from 10.3.5.5 give rise to R-bilinear pairings e i (K∞ /K, X) e j (K∞ /K, Y ) ∪π0 ,0,i,j : H ×H −→ H 0 (ωR ) ⊗R (Frac(R)/R) f,Iw f,Iw R−tors R−tors

(i + j = 4)

in (R Mod)/(pseudo − null) (or even in (R Mod), if R is Cohen-Macaulay). In the self-dual case, when Y = X, ∆(Y ) = ∆(X) and π ◦ s12 = c · π (c = ±1), the symmetry property (10.3.5.10.2) implies that π 0 ◦ s12 = c · π 0 . Applying Proposition 10.2.5, we deduce that x ∪π0 ,0,i,j y = c(−1)ij y ∪π 0 ,0,j,i x.

(i + j = 4)

In particular, the pairing ∪π0 ,0,2,2 is R-bilinear symmetric (resp., skew-symmetric) if c = +1 (resp., if c = −1). (10.4) Localized Cassels-Tate pairings In this section we drop the earlier assumptions on R (i.e. R can have embedded primes), but we fix a prime ideal p ∈ Spec(R) satisfying dim(Rp ) = depth(Rp ) = 1 and consider only Selmer complexes localized at p. • • • • (10.4.1) We fix ωR = σ≥0 ωR as usual and put ωR = (ωR )p – this is a bounded complex of injective p Rp -modules representing ωRp . Let Y1 , Y2 be bounded complexes of admissible Rp [GK,S ]-modules (in the sense of 3.7.2) with cohomology of finite type over Rp and • π(p) : Y1 ⊗Rp Y2 −→ ωR [n](1) p

(n ∈ Z)

a morphism of complexes of Rp [GK,S ]-modules. Assume that we are also given orthogonal local conditions 251

∆(Y1 ) ⊥π(p),hS (p) ∆(Y2 ) such that US+ (Yi ) (i = 1, 2) have cohomology of finite type over Rp . Under these assumptions, the Selmer e • (Yi ) = C e• (GK,S , Yi ; ∆(Yi )) (i = 1, 2) also have cohomology of finite type over Rp . complexes C f f (10.4.2) Example. localizations

• If π : X1 ⊗R X2 −→ ωR [n](1) and ∆(X1 ) ⊥π,hS ∆(X2 ) are as in 10.2.1, then the

(Yi , ∆(Yi ), π(p), hS (p)) = ((Xi )p , ∆(Xi )p , πp , (hS )p ) satisfy the assumptions of 10.4.1 and e • (Yi ) = C e• (Xi ) C f f p

(i = 1, 2).

(10.4.3) Under the assumptions of 10.4.1, a localized version of the cup products from 6.3.1 e• (Y1 ) ⊗Rp C e• (Y2 ) −→ ω • [n − 3] ∪π(p),r,h : C f f Rp

(r ∈ Rp )

together with the construction in 2.10.7-9 define cup products e i (Y1 ) ∪π(p),hS (p),i,j : H f R

p −tors

∼

e j (Y2 ) ⊗Rp H f R

p −tors

−→ H 0 (ωRp ) ⊗Rp (Frac(Rp )/Rp ) −→ IRp (i + j = 4 − n)

in (Rp Mod) (independent of r ∈ Rp ). If, in Example 10.4.2, R has no embedded primes, then ∪π(p),r,h (for r ∈ R) and ∪π(p),hS (p),i,j are obtained from the corresponding products ∪π,r,h and ∪π,hS ,i,j from 10.2.2 by localizing at p. If π(p) is a perfect duality in the sense that the map • adj(π(p)) : Y1 −→ Hom•Rp (Y2 , ωR [n](1)), p

or, equivalently, • adj(π(p) ◦ s12 ) : Y2 −→ Hom•Rp (Y1 , ωR [n](1)), p

is a quasi-isomorphism (cf. 6.2.6), then the adjoint map • • ef• (Y1 ) −→ HomR ef• (Y2 ), ωR γπ(p),r,hS (p) = adj(∪π(p),r,h ) : C (C [n − 3]) p p

appears in the following exact triangle in Df t (Rp Mod):   γπ(p),r,hS (p) gf (Y1 )−−−−−−→D g g RΓ Rp RΓf (Y2 ) [n − 3] −→ Err(∆(Y1 ), ∆(Y2 ), π(p)) −→ RΓf (Y1 )[1]. Again, in the situation of 10.4.2, we have γπ(p),r,hS (p) = (γπ,r,hS )p (if r ∈ R) and Err(∆(Y1 ), ∆(Y2 ), π(p)) = Err(∆(X1 ), ∆(X2 ), π)p . Lemma 2.10.11 gives isomorphisms of Rp -modules e j (Y2 ) DRp (H f R

∼

∼

e j (Y2 ) ) −→ HomRp (H , H 0 (ωRp ) ⊗Rp (Frac(Rp )/Rp )) −→ f Rp −tors   ∼ ∼ i e j (Y2 ), ωRp ) −→ g −→ Ext1Rp (H H D ( RΓ (Y ))[n − 3] , (i + j = 4 − n) R f 2 p f p −tors

Rp −tors

the composition of which with 252

e fi (Y1 ) adj(∪π(p),hS (p),i,j ) : H R

p −tors

e j (Y2 ) −→ DRp (H f R

p −tors

)

(i + j = 4 − n)

coincides, up to a sign, with the restriction of the map   gf (Y2 ))[n − 3] . e fi (Y1 ) −→ H i DRp (RΓ (γπ(p),r,hS (p) )∗ : H (10.4.4) Theorem. Under the assumptions of 10.4.1, let i + j = 4 − n and Err = Err(∆(Y1 ), ∆(Y2 ), π(p)). Assume that π(p) is a perfect duality. Then, in the category (Rp Mod), (i) Ker(adj(∪π(p),hS (p),i,j )) is isomorphic to a subquotient of H i−1 (Err). (ii) If H i−1 (Err) is Rp -torsion, then Coker(adj(∪π(p),hS (p),i,j )) (resp., Ker(adj(∪π(p),hS (p),i,j ))) is isomorphic to a submodule (resp., a quotient) of H i (Err) (resp., of H i−1 (Err)). ∼ In particular, if H i−1 (Err) = H i (Err) −→ 0, then adj(∪π,hS ,i,j ) is an isomorphism of Rp -modules. Proof. This follows from Proposition 2.10.12 applied to ∪π(p),r,h . (10.4.5) Under the assumptions of 10.4.1, the pairings ∪π(p),hS (p),i,j satisfy the obvious analogues of 10.2.47, with the same proofs. (10.5) Greenberg’s local conditions - localized version • • Let R, p, ωR and ωR be as in 10.4. p (10.5.1) A special case of the data from 10.4.1 is the following: Y1 = X(p), Y2 = Y (p) and π(p) : • X(p) ⊗Rp Y (p) −→ ωR (1) are as in 10.4.1, Sf = Σ ∪ Σ0 are as in 7.8, and for each Z = X(p), Y (p) and p v ∈ Σ we are given a bounded complex of admissible Rp [Gv ]-modules Zv+ with cohomology of finite type over Rp and a morphism of complexes of Rp [Gv ]-modules jv+ (Z) : Zv+ −→ Z

(v ∈ Σ);

we assume that + X(p)+ v ⊥π(p) Y (p)v

(v ∈ Σ).

These data define the usual local conditions (Z = X(p), Y (p)) ( • • Ccont (Gv , Zv+ ) −→ Ccont (Gv , Z), ∆v (Z) = • • Cur (Gv , Z) −→ Ccont (Gv , Z),

(v ∈ Σ) (v ∈ Σ0 )

satisfying ∆(X(p)) ⊥π(p),0 ∆(Y (p)), which admit transposition operators satisfying a localized version of 6.5.3.1-5. If π(p) is a perfect duality, then the error terms Errv = Errv (∆v (X(p)), ∆v (Y (p)), π(p)) in Df t (Rp Mod) are as follows: For v ∈ Σ, ∼

Errv −→ RΓcont (Gv , Wv ), where Wv sits in an exact triangle + Wv −→ X(p)− v −→ DRp (Y (p)v ) −→ Wv [1]

in D(ad Rp [Gv ] Mod), by a localized version of Proposition 6.7.6(iv). For v ∈ Σ0 , 253

h i fv −1 ∼ 0 1 0 1 Errv −→ H{p} (Hcont (Iv , X(p)))−−→H{p} (Hcont (Iv , X(p))) , where the complex on the R.H.S. is in degrees 1, 2, and ( Tamv (X(p), p),
i `Rp H (Errv ) = 0,

i = 1, 2 i 6= 1, 2,

by a localized version of 7.6.10.7. • (10.5.2) Example. Let π : X ⊗R Y −→ ωR (1) be as in 10.3.1; then the localized data + + + (X(p), Y (p), π(p), X(p)+ v , Y (p)v ) = (Xp , Yp , πp , (Xv )p , (Yv )p )

are as in 10.5.1 and Errv = Errv (∆(X), ∆(Y ), π)p

(v ∈ Sf ).

(10.5.3) Under the assumptions of 10.5.1, we obtain from 10.4 bilinear forms in (Rp Mod) e fi (X(p)) ∪π(p),0,i,j : H R

p −tors

e j (Y (p)) ⊗ Rp H f R

p −tors

−→ IRp

(i + j = 4)

satisfying the formula (10.3.2.1). The kernels and cokernels of the adjoint maps   e i (X(p)) e j (Y (p)) adj(∪π(p),0,i,j ) : H −→ DRp H f f R −tors R −tors p

p

(i + j = 4)

are as in Theorem 10.4.4. (10.5.4) Iwasawa theory Let Γ = Gal(K∞ /K) and R = R[[Γ]] be as in 10.3.3. Fix ϕ and ν as in 10.3.3.1. Let Rp ∈ Spec(R) be a prime ideal satisfying dim(Rp ) = depth(Rp ) = 1. One can obtain data of the type 10.5.1 over R as follows. (10.5.4.1) Assume that p ∈ Spec(R), p ⊃ R ∩ p and we are given the data from 10.5.1 (except that we do i not assume that dim(Rp ) = depth(Rp ) = 1) such that all Z(p)i , (Z(p)+ v ) (Z = X, Y , v ∈ Σ, i ∈ Z) are of finite type over Rp . Then a localized version of the recipe in 8.9.2 defines Greenberg’s local conditions for

FΓ (Z(p))p = Z(p) ⊗R R p < −1 >= Z(p) ⊗Rp Rp < −1 >     ι ι FΓ (Z(p))ιp = Z(p) ⊗R R < −1 >= Z(p) ⊗Rp Rp < −1 >, p

together with a pairing ι

• π(p) : FΓ (X(p))p ⊗R FΓ (Y (p))p −→ ωR (1), p

p

under which 

FΓ (X(p))p

+ v

 + ι ⊥π(p) FΓ (Y (p))p v

(v ∈ Σ).

Above, (−)ιp is a shorthand for ((−)ι )p . (10.5.4.2) Lemma. In the situation of 10.5.4.1, assume that the localization of π(p) at the prime ideal • q = R ∩ p ∈ Spec(R) is a perfect pairing π(p)q : X(p)q ⊗Rq Y (p)q −→ ωR (1) over Rq and (X(p)+ v )q ⊥⊥π(p)q q + (Y (p)v )q for all v ∈ Σ. Then π(p) is a perfect pairing and 

FΓ (X(p))p

+ v

 + ι ⊥⊥π(p) FΓ (Y (p))p v

Proof. This follows from the flatness of Rp over Rq and the fact that 254

(∀v ∈ Σ).

FΓ Z(p)+ v

FΓ (Z(p))p < 1 >= (Z(p)q ) ⊗Rq Rp ,


p


< 1 >= (Z(p)+ v )q ⊗Rq Rp

(Z = X, Y ).

• (10.5.4.3) Alternatively, given π : X ⊗R Y −→ ωR (1) and Zv+ (Z = X, Y , v ∈ Σ) as in 10.3.1, we can ι localize FΓ (X), FΓ (Y ) and π from 10.3.3.2 at p.

(10.5.5) Under the assumptions of 10.5.4.1 (resp., 10.5.4.3), we can apply 10.5.3 and obtain bilinear forms in (R Mod) (for i + j = 4) p

  e i (K∞ /K, X(p)) ∪π(p),0,i,j : H f,Iw p

(Rp )−tors

⊗R

p

  e j (K∞ /K, Y (p))ι H f,Iw p

(Rp )−tors

−→ IR

p

(10.5.5.1)

resp.,   e i (K∞ /K, X) ∪πp ,0,i,j : H f,Iw p

(Rp )−tors

⊗R



p

e j (K∞ /K, Y )ι H f,Iw p

 (Rp )−tors

−→ IR . p

(10.5.5.2)

All results in 10.3.3-5 hold (with the same proofs) for the pairings (10.5.5.1-2). If R has no embedded primes (and if the data 10.5.4.1 are of the kind considered in 10.5.2), then the pairings (10.5.5.1-2) are obtained from those in 10.3.3.3 by localizing at p. (10.6) Discrete valuation rings with involution (10.6.1) An involution on a (commutative) ring O is a ring isomorphism ι : O −→ O satisfying ι ◦ ι = id. A typical example is the standard involution on R = R[[Γ]]. (10.6.1.1) If O is a discrete valuation ring with prime element $ ∈ O, then each involution ι : O −→ O maps the maximal ideal $O to itself, hence induces an involution ιk on the residue field k = O/$O and ι($)/$ ∈ O∗ is a unit; denote by ε ∈ k ∗ its image in k ∗ , which satisfies ε · ιk (ε) = 1. If we replace $ by $0 = u$ (u ∈ O∗ ), then ε is replaced by ε0 = (ιk (u)/u)ε, where u ∈ k ∗ denotes the image of u in k ∗ . (10.6.1.2) If ιk = id, then ε = ±1, hence ι($) ≡ ±$ (mod $2 ). (10.6.1.3) Lemma. If ιk = id, ε = 1 and 2 ∈ O∗ , then ι = id. Proof. Assume that ι(u) 6= u for some u ∈ O∗ ; then ι(u) = u + v$n , where v ∈ O∗ and n ≥ 1. This implies that 0 = v$n + ι(v)ι($)n ≡ 2v$n (mod $n+1 ), which is impossible. Thus ι acts trivially on O∗ and also on $, as ι($) = ι(1 + $) − ι(1) = $. (10.6.1.4) If ιk 6= id, then k is a quadratic Galois extension of k+ = k ιk =1 , with Gal(k/k+ ) = {id, ιk }. Hilbert’s Theorem 90 implies that there is u ∈ k ∗ satisfying ε = u/ιk (u); replacing $ by $0 = u$ (where u ∈ O∗ is any lift of u ∈ k ∗ ), we have ι($0 ) ≡ $0 (mod $02 ). (10.6.2) A typical example of a discrete valuation ring with involution is the localization O = Rp of R = R[[Γ]] at a regular prime ideal p ∈ Spec(R) satisfying ht(p) = 1 and ι(p) = p. ∼

(10.6.3) In particular, assume that R is an integral domain, flat over Zp , and that Γ −→ Zp (γ 7→ 1). If f = (γ − 1)n + a1 (γ − 1)n−1 + · · · + an ∈ R[γ − 1] 255

(a1 , . . . , an ∈ m)

∼

is an irreducible distinguished polynomial of degree n ≥ 1, then p = (f ) is a prime ideal p ⊂ R = R[[Γ]] −→ R[[γ − 1]] satisfying ht(p) = 1 and p ∩ R = (0). The localization O = Rp = R[[γ − 1]](f ) is a discrete valuation ring with prime element f . Factorizing f into linear factors f = (γ − α1 ) · · · (γ − αn ) over a suitable finite extension of Frac(R), we have −1 p = ι(p) =⇒ Z(f ) := {α1 , . . . , αn } = {α−1 1 , . . . , αn }.

There are three mutually exclusive cases: (10.6.3.1) 1 ∈ Z(f ) =⇒ p = (γ − 1), k = Frac(R), ιk = id, ε = −1. (10.6.3.2) −1 ∈ Z(f ) =⇒ p = (γ + 1), p = 2, k = Frac(R), ιk = id, ε = −1 = 1. (10.6.3.3) {±1} ∩ Z(f ) = ∅ =⇒ γ 6≡ γ −1 (mod f O) =⇒ ιk 6= id. (10.6.4) Lemma. Let O be a discrete valuation ring with involution ι. Assume that either 2 ∈ O∗ , or that O is complete. Then there exist a prime element $ ∈ O and ε = ±1 satisfying ι($) = ε$. Proof. According to 10.6.1.2 and 10.6.1.4 there exist ε = ±1 and a prime element $0 ∈ O satisfying ι($0 ) ≡ ε$0 (mod $02 ). If 2 ∈ O∗ , then the elements x±ε := $0 ± ει($0 ) satisfy ι(x±ε ) = ±εx±ε , and at least one of them is a prime element of O. If O is complete, it is enough to construct a sequence of prime elements $1 , $2 , . . . satisfying $n ≡ $n−1 (mod $0n+1 ),

ι($n ) ≡ ε$n (mod $0n+2 )

(n ≥ 1),

as $ = lim $n will have the desired property. Given $n (n ≥ 0), put u = (ι($n ) − ε$n )/$nn+2 ∈ O. If u 6∈ O∗ , let $n+1 = $n . If u ∈ O∗ , denote by u ∈ k ∗ its image in k ∗ . The exactness of the sequence id−ι

id+ι

id−ι

id+ι

k k k k · · · −−−−→k −−−−→k −−−−→k −−−−→ ···

implies that there exists a ∈ k satisfying (εn−1 ιk − id)a = −εu. The element $n+1 = $n + a$nn+2 (where a ∈ O is any lift of a) has the desired properties, as $n+1 ≡ $n (mod $0n+2 ) and ι($n+1 ) − ε$n+1 = (u − εa)$nn+2 + ι(a)(ε$n + u$nn+2 )n+2 ≡ (u + εn ι(a) − εa)$nn+2 ≡ 0 (mod $0n+3 ). (10.6.5) Lemma. Let O be a discrete valuation ring with fraction field F , $ ∈ O a prime element and k = O/$O the residue field. Assume that ε ∈ {±1} and ι : O −→ O is an involution satisfying ι($) ≡ ε$ (mod $2 ). Let N be an O-module of co-finite type and h , i : N × N −→ F/O a skew-hermitian form, i.e. a bi-additive map satisfying hλx, µyi = λι(µ)hx, yi,

hy, xi = −ι(hx, yi)

(∀λ, µ ∈ O, ∀x, y ∈ N ).

Assume that the kernel of h , i is equal to Ndiv (= the maximal O-divisible submodule of N ). Then: (i) The k-vector space F 1 := N [$] has a canonical decreasing filtration F1 ⊇ F2 ⊇ ··· by the k-subspaces F j = N [$] ∩ $j−1 N . T ∞ j (ii) F := j≥1 F = N [$] ∩ Ndiv . (iii) The formula

j−1 
$ x, $j−1 y j,$ := $j hx, yi (mod $−j+1 ) ∈ O/$O = k 256

(x, y ∈ N [$j ])

defines an (−εj )-hermitian form

h , ij,$ : F j ⊗k F j −→ k,

i.e. a bi-additive map satisfying hλx, µyij,$ = λιk (µ)hx, yij,$ ,

hy, xij,$ = −εj ιk (hx, yij,$ )

(∀λ, µ ∈ k, ∀x, y ∈ F j ),

with kernel equal to F j+1 , hence a non-degenerate (−εj )-hermitian form h , ij,$ : grFj × grFj −→ k on grFj = F j /F j+1 . (iv) If we replace $ ∈ O by another prime element $0 satisfying ι($0 ) ≡ ε$ 0 (mod $02 ), then h , ij,$0 = u2−j h , ij,$ , where u ∈ k ∗ is the image of $0 /$ ∈ O∗ in k ∗ (it satisfies ιk (u) = u). (v) Assume that ιk = id and εj = 1. If 2 ∈ O∗ (more generally, if the pairing h , i is alternating, i.e. if hx, xi = 0 for all x ∈ N ), then h , ij,$ is a symplectic (= alternating and non-degenerate) form on grFj , hence dimk (F j ) ≡ dimk (F j+1 ) (mod 2). (vi) Assume that ι = id. If 2 ∈ O∗ (more generally, if the pairing h , i is alternating), then (∀j ≥ 1) In particular,

dimk (F j ) ≡ dimk (F ∞ ) (mod 2).

dimk (N [$])(= dimk (F 1 )) ≡ corkO (N )(= dimk (F ∞ )) (mod 2).

Proof. Elementary linear algebra. (10.6.6) Lemma (Dihedral case). In the situation of 10.6.5, assume that we are given a bijective additive map τ : N −→ N satisfying τ ◦ τ = id and τ (λx) = ι(λ)τ (x),

hτ x, τ yi = ι(hx, yi)

(∀λ ∈ O, ∀x, y ∈ N ).

Then: (i) The formula (x, y) = hx, τ yi defines an O-bilinear skew-symmetric pairing ( , ) : N × N −→ F/O with kernel Ndiv . (ii) The filtration F j on F 1 = N [$] is τ -stable and the pairings h , ij,$ satisfy hτ x, τ yij,$ = εj ιk (hx, yij,$ )

(x, y ∈ grFj ).

(iii) If 2 ∈ O∗ (more generally, if the pairing ( , ) is alternating), then (∀j ≥ 1) In particular,

dimk (F j ) ≡ dimk (F ∞ ) (mod 2).

dimk (N [$])(= dimk (F 1 )) ≡ corkO (N )(= dimk (F ∞ )) (mod 2).

Proof. (i) and (ii) follow from the definition of ( , ), while (iii) is a consequence of Lemma 10.6.5 applied to the pairing ( , ) and the trivial involution id. (10.6.7) The conclusion of Lemma 10.6.6(iii) can be proved without introducing the pairing ( , ), at least in the following special case. 257

(10.6.8) Lemma. In the situation of 10.6.6, assume that ε = −1, ιk = id and 2 ∈ O∗ . For η, η 0 = ± denote 0 j η j η0 j ± j τ =±1 by h , iηη . Let j ≥ 1; then: j,$ the restriction of h , ij,$ to (grF ) × (grF ) , where (grF ) = (grF ) ++ −− (i) If 2 - j, then h , ij,$ = h , ij,$ = 0 and j + j − h , i+− j,$ : (grF ) × (grF ) −→ k

is a non-degenerate k-bilinear form. In particular, dimk (grFj )+ = dimk (grFj )− ,

dimk (grFj ) = 2 dimk (grFj )+ ≡ 0 (mod 2).

−+ ++ −− (ii) If 2|j, then h , i+− j,$ = h , ij,$ = 0 and h , ij,$ (resp., h , ij,$ ) is a symplectic (= alternating and non-degenerate) form on (grFj )+ (resp., on (grFj )− ). In particular,

dimk (grFj )± ≡ 0 (mod 2), (iii) We have

dimk (F j ) ≡ dimk (F ∞ ) (mod 2).

(∀j ≥ 1) In particular,

dimk (grFj ) ≡ 0 (mod 2).

dimk (N [$])(= dimk (F 1 )) ≡ corkO (N )(= dimk (F ∞ )) (mod 2).

Proof. (i), (ii) As (x, y ∈ grFj )

hτ x, τ yij,$ = εj hx, yij,$ = (−1)j hx, yij,$ 0

0 j and −1 6= 1 in k, it follows that h , iηη j,$ = 0 if η 6= (−1) η. This implies, by non-degeneracy of h , ij,$ , that 0

0 j the pairings h , iηη j,$ are non-degenerate if η = (−1) η. Finally, if 2|j, then the pairing h , ij,$ (hence each ηη h , ij,$ ) is skew-symmetric by Lemma 10.6.5(iii), hence alternating (as 2 6= 0 in k). The statement (iii) is an immediate consequence of (i) and (ii).

(10.7) Skew-hermitian and symplectic pairings on generalized Selmer groups (10.7.1) In this section we investigate skew-hermitian and skew-symmetric pairings arising from the constructions in 10.1-5. In self-dual situations, the localizations of generalized Cassels-Tate pairings at regular prime ideals of height one give rise to pairings of the type considered in Lemma 10.6.5, from which one can deduce various parity results. As we shall see in 11.8, in the special case 10.6.3.1 the corresponding pairings on the graded pieces grFj can be identified with (higher) height pairings. (10.7.2) Symplectic pairings revisited. In the situation of 10.6.5, the O-module N is isomorphic to M ∼ N −→ (F/O)s ⊕ (O/$ i )ni , i≥1

with ni ≥ 0 and only finitely many ni non-zero, ∼

Ndiv −→ (F/O)s ,

s = corkO (N )

and dimk (N [$]) = s +

X

ni

i≥1

dimk (F j ) = s +

X

ni

(j ≥ 1).

i≥j

Assume that ι = id and that the pairing h , i is alternating (which is automatic if 2 ∈ O∗ ). The existence of the symplectic forms on the graded quotients grFj implies that 258

ni ≡ 0 (mod 2)

(∀i ≥ 1).

In particular, if h , i is non-degenerate, then s = 0 and ∼

N −→ M ⊕ M,

M=

M

(O/$ i )ni /2 .

i≥1

Note that M can be chosen to be a Lagrangian (= maximal isotropic) submodule of N . If the form h , i is merely skew-symmetric (i.e. hx, yi = −hy, xi for all x, y ∈ N ) and non-degenerate, then 2h , i is alternating, but possibly degenerate, with kernel N 0 ⊂ N annihilated by 2. By the previous remark, we have ∼

N/N 0 −→ M 0 ⊕ M 0 . Denoting the image of the map M 0 ,→ N/N 0 −→ N n + N 0 7→ 2n by M ⊂ N , then M ⊕ M ⊂ N,

2 · (N/(M ⊕ M )) = 0.

(10.7.3) The abstract linear algebra construction from Lemma 10.6.5 appears in several contexts, e.g. as the “Jantzen filtration” in representation theory [Jan] (as pointed out to us by L. Clozel). For us, the most important example comes from the classical descent theory on elliptic curves. Let E be an elliptic curve over a number field K and S a finite set of primes of K, which contains all primes above p, all archimedean primes and all primes of bad reduction of E. The classical Selmer group for the pn -descent on E (over K) is defined by the following cartesian diagram:

L

Sel(E/K, pn)   y v∈S

E(Kv ) ⊗ Z/pn Z

,→

,→

H 1 (GK,S , E[pn ])   y L 1 n v∈S H (Gv , E[p ]).

Passing to the limit, n Sel(E/K, p∞) = lim −→ Sel(E/K, p ) n

is a Zp -module of co-finite type sitting in an exact sequence 0 −→ E(K) ⊗ Qp /Zp −→ Sel(E/K, p∞ ) −→ X(E/K)[p∞ ] −→ 0, ˇ where X(E/K) is the Tate-Safareviˇ c group of E over K. The image of the canonical map g Im [Sel(E/K, pn ) −→ Sel(E/K, p∞)] =: Sel(E/K, pn ) coincides with Sel(E/K, p∞ )[pn ] and sits in an exact sequence g 0 −→ (E(K)/tors) ⊗ Z/pn Z −→ Sel(E/K, pn ) −→ X(E/K)[pn ] −→ 0. Putting r = rkZ (E(K)),

sn = max {j ≥ 0 | (Z/pn Z)j ⊆ g Sel(E/K, pn )}

(for each n ≥ 1), then 259

s1 ≥ s2 ≥ · · · ≥ sn0 = sn0 +1 = · · · = s∞ ≥ r, M ∼ Sel(E/K, p∞) −→ (Qp /Zp )s∞ ⊕ (Z/pi Z)si −si+1 , i≥1

s∞ − r = corkZp (X(E/K)[p∞ ]) (as X(E/K) is conjecturally finite, it is expected that s∞ = r). The integer sn (for fixed p) is classically called “the number of n-th descents”. It was first observed by Selmer that, in all available examples, the integers si − si+1 (as well as si − r) always turned out to be even. Selmer’s observation concerning si − si+1 was conceptually explained by Cassels [Ca 2], who constructed a non-degenerate alternating pairing (subsequently generalized by Tate) h , i : Sel(E/K, p∞) × Sel(E/K, p∞ ) −→ Qp /Zp with kernel Sel(E/K, p∞)div . The filtration F j on F 1 = Sel(E/K, p∞ )[p] = g Sel(E/K, p) (constructed as in Lemma 10.6.5, with O = Zp , ι = id and $ = p) then coincides with h i F j = Im g Sel(E/K, pj ) −→ g Sel(E/K, p) , (10.7.3.1) i.e. sj = dimZ/pZ (F j ) is equal to the number of first descents that can be “extended” to j-th descents. The fact that F 2 (defined by (10.7.3.1)) is the kernel of a suitable alternating form on F 1 was first established in a special case in [Ca 1]; this was the starting point of the general construction of h , i, given in [Ca 2]. (10.7.4) Lemma. Let N be an R-module of finite length and h , i : N × N −→ IR a symplectic pairing (i.e. hx, xi = 0 for all x ∈ N and the adjoint map j = adj(h , i) : N −→ D(N ) is an isomorphism). Then: (i) For each Lagrangian (maximal isotropic) submodule M ⊂ N , the map j induces an isomorphism ∼

N/M −→ D(M ). (ii) `R (N ) ≡ 0 (mod 2). Proof. (i) Lagrangian submodules exist (since N is noetherian); let i : M ,→ N be one of them. As M is isotropic, the (surjective) map j

D(i)

N −−→D(N )−−→D(M ) factors through j 0 : N/M −→ D(M ). If x + M ∈ Ker(j 0 ) and x 6∈ M , then M + Rx ) M is also isotropic - contradiction. Thus j 0 is injective. (ii) `R (N ) = `R (M ) + `R (D(M )) = 2 `R (M ). (10.7.5) Abstract self-dual case. Pairings of the kind considered in 10.6.5 and 10.7.4 naturally arise in the following context. 260

Consider the following assumptions: (10.7.5.1) p ∈ Spec(R) satisfies dim(Rp ) = depth(Rp ) = 1. • (10.7.5.2) In 10.4.1, we have Yi = Y , ∆(Yi ) = ∆(Y ) (i = 1, 2), π(p) : Y ⊗Rp Y −→ ωR (1) satisfies π(p) ◦ s12 = p −π(p) and the local conditions ∆(Y ) ⊥π(p),hS (p) ∆(Y ) admit (a localized version of) the transposition data as in 6.6.2. (10.7.5.3) H i (Err(∆(Y ), ∆(Y ), π(p))) = 0 for i = 1, 2. (10.7.5.4) 2 is invertible in Rp . (10.7.5.5) Rp is a discrete valuation ring. (10.7.6) Proposition. (i) Under the assumptions 10.7.5.1-3, the pairing h , i = ∪π(p),hS (p),2,2 on N = e 2 (Y ) H is a non-degenerate skew-symmetric Rp -bilinear form f R −tors p

h , i : N × N −→ IRp , i.e. hx, yi = −hy, xi for all x, y ∈ N and the adjoint map adj(h , i) : N −→ DRp (N ) is an isomorphism. (ii) Under the assumptions 10.7.5.1-4, h , i is a symplectic pairing as in Lemma 10.7.4 (over Rp ) and `Rp (N ) ≡ 0 (mod 2). (iii) Under the assumptions 10.7.5.2-5, h , i is a symplectic pairing as in Lemma 10.6.5 (over O = Rp ) with Ndiv = 0 and ∼ N −→ M ⊕ M for a suitable Lagrangian submodule M ⊂ N . (iv) Under the assumptions 10.7.5.n with n = 2, 3, 5, there is an Rp -submodule M ⊂ N such that M ⊕ M ⊂ N,

2 · (N/(M ⊕ M )) = 0.

Proof. (i), (ii) A localized version of Proposition 10.2.5 implies that the pairing h , i is skew-symmetric (hence alternating, under the assumption 10.7.5.4). It is also non-degenerate, by Theorem 10.4.4 and the assumption 10.7.5.3. The remaining statements follow by applying Lemma 10.6.5, 10.7.4 and the discussion in 10.7.2. (10.7.7) Self-dual Greenberg’s local conditions. Consider the following assumptions: (10.7.7.1) p ∈ Spec(R) satisfies dim(Rp ) = depth(Rp ) = 1. • + (10.7.7.2) In 10.5.1, we have X(p) = Y (p), X(p)+ v = Y (p)v (v ∈ Σ), π(p) : X(p) ⊗Rp X(p) −→ ωRp (1) satisfies + + π(p) ◦ s12 = −π(p) and X(p)v ⊥π(p) X(p)v (v ∈ Σ). i (10.7.7.3) Hcont (Gv , Wv ) = 0 (i = 1, 2; v ∈ Σ), Tamv (X(p), p) = 0 (v ∈ Σ0 ). (10.7.7.4) 2 is invertible in Rp . (10.7.7.5) Rp is a discrete valuation ring. i (10.7.7.6) In each degree i ∈ Z, the Rp -modules X(p)i , (X(p)+ v ) (v ∈ Σ) are torsion-free. (10.7.8) Under the assumptions 10.7.7.2-6, fix a prime element $ ∈ Rp . Then X[p] := X(p) ⊗Rp Rp /$ ad is a complex in (ad R[GK,S ] Mod), which can also be viewed as a complex in ((R/p)[GK,S ] Mod). The complex of • continous cochains Ccont (GK,S , X[p] ) is the same in both cases, by Proposition 3.5.10. The same is true for

X[p]


+ v

:= X(p)+ v ⊗Rp Rp /$

as complexes of Gv -modules (v ∈ Σ). The maps jv+ (X[p] ) : X[p]


+ v

−→ X[p] 261

(v ∈ Σ),

e • (X[p] ) and the induced by jv+ (X(p)), define Greenberg’s local conditions for X[p] , the Selmer complex C f gf (X[p] ) ∈ D(κ(p) Mod), where κ(p) = Rp /$ = Frac(R/p) is the residue field of p. corresponding object RΓ By Proposition 7.6.7(iv), the assumptions 10.7.7.3 and 10.7.7.6 give an exact triangle $ g gf (X(p))−→ gf (X[p] ) −→ RΓ gf (X(p))[1] RΓ RΓf (X(p)) −→ RΓ

in D(Rp Mod). The corresponding cohomology sequence yields short exact sequences e i (X(p)) ⊗Rp Rp /$ −→ H e i (X[p] ) −→ H e i+1 (X(p))[$] −→ 0, 0 −→ H f f f

(10.7.8.1)

gf (X[p] ) ∈ D (κ(p) Mod). which imply that RΓ ft (10.7.9) Proposition. Under the assumptions 10.7.7.2-6, e 1 (X[p] ) has a canonical decreasing filtration by subspaces F 1 ⊇ F 2 ⊇ · · · (i) The κ(p)-vector space F 1 := H f satisfying \ e f1 (X(p)) ⊗Rp Rp /$. F ∞ := Fj = H j≥1

(ii) There are natural symplectic pairings h , ij,$ : grFj × grFj −→ κ(p) depending on $, with 

2−j

h , ij,$0 = ($0 /$ (mod $))

 e 1 (X[p] ) ≡ dimκ(p) (F ∞ ) (mod 2). H f

h , ij,$ .

(iii) dimκ(p) e 0 (X[p] ) = 0, then (iv) If H f

    e f1 (X[p] ) ≡ rkRp H e f1 (X(p)) (mod 2). dimκ(p) H Proof. By Proposition 10.7.6(iii), h , i = ∪π(p),0,2,2 is a non-degenerate alternating pairing h , i : N × N −→ Frac(Rp )/Rp on   e 2 (X(p)) N= H f

Rp −tors

.

Applying Lemma 10.6.5 to this pairing, we obtain a filtration F j on N [$] and symplectic pairings on its graded quotients. Taking the pull-back of these objects by the canonical map e 1 (X[p] ) −→ H e 2 (X(p))[$] = N [$] H f f e 0 (X[p] ) = 0 implies from (10.7.8.1), we obtain the statements (i)-(iii). As regards (iv), the assumption H f 1 e (X(p)) is torsion-free, hence free, over Rp ; thus (again by (10.7.8.1)) that H f   e f1 (X(p)) . dimκ(p) (F ∞ ) = rkRp H (10.7.10) Note that p contains a unique minimal prime ideal q ( p, and     e f1 (X(p)) = dimFrac(R ) H e f1 (X(p) ⊗Rp Frac(Rp )) rkRp H p depends only on 262

X(q) := X(p) ⊗Rp Frac(Rp ) = X(p)q (a complex of admissible Rq [GK,S ]-modules) and + X(q)+ v := X(p)v




(v ∈ Σ).

q

(10.7.11) Self-dual dihedral case in Iwasawa theory. Consider the following assumptions: (10.7.11.1) We are given K + ⊂ K ⊂ K∞ as in 10.3.5.1. • (10.7.11.2) In 10.3.3.2, we have X = Y , Xv+ = Yv+ (v ∈ Σ), π : X ⊗R X −→ ωR (1) satisfies π ◦ s12 = −π, Xv+ ⊥π Xv+ (∀v ∈ Σ) and the conditions 10.3.5.1.1-5 are satisfied for X, Xv+ (v ∈ Σ). (10.7.11.3) p ∈ Spec(R) is a prime ideal satisfying dim(Rp ) = depth(Rp ) = 1; put q0 = R[∆] ∩ p ∈ Spec(R[∆]) and q = q0 ∩ R ∈ Spec(R). (10.7.11.4) If q is not a minimal prime ideal of R (=⇒ ht(q) = 1), then Tamv (F∆ (X), q0 ) = 0 for all v ∈ Σ0 satisfying Γv ⊂ ∆. (10.7.11.5) 2 is invertible in Rp . (10.7.11.6) Rp is a discrete valuation ring. • (10.7.11.7) The localization of π at q is a perfect pairing πq : Xq ⊗Rq Xq −→ ωR (1) over Rq and (Xv+ )q ⊥⊥πq (Xv+ )q q for all v ∈ Σ. • (10.7.11.8) The localization of π at q is a perfect pairing πq : Xq ⊗Rq Xq −→ ωR (1) over Rq . q ∼

(10.7.11.9) (∀v ∈ Σ) RΓcont (Gv , F∆ (Wv ))q0 −→ 0 in Dfb t (R[∆]q0 Mod), where Wv sits in an exact triangle in Db (ad Rq [Gv ] Mod)
Wv −→ (Xv− )q −→ DRq (Xv+ )q −→ Wv [1]. ∼

(10.7.11.10) Γ = Γ00 × Γ000 × ∆, Γ00 = hγi −→ Zp , p = q0 R + (γ − 1)R, q0 ∈ Spec(R[∆]), ht(q0 ) = 0, q = q0 ∩ R ∈ Spec(R). (10.7.12) Proposition. (i) Under the assumptions 10.7.11.1-5,7 (resp., 10.7.11.1-7), there is a symplectic pairing over Rp h , i : N × N −→ IR (resp., N × N −→ Frac(Rp )/Rp ) p

on



 2 e f,Iw N= H (K∞ /K, X)

(Rp )−tors

  e f1 (KS /K∞ , D(X)(1))ι ))p = (D(H

(Rp )−tors

as in Lemma 10.7.4 (resp., in Lemma 10.6.5, with O = Rp and Ndiv = 0) and `R (N ) ≡ 0 (mod 2) (resp., p

there is a submodule M ⊂ N satisfying M ⊕ M = N ). (ii) Under the assumptions 10.7.11.n for n = 1, 2, 4, 6, 7 (resp., n = 1, 2, 8, 9, 10 (=⇒ n = 6)), there is a non-degenerate skew-symmetric pairing h , i : N × N −→ Frac(Rp )/Rp and a submodule M ⊂ N satisfying M ⊕ M ⊂ N,

2 · (N/(M ⊕ M )) = 0.

Proof. A localized version of Proposition 10.3.5.8 (and a choice of τ, τ as in 10.3.5.2) defines a skew-symmetric pairing h , i = ( , )2,2 : N × N −→ IR , p

which is non-degenerate by Theorem 10.4.4, Theorem 8.9.8 and the relevant assumptions 10.7.11.n (using Lemma 10.5.4.2). The rest follows from Lemma 10.6.5, 10.7.4 and the discussion in 10.7.2. (10.7.13) Consider the following global version of the assumptions from 10.7.11: (10.7.13.1) = (10.7.11.1) 263

(10.7.13.2) (10.7.13.3) (10.7.13.4) (10.7.13.5) (10.7.13.6) (10.7.13.7) (10.7.13.8)

= (10.7.11.2) R (hence also R) has no embedded primes. Γv is infinite for each v ∈ Σ0 . 2 is invertible in R. R (hence also R) satisfies (R1 ). The condition (10.7.11.7) is satisfied for all prime ideals q ∈ Spec(R) with ht(q) = 0. The condition (10.7.11.7) is satisfied for all prime ideals q ∈ Spec(R) with ht(q) = 1.

(10.7.14) Proposition. Under the assumptions 10.7.13.1-8 (resp., 10.7.13.n for n = 1, 2, 3, 6, 7), the Rmodule of finite type     e 2 (K∞ /K, X) e 1 (KS /K∞ , D(X)(1))ι )) N= H = (D(H f,Iw

f

R−tors

R−tors

has a subobject M ,→ N in (R Mod)/(pseudo − null) such that ∼

M ⊕ M −→ N in (R Mod)/(pseudo − null) (resp., ∼

2 · (N/(M ⊕ M ⊕ N 0 )) = 0,

N 0 −→

MM

(R/pi )a(p,i)

p∈A i≥1

in (R Mod)/(pseudo − null)), where A = {p ∈ Spec(R) | ht(p) = 1, (∃p ∈ Spec(R)) ht(p) = 1, pR ⊆ p}. ∼

In particular, if Γ = Γ0 −→ Zrp , then A = {pR | p ∈ Spec(R), ht(p) = 1}. Proof. Assume that 10.7.13.1-8 hold. By Corollary 2.10.19, N is isomorphic in (R Mod)/(pseudo − null) to M M ∼ N −→ (R/pi )n(p,i) . p∈Spec(R) ht(p)=1

i≥1

For each n = 1, . . . , 6 (resp., n = 7), the condition 10.7.11.n follows from 10.7.13.n (resp., from 10.7.13.7-8 applied to q = p ∩ R). As M ∼ Np −→ (Rp /pi Rp )n(p,i) , i≥1

Proposition 10.7.12(i) implies that each exponent n(p, i) is even, hence ∼

N −→ M ⊕ M in (R Mod)/(pseudo − null), where M=

M p∈Spec(R) ht(p)=1

M

(R/pi )n(p,i)/2 .

i≥1

If we only assume 10.7.13.1-3 and 10.7.13.6-7, then Proposition 10.7.12(ii) shows that, for each p 6∈ A, there is an (Rp )-submodule M (p) ⊕ M (p) ⊂ Np 264

satisfying
2 · Np /(M (p) ⊕ M (p)) = 0. Writing M (p) =

M

(Rp /pi Rp )m(p,i) ,

i≥1

Proposition 2.10.18 and Corollary 2.10.19 then show the existence of the required M ,→ N , isomorphic to M ∼ M −→ (R/pi )m(p,i) . ht(p)=1 p6∈A

(10.7.15) Theorem. Assume that 10.7.11.1-2 hold and that X = H 0 (X) and Xv+ = H 0 (Xv+ ) (∀v ∈ Σ) are concentrated in degree zero. Let q0 ∈ Spec(R[∆]) be a minimal prime ideal such that 10.7.11.8-9 hold for q = q0 ∩ R ∈ Spec(R). Assume, in addition, that κ(q0 ) = R[∆]q0 is a field of characteristic char(κ(q0 )) 6= 2 Γ0 e 0 (K0 , X)q0 = 0, where K0 := K∞ e j (K0 , X) := H e j (K, F∆ (X)). Denote by q = q0 R the unique and H and H f f f minimal prime ideal of R containing q0 (=⇒ κ(q) = Rq is a field containing κ(q0 )). Then: (i) Denoting by ι : R[∆] −→ R[∆] (resp., ι : R −→ R) the standard involution, then e f1 (K0 , X)q0 = dimκ(q ) H e f2 (K0 , X)q0 = dimκ(q ) (H e f1 (K0 , X)ι )q0 = dimκ(q ) (H e f2 (K0 , X)ι )q0 dimκ(q0 ) H 0 0 0 e 1 (K∞ /K, X)q = dimκ(q) H e 2 (K∞ /K, X)q = dimκ(q) (H e 1 (K∞ /K, X)ι )q = dimκ(q) H f,Iw f,Iw f,Iw ι 2 e f,Iw = dimκ(q) (H (K∞ /K, X) )q .

(ii) The dimensions from (i) satisfy e 2 (K0 , X)q0 ≥ dimκ(q) H e 2 (K∞ /K, X)q dimκ(q0 ) H f f,Iw e 2 (K0 , X)q0 ≡ dimκ(q) H e 2 (K∞ /K, X)q (mod 2). dimκ(q0 ) H f f,Iw In particular, if ∆ = 0 (=⇒ K0 = K, q0 = q) and R is a domain (=⇒ q = (0)), then         1 2 e f1 (X) = rkR H e f2 (X) , e f,Iw e f,Iw rkR H rkR H (K∞ /K, X) = rkR H (K∞ /K, X)         2 2 e 2 (X) ≥ rk H e 2 (K∞ /K, X) , e e rkR H rk H (X) ≡ rk H (K /K, X) (mod 2). R ∞ f f,Iw f f,Iw R R Γ0

(iii) Let Γ00 ⊂ Γ0 be an open subgroup, K00 = K∞0 , ∆0 = Γ0 /Γ00 and q00 ∈ Spec(R[∆ × ∆0 ]) a minimal prime e 0 (K 0 , X)q0 = 0, where H e j (K 0 , X) := H e j (K, F∆×∆0 (X)). Assume ideal above q0 ∈ Spec(R[∆]) such that H 0 0 f f f 0 ∼ that (∀v ∈ Σ) RΓcont (Gv , F∆×∆0 (Wv ))q0 −→ 0 in Dfb t (R[∆×∆0 ]q0 Mod). Assume, in addition, that the field 0

0

κ(q0 ) has characteristic zero. Then κ(q00 ) := R[∆ × ∆0 ]q00 is a field of finite degree over κ(q0 ) and e f1 (K00 , X)q0 = dimκ(q0 ) H e f2 (K00 , X)q0 dimκ(q00 ) H 0 0 0 2 e f2 (K00 , X)q0 ≡ dimκ(q) H e f,Iw e f2 (K0 , X)q0 (mod 2), dimκ(q00 ) H (K∞ /K, X)q ≡ dimκ(q0 ) H 0

e 2 (K 0 , X)q0 ≥ dimκ(q) H e 2 (K∞ /K, X)q . dimκ(q00 ) H f 0 f,Iw 0 Proof. (i) The assumption 10.7.11.1 yields, by 10.3.5.4-5, isomorphisms of R[∆]-modules (resp., of Rmodules) ∼ ej e j (K0 , X)ι −→ H Hf (K0 , X), f

∼ ej e j (K∞ /K, X)ι −→ H Hf,Iw (K∞ /K, X). f,Iw

According to Theorem 8.9.8, localizing the morphism 265

(10.7.15.1)

gf (F∆ (X)) −→ RHomR[∆] (RΓ gf (F∆ (X)), ωR[∆] )ι [−3] γπ∆ : RΓ at q0 , we obtain isomorphisms ∼ e j (K0 , X)q0 −→ e 3−j (K0 , X))ι )q0 , κ(q0 )), H Homκ(q0 ) ((H f f

(10.7.15.2)

which proves the first half of (i). In order to prove the second half, fix a chain of subgroups Γ0 ⊃ Γ1 ⊃ · · · ⊃ ∼ Γi Γr = 0 satisfying Γi /Γi+1 −→ Zp (i = 0, . . . , r − 1) and put K∞,i = K∞ (K0 = K∞,0 ⊂ · · · ⊂ K∞,r = K∞ ), Ri = R[[Γ/Γi ]] = R[[Γ0 /Γi ]][∆], qi = q0 Ri (R0 = R[∆], Rr = R). For each i = 0, . . . , r − 1, we denote by pi+1 ∈ Spec(Ri+1 ) the inverse image of qi under the augmentation map Ri+1 = Ri [[Γi /Γi+1 ]]  Ri . Then Si+1 := (Ri+1 )pi+1 is a discrete valuation ring with residue field κ(qi ) := (Ri )qi (and uniformizing element γi − 1, for any topological generator γi of Γi /Γi+1 ). According to Theorem 8.9.8, the localization of the morphism gf,Iw (K∞ /K, X) −→ RHom (RΓ gf,Iw (K∞ /K, X), ω )ι [−3] γπ : RΓ R R at pr ∈ Spec(R) is an isomorphism. Localizing further at the minimal prime ideal q ⊂ pr , we obtain isomorphisms ∼ e j (K∞ /K, X)q −→ e 3−j (K∞ /K, X)ι)q , κ(q)); H Homκ(q) ((H f,Iw f,Iw

the second half of (i) follows. (ii) Denote, for each i = 0, . . . , r, e 2 (K∞,i /K, X) ∈ (Ri Mod) . Ni = H f,Iw ft A localized version of Proposition 10.3.5.8 yields, for each i = 0, . . . , r − 1, a skew-symmetric (hence alternating, as char(κ(qi )) 6= 2) Si+1 -bilinear form (Ni+1 )pi+1


Si+1 −tors

× (Ni+1 )pi+1


Si+1 −tors

−→ Frac(Si+1 )/Si+1 .

(10.7.15.3)

The assumptions on q0 imply, by Lemma 10.5.4.2, that FΓ/Γi+1 (Xv+ )pi+1 ⊥πi+1 FΓ/Γi+1 (Xv+ )pi+1 .

(∀i = 0, . . . , r − 1) (∀v ∈ Σ) The cohomology sequences of γi −1

0 −→ FΓ/Γi+1 (Wv )−−→FΓ/Γi+1 (Wv ) −→ FΓ/Γi (Wv ) −→ 0 give rise to exact sequences j j 0 −→ Hcont (Gv , FΓ/Γi+1 (Wv ))pi+1 /pi+1 −→ Hcont (Gv , FΓ/Γi (Wv ))qi −→ j+1 −→ Hcont (Gv , FΓ/Γi+1 (Wv ))pi+1 [pi+1 ] −→ 0.

According to the assumptions, the middle term vanishes for i = 0 (and all j). Induction on i, together with the Nakayama Lemma, imply that (∀i = 0, . . . , r − 1)

RΓcont (Gv , FΓ/Γi+1 (Wv ))p

∼

i+1

−→ 0,

hence M

Errv (∆(FΓ/Γi+1 (X)), ∆(FΓ/Γi+1 (X)), πi+1 )pi+1 = 0.

v∈Σ

As pi+1 6= pRi+1 for any p ∈ Spec(Ri ), we also have 266

M

Errur v (∆(FΓ/Γi+1 (X)), ∆(FΓ/Γi+1 (X)), πi+1 )pi+1 = 0.

v∈Σ0

It follows from Theorem 10.4.4 that the pairing (10.7.15.3) is non-degenerate, hence (Ni+1 )pi+1


tors

∼

−→ Mi+1 ⊕ Mi+1

e 3 (K0 , X)q0 = 0 (by (10.7.15.2) for j = 0 and (10.7.15.1)), for some Si+1 -module Mi+1 of finite length. As H f Proposition 8.10.11 implies that the canonical map (Ni+1 )Γi /Γi+1 −→ Ni becomes an isomorphism after localizing at qi , hence ∼

(Ni+1 )pi+1 /pi+1 (Ni+1 )pi+1 −→ (Ni )qi . As ∼

⊕a

(Ni+1 )pi+1 −→ Si+1i+1 ⊕ Mi+1 ⊕ Mi+1 , where ai+1 := rkSi+1 (Ni+1 )pi+1 = dimκ(qi+1 ) (Ni+1 )qi+1 , it follows that ai = dimκ(qi ) (Ni )qi = ai+1 + 2 dimκ(qi ) (Mi+1 /pi+1 Mi+1 ) ≡ ai+1 (mod 2)

(10.7.15.4)

and ai ≥ ai+1 . By induction, we deduce that a0 ≥ a 1 ≥ · · · ≥ a r 2 2 e e f,Iw a0 = dimκ(q0 ) Hf (K0 , X)q0 ≡ ar = dimκ(q) H (K∞ /K, X)q (mod 2), proving (ii).
∗ (iii) The prime ideal q000 = q00 ∩ R[∆0 ] ∈ Spec(R[∆0 ]) defines a tautological character χ : ∆0 −→ R[∆0 ]q000 . Replacing K00 by the fixed field of Ker(χ) and using Lemma 8.6.4.4(iv) (which applies, since [K00 : K0 ] is invertible in κ(q0 ) = R[∆]q0 ), we reduce to the case when ∆0 is cyclic. In this case we have K ⊂ K0 ⊂ K00 ⊂ K∞,1 ⊂ K∞ for a suitable choice of Γ1 , . . . , Γr−1 as in the proof of (ii). Let p0 ∈ Spec(R1 ) be the inverse image of q00 under the canonical surjection R1 = R[[Γ/Γ1 ]] = R[[Gal(K∞,1 /K)]]  R[[Gal(K00 /K)]] = R[[Γ/Γ00 ]] = R[∆ × ∆0 ]; then S10 := (R1 )p0 is a discrete valuation ring with residue field R[∆ × ∆0 ]q00 = κ(q00 ). The same argument as in the proof of (ii) then shows that ∼

((N1 )p0 )S 0 −tors −→ M10 ⊕ M10 1

and ∼

e 2 (K 0 , X)q0 , (N1 )p0 /p0 (N1 )p0 −→ H f 0 0 hence 267

e f2 (K00 , X)q0 = dimκ(q ) (N1 )q1 + 2 dimκ(q ) (M10 /p0 M10 ) ≡ dimκ(q ) (N1 )q1 = dimκ(q00 ) H 1 1 1 0 2 e f,Iw = a1 ≡ ar = dimκ(q) H (K∞ /K, X)q (mod 2)

and 2 e f2 (K00 , X)q0 ≥ a1 ≥ ar = dimκ(q) H e f,Iw dimκ(q00 ) H (K∞ /K, X)q , 0

by (10.7.15.4). (10.7.16) An important special case in which the results of 10.7.14-15 apply is the following: R = O is a ∼ discrete valuation ring with fraction field F as in 9.1.1, R = O[[Γ]] = Λ −→ O[∆][[X1 , . . . , Xr ]], X = T is as in 9.1.4, Greenberg’s local conditions are given by exact sequences of O[Gv ]-modules 0 −→ Tv+ −→ T −→ Tv− −→ 0

(v ∈ Σ)

(with Tv− free over O) and π comes from a skew-symmetric bilinear form j : T ⊗O T −→ O(1) satisfying j(Tv+ ⊗O Tv+ ) = 0 (for all v ∈ Σ) and inducing an isomorphism ∼

adj(j) ⊗ id : V −→ HomF (V, F )(1) = V ∗ (1), where V = T ⊗O F . For each v ∈ Σ, this isomorphism induces an exact sequence of F [Gv ]-modules 0 −→ Wv −→ Vv− −→ HomF (Vv+ , F )(1) −→ 0,

(10.7.16.1)

where Vv± = Tv± ⊗O F . As in 8.9.4, define A∗ (1) = HomO (T, F/O)(1) = T ∗ (1) ⊗O F/O,

A = HomO (T ∗ (1), F/O)(1) = T ⊗O F/O

and, for each v ∈ Σ, Greenberg’s local conditions ∓ ∗ − + T ∗ (1)± A+ v = HomO (Tv , O)(1), v = HomO (T (1)v , F/O)(1) = Tv ⊗O F/O ⊂ A − ∗ + ∗ + ∗ ∗ + A∗ (1)+ A− A∗ (1)− v = HomO (Tv , F/O)(1) = T (1)v ⊗O F/O ⊂ A (1), v = A/Av , v = A (1)/A (1)v

for A, T ∗ (1), A∗ (1). The map adj(j) : T −→ T ∗ (1) is injective and its cokernel is O-torsion. It induces maps ± ∗ ± A −→ A∗ (1) and, for each v ∈ Σ, Tv± −→ T ∗ (1)± v , Av −→ A (1)v , hence also gf (X) −→ RΓ gf (X ∗ (1)), RΓ

gf,Iw (K∞ /K, Y ) −→ RΓ gf,Iw (K∞ /K, Y ∗ (1)) RΓ

(X = T, A; Y = T ).

The exact sequences (10.7.16.1) give rise to exact triangles ∗ Vv+ −→ V ∗ (1)+ v −→ Wv (1) b

(10.7.16.2)

in D (F [Gv ] Mod) (v ∈ Σ). It follows that, for each finite subextension L/K of K∞ /K, there is an exact triangle MM gf (L, V ) −→ RΓ gf (L, V ∗ (1)) −→ RΓ RΓcont (Gw , Wv∗ (1)) (10.7.16.3) v∈Σ w|v

in Db (F Mod) (where Gw = Gal(K v /Lw )), and an exact triangle 268

gf,Iw (K∞ /K, V ) −→ RΓ gf,Iw (K∞ /K, V ∗ (1)) −→ RΓ

M

RΓcont (Gv , FΓ (Wv∗ (1)))

(10.7.16.4)

v∈Σ

gf,Iw (K∞ /K, V ) = RΓ gf,Iw (K∞ /K, T ) ⊗O F (and similarly for V ∗ (1)). in Dfb t (Λ⊗Q Mod), where RΓ We are going to apply Theorem 10.7.15 to X = T , Xv+ = Tv+ and π = j, assuming that the conditions 10.7.11.1-2 are satisfied. Γ0 More precisely, let Γ00 ⊂ Γ0 be an open subgroup, K00 = K∞0 and χ : ∆ −→ (O0 )∗ ,

χ0 : ∆0 = Γ0 /Γ00 −→ (O0 )∗

(10.7.16.5)

a pair of characters with values in the ring of integers O0 of a finite extension F 0 of F . For every O[∆]-module M (resp., every O[∆ × ∆0 ]-module N ) put M (χ) = M ⊗O[∆],χ O0 ,

0

N (χ×χ ) = N ⊗O[∆×∆0],χ×χ0 O0 .

Then q0 := Ker(χ : O[∆] −→ O0 ),

q00 := Ker(χ × χ0 : O[∆ × ∆0 ] −→ O0 )

are minimal prime ideals as in Theorem 10.7.15, satisfying (by Proposition 8.8.7) e j (K0 , V )q0 ⊗κ(q ) F 0 = H e j (K0 , V )(χ) = H e j (K, V ⊗ χ−1 ), H 0 f f f

(10.7.16.6)

e j (K 0 , V )q0 ⊗κ(q0 ) F 0 = H e j (K 0 , V )(χ×χ0 ) = H e j (K, V ⊗ (χ × χ0 )−1 ). H 0 0 f f f 0 0 (10.7.17) Theorem. In the situation of 10.7.16, assume that 10.7.11.1-2 hold and (∀v ∈ Σ) (∀v0 | v in K0 )

0 0 Hcont (Gv0 , Wv ) = Hcont (Gv0 , Wv∗ (1)) = 0.

Then, for any pair of characters χ, χ0 as in (10.7.16.5), we have: (i) For each j ∈ Z, the canonical maps e j (K0 , V )(χ) −→ H e j (K0 , V ∗ (1))(χ) H f f e j (K∞ /K, V )(χ) ⊗Λ(χ) Frac(Λ(χ) ) −→ H e j (K∞ /K, V ∗ (1))(χ) ⊗Λ(χ) Frac(Λ(χ) ) H f,Iw f,Iw are isomorphisms. (ii) We have −1

e 1 (K0 , V )(χ) = dimF 0 H e 2 (K0 , V )(χ) = dimF 0 H e 1 (K0 , V )(χ dimF 0 H f f f (χ)

1 e f,Iw rkΛ(χ) H (K∞ /K, T )

(χ)

2 e f,Iw = rkΛ(χ) H (K∞ /K, T )

)

−1

e 2 (K0 , V )(χ = dimF 0 H f (χ−1 )

1 e f,Iw = rkΛ(χ−1 ) H (K∞ /K, T ) −1

(χ 2 e f,Iw = rkΛ(χ−1 ) H (K∞ /K, T )

)

.

e 0 (K0 , V )(χ) = 0, then (iii) If H f (χ) 1 e f1 (K0 , V )(χ) ≡ rkΛ(χ) H e f,Iw dimF 0 H (K∞ /K, T ) (mod 2), (χ) 1 e f1 (K0 , V )(χ) ≥ rkΛ(χ) H e f,Iw dimF 0 H (K∞ /K, T ) .

In particular, if ∆ = 0, then e 1 (T ) = rkO H e 2 (T ), rkO H f f

e 1 (K∞ /K, T ) = rkΛ H e 2 (K∞ /K, T ) rkΛ H f,Iw f,Iw

e 1 (T ) ≡ rkΛ H e 1 (K∞ /K, T ) (mod 2), rkO H f f,Iw 269

e 1 (T ) ≥ rkΛ H e 1 (K∞ /K, T ). rkO H f f,Iw

=

)

(iv) Assume that (∀v ∈ Σ) (∀v00 | v in K00 )

0 0 Hcont (Gv00 , Wv ) = Hcont (Gv00 , Wv∗ (1)) = 0.

Then, for each j ∈ Z, the canonical map e j (K 0 , V )(χ×χ0 ) −→ H e j (K 0 , V ∗ (1))(χ×χ0 ) H 0 0 f f e 0 (K 0 , V )(χ×χ0 ) = 0, then is an isomorphism. If, in addition, H 0 f (χ×χ0 )

e 1 (K 0 , V ) dimF 0 H f 0

(χ×χ0 )

e 2 (K 0 , V ) = dimF 0 H f 0

(χ×χ0 )−1

e 1 (K 0 , V ) = dimF 0 H f 0

(χ×χ0 )−1

e 2 (K 0 , V ) = dimF 0 H f 0

and 0

e 1 (K 0 , V )(χ×χ ) ≡ rkΛ(χ) H e 1 (K∞ /K, T )(χ) ≡ dimF 0 H e 1 (K0 , V )(χ) (mod 2) dimF 0 H f 0 f,Iw f 0

(χ) 1 e f1 (K00 , V )(χ×χ ) ≥ rkΛ(χ) H e f,Iw dimF 0 H (K∞ /K, T ) .

e 0 (K00 , V )(χ) = 0, dimF 0 H e 1 (K0 , V )(χ) ≡ 1 (mod 2) and (v) Assume that H f f (∀v ∈ Σ) (∀v00 | v in K00 )

0 0 Hcont (Gv00 , Wv ) = Hcont (Gv00 , Wv∗ (1)) = 0;

e 1 (K00 , V )(χ) ≥ [K00 : K0 ]. then dimF 0 H f Proof. This is a special case of Theorem 10.7.15. However, in view of the importance of the congruence 0

e f1 (K00 , V )(χ×χ ) ≡ dimF 0 H e f1 (K0 , V )(χ) (mod 2) dimF 0 H for the arithmetic applications proved in Chapter 12 below, we repeat the key arguments in this simplified setting. 0 (i) Fix a prime v0 of K0 above v ∈ Σ. By assumption, the cohomology groups Hcont (Gv0 , Wv∗ (1)) and ∼ 2 ∗ 0 ∗ Hcont (Gv0 , Wv (1)) −→ Hcont (Gv0 , Wv ) vanish; the Euler characteristic formula (Theorem 4.6.9 and 5.2.11) 2 X

q (−1)q dimF Hcont (Gv0 , Wv∗ (1)) = 0

q=0 1 Hcont (Gv0 , Wv∗ (1))

implies that also vanishes. The first half of (i) then follows from the exact triangle (10.7.16.3) for L = K0 . Proposition 8.4.8.5 applied to the extension K∞ /K0 , R = O, p = (0) and T = F∆ (T (Wv )∗ (1)) (where T (Wv )∗ (1) ⊂ Wv∗ (1) is an arbitrary Gv -stable O-lattice) shows that ∼ RΓcont (Gv , FΓ (Wv∗ (1))) ⊗Λ Frac(Λ) −→ 0; the second half of (i) then follows from (10.7.16.4). (ii) This follows from (i), the duality isomorphisms ∼ e j (K0 , V )(χ) −→ e 3−j (K0 , V ∗ (1))(χ−1 ) , F 0 ) H HomF 0 (H f f ∼ e j (K∞ /K, T ) ⊗Λ Frac(Λ) −→ e 3−j (K∞ /K, T ∗ (1)), Λ)ι ⊗Λ Frac(Λ) H HomΛ (H f,Iw f,Iw

and the fact that the action of τ (= a lift of the non-trivial element of Gal(K/K + )) interchanges the eigenspaces for χ and χ−1 . (iv) As χ0 (Γ0 ) is a finite cyclic group, χ0 factors through a quotient Γ0 of Γ0 , which is isomorphic to Zp . We ∼ can then replace K∞ by the fixed field of Ker(Γ0 −→ Γ0 ) and assume that Γ0 −→ Zp , hence Ker(χ0 ) = pn Γ0 0 0 0 0 for some n ≥ 0. Let p ∈ Spec(Λ ) be the augmentation ideal of Λ := O [[Γ0 ]]. Proposition 10.7.12(ii) implies that there is an exact sequence of Λ0p0 -modules 270

0 −→ X ⊕ X −→

  (χ) 2 e f,Iw H (K∞ /K, T )

p0

−→ Λ0p0


⊕r

−→ 0,

(10.7.17.1)

where r = rk

Λ0 0 p

  (χ) 2 e Hf,Iw (K∞ /K, T )

p0

≥ 0.

Thanks to the assumptions 0

e 0 (K0 , V )(χ) = H e 0 (K 0 , V )(χ×χ ) = 0, H f f 0 Proposition 8.10.4 applies with R = O and S = R − {0} to the extensions K∞ /K0 and K∞ /K00 ; one obtains isomorphisms 

 (χ) ∼ e2 e 2 (K∞ /K, T )(χ) ⊗O0 F 0 −→ H Hf (K0 , V ) f,Iw Γ0   (χ×χ0 ) (χ×χ0 ) ∼ e2 2 0 e f,Iw H (K∞ /K, T )pn Γ ⊗O0 F 0 −→ H . f (K0 , V ) 0

The exact sequence (10.7.17.1) then implies that e f2 (K0 , V )(χ) = r + 2 dimF 0 (XΓ0 ) dimF 0 H 0

e 2 (K 0 , V )(χ×χ ) = r + 2 dimF 0 (Xpn Γ0 )(χ0 ) , dimF 0 H f 0 hence 0

e f2 (K0 , V )(χ) ≡ r ≡ dimF 0 H e f2 (K00 , V )(χ×χ ) (mod 2). dimF 0 H e 2 (K0 , V )(χ) is dual to As H f −1

e f1 (K0 , V ∗ (1))(χ H

)

(χ−1 )

e f1 (K0 , V ) =H

(χ)

∼

e f1 (K0 , V ) −→ H 0

e 2 (K00 , V )(χ×χ ) is dual to (the last isomorphism being given by the action of τ ) and H f 0 −1

e f1 (K00 , V ∗ (1))(χ×χ ) H

0 −1

e f1 (K00 , V )(χ×χ ) =H

0

(χ×χ ) ∼ e1 0 −→ H , f (K0 , V )

we obtain the desired congruence 0

e 1 (K0 , V )(χ) ≡ dimF 0 H e 1 (K 0 , V )(χ×χ ) (mod 2). dimF 0 H f f 0 (v) We can assume that F 0 contains the values of all characters of ∆0 = Gal(K00 /K0 ); then e f1 (K00 , V )(χ) = H

M

0

e f1 (K00 , V )(χ×χ ) , H

χ0 0

0

where χ runs through all characters of ∆ . As 0

e 1 (K 0 , V )(χ×χ ) ≡ dimF 0 H e 1 (K0 , V )(χ) ≡ 1 (mod 2) dimF 0 H f 0 f by (iv), it follows that e 1 (K 0 , V )(χ) ≥ H f 0

X

1 = |∆0 | = [K00 : K0 ],

χ0

271

as claimed. (10.7.18) For example, if K is an imaginary quadratic field and K∞ /K its anticyclotomic Zp -extension, then K + = Q and we are in the situation considered in [Ne 3]. If Σ+ = {p} and Sf+ consists of all rational primes dividing pN , where N ≥ 1 is an integer not divisible by p such that all primes dividing N are unramified (resp., split) in K/Q, then the condition 10.3.5.1.5 is satisfied (resp., the condition 10.3.5.1.5 is satisfied and no prime in Σ0 splits completely in K∞ /K). If E is an elliptic curve over Q with good ordinary reduction at p and N is the conductor of E, then T = Tp (E) and A = E[p∞ ] are as in 10.7.16 (for O = Zp , F = Qp ), with j given by the Weil pairing and Tp− = Tp (E ⊗Zp Fp ), where E is a proper smooth model of E over Zp . e 1 (A) (resp., the Λ-module H e 1 (KS /K∞ , A)) differs from Sel(E/K, p∞) (resp., from The Selmer group H f f ∞ Sel(E/K∞ , p∞ ) = − lim → Sel(E/Kα, p ), α

where Kα /K are the finite subextensions of K∞ /K) by a finite group (resp., by a Λ-module killed by a power of p), by Proposition 9.6.6 combined with Proposition 9.6.7.5. As the group H 0 (GK,S , A) = E(K)[p∞ ] is finite, the assumptions of Theorem 10.7.17(i)–(iii) are satisfied and the congruence in 10.7.17(iii) becomes corkZp (Sel(E/K, p∞ )) ≡ corkΛ (Sel(E/K∞ , p∞ )) (mod 2),

(10.7.18.1)

as in [Ne 3], Thm. B (assuming that each prime dividing N is unramified in K/Q). More precisely, if all primes dividing N are split in K/Q, then Proposition 10.7.12 implies that ∼

(DΛ (Sel(E/K∞, p∞ )))Λ−tors −→ M ⊕ M ⊕ N 0 ∼

in (Λ Mod)/(pseudo − null), where 2n · N 0 = 0 for some n ≥ 0 (in particular, N 0 −→ 0 in (Λ Mod)/(pseudo − null) if p 6= 2); this proves ([Ne 3], Lemma 2.5). It may be useful to spell out explicitly the following special case of Theorem 10.7.17 (which generalizes the congruence (10.7.18.1)) in a situation considered by Mazur and Rubin [M-R 3]. ∼

(10.7.19) Proposition. Let K + ⊂ K ⊂ K∞ be as in 10.3.5.1, with Γ = Gal(K∞ /K) −→ Zrp ; set Λ = Zp [[Γ]]. If E is an elliptic curve over K + with good ordinary reduction at all primes above p, then corkZp Sel(E/K, p∞) ≡ corkΛ Sel(E/K∞ , p∞ ) (mod 2) corkZp Sel(E/K, p∞) ≥ corkΛ Sel(E/K∞ , p∞ ). If, in addition, corkZp Sel(E/K, p∞) ≡ 1 (mod 2), then corkZp Sel(E/K 0 , p∞ ) ≥ [K 0 : K], for each finite subextension K 0 /K of K∞ /K (cf. [M-R 3], Thm. 3.1). Proof. Fix a finite set S + of primes of K + containing all archimedean primes, all primes above p and all primes at which E has bad reduction. Consider Greenberg’s local conditions attached to the data (10.3.5.1.24) with R = Zp , X = Y = Tp (E) = T , π : Tp (E) ⊗Zp Tp (E) −→ Zp (1) given by the Weil pairing, Σ+ = Sf+ and, for each v + ∈ Sf+ , ( Tp (E)+ v+ | p v + defined in (9.6.7.2) , + Tp (E)v+ = 0, v + - p. ∼

As Σ0 = ∅, we do not have to worry about the condition (10.3.5.1.5). As V = T ⊗Zp Qp = Vp (E) −→ V ∗ (1) satisfies the monodromy-weight conjecture at each non-archimedean prime not dividing p ([Ja 2], Sect. 5,7), we have H 0 (Gw , V ) = H 0 (Gw , V ∗ (1)) = 0, 272

(Gw = Gal(K v /Lw ))

for each finite extension L/K and each non-archimedean prime w | v - p of L. In particular, the assumption of Theorem 10.7.17 is satisfied for K0 = K and each v ∈ Σ (note that Wv = 0 if v | p, while Wv = Vv if v - p). e 0 (K, V ) ⊂ H 0 (GK , V ) = 0), we obtain, using the notation Applying Theorem 10.7.17(iii) (observing that H f from 9.7.8, e 1 (K, Tp (E)) ≡ rkΛ H e1 rkZp H f,Σ f,Iw,Σ (K∞ /K, Tp (E)) (mod 2), e 1 (K, Tp (E)) ≥ rkΛ H e1 rkZp H f,Σ f,Iw,Σ (K∞ /K, Tp (E)). On the other hand, Proposition 9.7.9(i) together with Lemma 9.6.7.6(iii) show that e 1 (K, Tp (E)) = rkZp H e 1 (K, Tp (E)) = corkZp Sel(E/K, p∞ ). rkZp H f,Σ f Similarly, Proposition 9.7.9(iii) together with (8.9.6.4.3), (9.6.5.2) and Proposition 9.6.7.8 imply that e1 e1 rkΛ H f,Iw,Σ (K∞ /K, Tp (E)) = rkΛ Hf,Iw (K∞ /K, Tp (E)) = e 1 (KS /K∞ , E[p∞ ]) = corkΛ S str (K∞ ) = corkΛ Sel(E/K∞ , p∞ ). = corkΛ H f A If corkZp Sel(E/K, p∞) ≡ 1 (mod 2), then Theorem 10.7.17(v) combined with the previous discussion yields the inequality corkZp Sel(E/K 0 , p∞ ) ≥ [K 0 : K]. (10.7.20) In Chapter 12, we generalize Proposition 10.7.19, as well as the main parity result of [Ne 3], to the case of Hilbert modular forms. (10.8) Comparison with the Flach pairing For simplicity, assume that the condition (P ) from 5.1 holds. (10.8.1) Let R = O, F = Frac(O) and T, A, T ∗ (1), A∗ (1) be as in 9.1.1-4. Set V = T ⊗O F , V ∗ (1) = ∗ T ∗ (1) ⊗O F and assume that we are given, for each v ∈ Sf , F [Gv ]-submodules Vv+ ⊂ V , V ∗ (1)+ v ⊂ V (1) + ∗ + + + ∗ + ∗ ∗ + satisfying Vv ⊥ev2 V (1)v ; set Tv = T ∩ Vv , T (1)v = T (1) ∩ V (1)v . We denote the inclusions T ,→ V , T ∗ (1) ,→ V ∗ (1) (resp., Xv+ ,→ X, for X = T, V, T ∗ (1), V ∗ (1)) by i (resp., ◦ by i+ v ). For each function f with values in V , we denote by f the corresponding function with values in ∗ ∗ ∗ ∗ V /T = A (and similarly for V (1) and V (1)/T (1) = A (1), resp., F and F/O). (10.8.2) These data define Greenberg style Selmer groups M   S(X) = Ker H 1 (GK,S , X) −→ H 1 (Gv , X)/L+ (X = A, A∗ (1)) v (X) , v∈Sf

where
1 + 1 L+ v (A) = Im Hcont (Gv , Vv ) −→ H (Gv , A) ,


∗ 1 ∗ + 1 ∗ L+ v (A (1)) = Im Hcont (Gv , V (1)v ) −→ H (Gv , A (1)) . (10.8.3) For example, assume that O = Zp , B is an abelian variety over K with good reduction outside b A = B[p∞ ] and Sf satisfying {v | p} = Σo ∪ Σt (in the notation of 9.6.7.2), T = Tp (B), T ∗ (1) = Tp (B), ∗ ∞ + ∗ + + ∗ + b A (1) = B[p ]. If we define Vv , V (1)v as in 9.6.7.2 if v | p (resp., set Vv = V (1)v = 0 if v - p), then we have, for each v ∈ Sf ,
1 L+ v (A) = Im B(Kv ) ⊗ Qp /Zp ,→ H (Gv , A) b hence (and similarly for A∗ (1) and B), S(A) = Sel(B/K, p∞ ),

b S(A∗ (1)) = Sel(B/K, p∞ ). 273

(10.8.4) The data from 10.8.1 also give rise, for each X = T, V, T ∗ (1), V ∗ (1), to Selmer complexes ef• (X) := C ef• (GK,S , X; ∆(X)) C attached to the local conditions • • ∆v (X) : Ccont (Gv , Xv+ ) −→ Ccont (Gv , X).

e j (X). We denote their cohomology groups by H f (10.8.5) Flach’s pairing. Flach’s construction [Fl 1] yields an O-bilinear pairing h , iFlach : S(A) × S(A∗ (1)) −→ F/O. In order to define this pairing, fix cohomology classes [α] ∈ S(A),

[α0 ] ∈ S(A∗ (1))

represented by 1-cocycles α ∈ C 1 (GK,S , A),

dα = 0,

α0 ∈ C 1 (GK,S , A∗ (1)),

dα0 = 0.

Lift these cocycles to 1-cochains • a1 ∈ Ccont (GK,S , V ),

• b1 ∈ Ccont (GK,S , V ∗ (1));

then da1 = −i(a2 ),

db1 = −i(b2 ),

2 a2 ∈ Ccont (GK,S , T ),

2 b2 ∈ Ccont (GK,S , T ∗ (1)).

The definition of S(X) implies that there exist, for each v ∈ Sf , elements 1 + a+ 1,v ∈ Ccont (Gv , Vv ),

0 A0,v ∈ Ccont (Gv , V ) = V,

1 A1,v ∈ Ccont (Gv , T )

1 ∗ + b+ 1,v ∈ Ccont (Gv , V (1)v ),

0 B0,v ∈ Ccont (Gv , V ∗ (1)) = V ∗ (1),

1 B1,v ∈ Ccont (Gv , T ∗ (1))

(10.8.5.1)

satisfying + resv (a1 ) = −i+ v (a1,v ) + dA0,v + i(A1,v ),

da+ 1,v = 0

+ resv (b1 ) = −i+ v (b1,v ) + dB0,v + i(B1,v ),

db+ 1,v = 0.

The coboundary of the 3-cochain 3 da1 ∪ b1 ∈ Ccont (GK,S , F (1))

satisfies
4 d(da1 ∪ b1 ) = i(a2 ∪ b2 ) ∈ i Ccont (GK,S , O(1)) , which means that (da1 ∪ b1 )◦ (= the image of da1 ∪ b1 modulo cochains with values in O(1)) is a 3-cocycle: 3 (da1 ∪ b1 )◦ ∈ Ccont (GK,S , F/O(1)),

d(da1 ∪ b1 )◦ = 0.

2 As H 3 (GK,S , F/O(1)) = 0, there exists ε ∈ Ccont (GK,S , F/O(1)) such that

−a2 ∪ b◦1 = (da1 ∪ b1 )◦ = dε. For each v ∈ Sf , the 1-cocycle 274

+ 1 ∗ βev := −i+ v (b1,v ) + dB0,v ∈ Ccont (Gv , V (1)),

dβev = 0

1 is a lift of resv (b1 )◦ ∈ Ccont (Gv , V ∗ (1)/T ∗ (1)). As

d(resv (a1 ) ∪ βev )◦ = −resv (a2 ) ∪ βev◦ = −resv (a2 ∪ b◦1 ) = resv (dε) = d(resv (ε)), the 2-cochain 2 cv := (resv (a1 ) ∪ βev )◦ − resv (ε) ∈ Ccont (Gv , F (1)/O(1)),

dcv = 0

is a 2-cocycle. Flach defined X

h[α], [α0 ]iFlach =

invv (cv ) ∈ F/O

(10.8.5.2)

v∈Sf

(and showed that this element depends only on [α] and [α0 ]). (10.8.6) The pairing from 10.2.2. The choice of the cocycles in (10.8.5.1) determines 2-cochains e2 (GK,S , T ), x = (a2 , 0, (A1,v )) ∈ C f

e 2 (GK,S , T ∗ (1)), y = (b2 , 0, (B1,v )) ∈ C f

dx = 0,

dy = 0,

where we have used the notation from 1.3.1 and 6.3.1: M M n−1 n n efn (GK,S , T ) = Ccont C (GK,S , T ) ⊕ Ccont (Gv , Tv+ ) ⊕ Ccont (Gv , T ), v∈Sf

v∈Sf

with the differentials given by + + + d(tn , (t+ n,v ), (tn−1,v )) = (dtn , (dtn,v ), (−resv (tn ) + iv (tn,v ) − dtn−1,v )).

The formula + i(x) = (i(a2 ), 0, (i(A1,v ))) = (−d(a1 ), 0, (resv (a1 ) + i+ v (a1,v ) − dA0,v )) = dX,

X = (−a1 , (a+ 1,v ), (A0,v ))

e1 (GK,S , V ) is a coboundary, hence the class of x is contained in shows that i(x) ∈ d C f   e f2 (T ) −→ H e f2 (V ) = H e f2 (T )tors . [x] ∈ Ker H e 2 (T ∗ (1))tors . We wish to relate the product of [x] and [y] under the Similarly, the class of y satisfies [y] ∈ H f pairing e f2 (T )tors × H e f2 (T ∗ (1))tors −→ F/O ∪2,2 = ∪ev2 ,0,2,2 : H (which was defined in 10.2.2) to the product (10.8.5.2). (10.8.7) Proposition. In the situation of 10.8.5-6, [x] ∪2,2 [y] = −h[α], [α0 ]iFlach . −i

Proof. Let C • = [O−→F ] be the complex from Lemma 2.10.7(i) (in degrees 0,1). The tensor product e• (T ) ⊗O C • has differentials complex C f efn (T ) ⊕ C en−1 (V ) −→ C en+1 (T ) ⊕ C efn (V ) C f f (xn , Xn−1 ) 7→ (dxn , (−1)n−1 i(xn ) + dXn−1 ), 275

which means that the pair e • (T ) ⊗O C • )2 a = (x, X) ∈ (C f e 2 (T ). Similarly, is a 2-cocyle lifting x ∈ C f ef• (T ∗ (1)) ⊗O C • )2 , b = (y, Y ) ∈ (C

Y = (−b1 , (b+ 1,v ), (B0,v )),

e 2 (T ∗ (1)). In order to compute [x] ∪2,2 [y], we must first determine v(s23 (a ∪ b)), is a 2-cocyle lifting y ∈ C f where v is the map defined in Lemma 2.10.7(i) and the cup product a ∪ b is computed using the product e• (T ) ⊗O C e• (T ∗ (1)) −→ C • (O(1)), ∪r,0 : C f f c from Proposition 1.3.2(i), for any fixed value r ∈ O: v(s23 (a ∪ b)) = (x ∪r,0 y, x ∪r,0 Y ) ∈ (Cc• (O(1)) ⊗O C • )4 . Taking r = 1, we compute 4 x ∪1,0 y = (a2 , 0, (A1,v )) ∪1,0 (b2 , 0, (B1,v )) = (a2 ∪ b2 , (A1,v ∪ resv (bv ))) ∈ Cc,cont (GK,S , O(1)) 3 x ∪1,0 Y = (a2 , 0, (A1,v )) ∪1,0 (−b1 , (b+ 1,v ), (B0,v )) = (−a2 ∪ b1 , (−A1,v ∪ resv (b1 ))) ∈ Cc,cont (GK,S , F (1)).

The next step is to reduce v(s23 (a ∪ b)) modulo cochains with values in O(1), obtaining a 3-cocycle 3 v(s23 (a ∪ b))◦ = (−a2 ∪ b1 , (−A1,v ∪ resv (b1 )))◦ ∈ Cc,cont (GK,S , F (1)/O(1)),

d(v(s23 (a ∪ b))◦ ) = 0,

whose image under the isomorphism ∼

3 Hc,cont (GK,S , F (1)/O(1)) −→ F/O

gives the desired product [x] ∪2,2 [y]. As (−a2 ∪ b1 , (−A1,v ∪ resv (b1 )))◦ = (−a2 ∪ b◦1 , (−A1,v ∪ βev )◦ ) = d(ε, 0) + (0, (resv (ε) − (A1,v ∪ βev )◦ )), we have [x] ∪2,2 [y] =

X

invv (ev ),

v∈Sf

where 2 ev = resv (ε) − (A1,v ∪ βev )◦ ∈ Ccont (Gv , F (1)/O(1)),

dev = 0.

However, (∀v ∈ Sf )

 ◦ + + + + + e ◦ cv + ev = ((dA0,v − i+ = v (a1,v )) ∪ βv ) = (dA0,v − iv (a1,v )) ∪ (dB0,v − iv (b1,v ))  ◦ + + = d(A0,v ∪ (dB0,v − iv (b1,v )))

is a coboundary, hence invv (cv + ev ) = 0, which implies that [x] ∪2,2 [y] = −h[α], [α0 ]iFlach , as claimed. 276

11. R-valued height pairings This chapter is devoted to various generalizations of p-adic height pairings in the context of the Greenberg local conditions. The pairings are introduced and studied in the full generality in Sect. 11.1-2. They are compared to the classical p-adic height pairings for Galois representations in Sect. 11.3-4. Higher (or “derived”) height pairings are studied in Sect. 11.5. An abstract descent formalism for Zp -extensions is developed in Sect. 11.6; it is applied to formulas of the Birch and Swinnerton-Dyer type in Sect. 11.7. For Zp -extensions, the higher height pairings can be related to the generalized Cassels-Tate pairing from Chapter 10 (Sect. 11.8). In the self-dual dihedral case one deduces various parity results in Sect 11.9. (11.1) Definition of the height pairing (11.1.1) Let K be a number field, S a finite set of primes of K containing all primes above p and all ∼ archimedean primes, and K∞ /K a subextension of KS /K with Gal(K∞ /K) = Γ = Γ0 × ∆, where Γ0 −→ Zrp (r ≥ 0) and ∆ is a finite abelian p-group. We assume that the condition (P ) from 5.1 holds (i.e. K has no real primes if p = 2) and that (F l(Γ)) ΓR := Γ ⊗Zp R is a flat (hence free) R-module. The condition (F l(Γ)) is automatically satisfied if ∆ = 0, while in the case ∆ 6= 0 it simply says that R is annihilated by the exponent of ∆. (11.1.2) Denote by J ⊂ R the augmentation ideal of R = R[[Γ]]. The first two graded quotients of the ∼ J-adic filtration of R are isomorphic to R/J −→ R and ∼

J/J 2 −→ ΓR ,

γ − 1 (mod J 2 ) 7→ γ ⊗ 1

(γ ∈ Γ).

It follows from (F l(Γ)) that R/J, J/J 2 and R/J 2 are flat R-modules. (11.1.3) Let X, Y be complexes of admissible R[GK,S ]-modules of finite type over R, and π : X ⊗R Y −→ • • • ωR (1) a morphism of complexes (where ωR = σ≥0 ωR is a complex of injective R-modules representing ωR ). We assume that X, Y are equipped with Greenberg’s local conditions 7.8.2 attached to Xv+ −→ X,

Yv+ −→ Y

(v ∈ Σ ⊂ Sf ),

where Xv+ , Yv+ are complexes of admissible R[Gv ]-modules of finite type over R, satisfying Xv+ ⊥π Yv+ (v ∈ Σ). We also assume that the condition (U ) from 8.8.1 holds, i.e. each prime v ∈ Σ0 = Sf − Σ is unramified in K∞ /K; this is automatic if ∆ = 0. Fixing a quasi-isomorphism • • ωR ⊗R R −→ ωR

as in 10.3.3.1, we obtain a morphism of complexes of admissible R[GK,S ]-modules FΓ (π)

• • π : FΓ (X) ⊗R FΓ (Y )ι −−−−→ωR (1) ⊗R R −→ ωR (1).

As FΓ (Z) = (Z ⊗R R) < −1 > for Z = X, Y , the sequence of complexes 0 −→ FΓ (Z) ⊗R J/J 2 −→ FΓ (Z) ⊗R R/J 2 −→ FΓ (Z) ⊗R R/J −→ 0 in (ad R[G

K,S ]

(Z = X, Y )

Mod)R−f t is exact and isomorphic to 0 −→ Z ⊗R ΓR −→ (Z ⊗R R/J 2 ) < −1 >−→ Z −→ 0,

using (F l(Γ)) and observing that Im(χΓ ) ∈ 1 + J, hence ∼

Z < n > ⊗R J r /J r+1 −→ Z ⊗R J r /J r+1 277

(r ≥ 0, n ∈ Z).

(11.1.3.1)

The first two terms in (11.1.3.1) inherit Greenberg’s local conditions from those for Z, as in 8.9.2. The assumptions (U ) and (F l(Γ)) imply that 0 −→ US+ (Z ⊗R ΓR ) −→ US+ ((Z ⊗R R/J 2 ) < −1 >) −→ US+ (Z) −→ 0

(Z = X, Y )

is an exact sequence of complexes and US+ (Z ⊗R ΓR ) = US+ (Z) ⊗R ΓR

(Z = X, Y ).

It follows that e • (Z ⊗R ΓR ) −→ C e • ((Z ⊗R R/J 2 ) < −1 >) −→ C e• (Z) −→ 0 0 −→ C f f f

(Z = X, Y )

is also an exact sequence of complexes and e• (Z ⊗R ΓR ) = C e • (Z) ⊗R ΓR C f f

(Z = X, Y ).

As a result, we obtain (for Z = X, Y ) exact triangles in Df t (R Mod) β g gf (Z) ⊗R ΓR −−→RΓ gf ((Z ⊗R R/J 2 ) < −1 >)−−→RΓ gf (Z)−−→ RΓ RΓf (Z)[1] ⊗R ΓR .

(11.1.3.2)

These triangles can be constructed more directly, by applying the derived tensor product L

gf,Iw (K∞ /K, Z)⊗ (−) RΓ R to the exact triangle J/J 2 −→ R/J 2 −→ R −→ J/J 2 [1] (using Proposition 8.10.1 and the assumption (F l(Γ))). (11.1.4) The “height pairing” attached to the data (X, Y, π, Xv+ , Yv+ ) is defined as the following morphism in Df t (R Mod): γπ,0 [1]

β

gf (X)−−→RΓ gf (X)[1] ⊗R ΓR −−−−→RHomR (RΓ gf (Y ), ωR [−2]) ⊗R ΓR = D−2 (RΓ gf (Y )) ⊗R ΓR . h = hπ : RΓ This induces maps on cohomology gf (Y ))) ⊗R ΓR −→ HomR (H e fi (X) −→ H i (D−2 (RΓ e j (Y ), H i+j−2 (ωR ) ⊗R ΓR ), H f i.e. bilinear maps e e fi (X) ⊗R H e j (Y ) −→ H i+j−2 (ωR ) ⊗R ΓR . hπ,i,j : H f The only (?) interesting case is that of i + j = 2, when we obtain pairings e e fi (X) ⊗R H e j (Y ) −→ H 0 (ωR ) ⊗R ΓR hπ,i,j : H f

(i + j = 2),

of which e hπ,1,1 is the most important one. e • (X), C e• (Y )) are cohomologically bounded above, then h = If all complexes X, Y, Xv+, Yv+ (hence also C f f adj(e h), where ∪π,0 [1] β⊗id e gf (X)⊗R RΓ gf (Y )−−−−→ gf (X)[1] ⊗R RΓ gf (Y ) ⊗R ΓR −−−−→ω h=e hπ : RΓ RΓ R [−2] ⊗R ΓR L

L

278

is a pairing in Df−t (R Mod). 0 0 (11.1.5) If K ⊂ K∞ ⊂ K∞ is a subextension such that Γ0R := Gal(K∞ /K) ⊗Zp R is flat over R, then the 0 0 various height “pairings” h attached to K∞ /K are obtained from h by applying the canonical projection ΓR  Γ0R . (11.1.6) The above constructions apply, in particular, in the case when R, X, Y are of the form R = R0 [[Γ0 ]], 0 ∼ 0 X = FΓ0 (X0 ), Y = FΓ0 (Y0 ) for suitable Γ0 = Gal(K∞ /K) −→ Zrp . This is the situation considered in [PR 3], 0 where K is an imaginary quadratic field and K∞ (resp., K∞ ) is the cyclotomic (resp., anti-cyclotomic) Zp extension of K. (11.1.7) More generally, let L/K be any subextension of K∞ /K. As in 9.7.1, put ΓL = Gal(K∞ /L), ΓL = Gal(L/K) = Γ/ΓL and J L = Ker(R[[ΓL ]] −→ R),

JL = Ker(R[[Γ]] −→ R[[ΓL ]]).

For each r ≥ 0, there is a canonical isomorphism of R[[ΓL ]]-modules ∼

JLr /JLr+1 −→ (J L )r /(J L )r+1 ⊗R R[[ΓL ]].

(11.1.7.1)

In particular, for r = 1, ∼

J L /(J L )2 −→ ΓL ⊗Zp R = ΓL R

=⇒

∼

JL /JL2 −→ ΓL R ⊗R R[[ΓL ]].

If we replace the flatness condition (F l(Γ)) by L (F l(ΓL )) ΓL R = Γ ⊗Zp R is a flat (hence free) R-module ∼

(which is automatic if Γ −→ Zrp ), then the construction from 11.1.3 works if one replaces everywhere J by JL (with the proviso that JLr /JLr+1 is no longer isomorphic to JLr /JLr+1 < n > as a Galois module): using (8.10.10.1), one obtains an exact sequence 2 0 −→ FΓL (Z) ⊗R ΓL R −→ FΓ (Z)/JL −→ FΓL (Z) −→ 0

(11.1.7.2)

generalizing (11.1.3.1), an exact triangle
β gf,Iw (L/K, Z) ⊗R ΓL −→ RΓ gf (Z ⊗R R)/J 2 < −1 > −→ RΓ gf,Iw (L/K, Z)−→ gf,Iw (L/K, Z)[1] ⊗R ΓL RΓ RΓ R L R (11.1.7.3) generalizing (11.1.3.2) and pairings L

β⊗id e gf,Iw (L/K, X) ⊗R[[Γ ]] RΓ gf,Iw (L/K, Y )ι −−−−→ hπ,L/K : RΓ L ∪π,0 [1]

L

gf,Iw (L/K, X)[1] ⊗R[[Γ ]] RΓ gf,Iw (L/K, Y )ι ⊗R ΓL −−−−→ ωR[[Γ ]] [−2] ⊗R ΓL −−→RΓ R R L L i e e f,Iw e j (L/K, Y )ι −→ H i+j−2 (ωR ) ⊗R R[[ΓL ]] ⊗R ΓL hπ,L/K,i,j : H (L/K, X) ⊗R[[ΓL ]] H R. f,Iw

(11.1.7.4)

(11.1.7.5)

As in 11.1.3, the exact triangle (11.1.7.3) can also be obtained by applying L

gf,Iw (K∞ /K, Z)⊗ (−) RΓ R to the exact triangle JL /JL2 −→ R/JL2 −→ R[[ΓL ]] −→ JL /JL2 [1]. (11.1.8) What is the relation between (11.1.7.4-5) and the pairings constructed in 11.1.4? In the special case when the exact sequence 279

0 −→ ΓL −→ Γ −→ ΓL −→ 0

(11.1.8.1)

L

splits (i.e. when Γ = Γ × ΓL ), then the construction from 11.1.7 reduces to that in 11.1.3, applied to • π : FΓL (X) ⊗R[[ΓL ]] FΓL (Y )ι −→ ωR (1) ⊗R R[[ΓL ]]

0 ΓL 0 • and K∞ = K∞ instead of π : X ⊗R Y −→ ωR (1) and K∞ , as Gal(K∞ /K) = ΓL ; cf. 11.1.6. If the sequence (11.1.8.1) does not split, then the following Proposition shows that at least the height pairings on cohomology (11.1.7.5) reduce to those from 11.1.4.

(11.1.9) Proposition. Let {Lα /K} be the set of all finite subextensions of L/K. Then the height pairing e hπ,L/K,i,j :

   ι
j i e e lim Hf (Lα , X) ⊗R[[ΓL ]] lim Hf (Lα , Y ) −→ lim H i+j−2 (ωR ) ⊗R ΓL R ⊗R R[Gal(Lα /K)] ← ← ← α− α− α−

is given by the formula  e hπ,L/K,i,j ((xα ), (yα )) = 

X

 e hπα ,i,j (xα , σyα ) ⊗ [σ] ,

σ∈Gal(Lα /K)

where

α

e e i (Lα , X) ⊗R H e j (Lα , Y ) −→ H i+j−2 (ωR ) ⊗R (Gal(K∞ /Lα ) ⊗Zp R) hπα ,i,j : H f f

is the height pairing 11.1.4 attached to the extension K∞ /Lα (if Lα is sufficiently large, then the flatness condition (F l(Gal(K∞ /Lα )) holds, thanks to the assumption (F l(ΓL ))). Proof. This follows from the definitions and the formula proved in Proposition 8.9.14. (11.2) Symmetry of the height pairing (11.2.1) Theorem. (i) If all complexes X, Y, Xv+ , Yv+ are cohomologically bounded above, then the diagram e hπ :

L

gf (X)⊗R RΓ gf (Y ) −−−−→ RΓ  s y 12 L

e gf (Y )⊗R RΓ gf (X) −−−−→ hπ◦s12 : RΓ is commutative in Df−t (R Mod). (ii) The pairings e hπ,i,j satisfy

ωR [−2] ⊗R ΓR k ωR [−2] ⊗R ΓR

e hπ,i,j (x, y) = (−1)ij e hπ◦s12 ,j,i (y, x).

(11.2.2) Corollary. If X = Y , Xv+ = Yv+ (v ∈ Σ) and π ◦ s12 = c · π with c = ±1, then the bilinear form e e f1 (X) ⊗R H e f1 (X) −→ H 0 (ωR ) ⊗R ΓR hπ,1,1 : H is symmetric (resp., skew-symmetric) if c = −1 (resp., c = +1). Proof of Theorem 11.2.1. It is enough to prove (i), since (ii) follows from (i) applied to τ≤N Z,τ≤N Zv+ (Z = X, Y ) for big enough N >> 0. The first step is to observe that the “Bockstein map” gf (Z) −→ RΓ gf (Z)[1] ⊗R ΓR β : RΓ

(Z = X, Y )

in Df t (R Mod) can be represented by a canonical morphism of complexes e• (Z) −→ C e • (Z)[1] ⊗R ΓR . βE : C f f 280

This follows from the construction 1.3.11 applied to the complexes M

C= 0

0

0

A = Ccont (GK,S , Z ) • A00 = Ccont (GK,S , Z 00 ) •

v∈Sf

B = US+ (Z)

• A = Ccont (GK,S , Z)

B = B 00 =

• Ccont (Gv , Z)

M

0

C =

US+ (Z 0 ) US+ (Z 00 )

• Ccont (Gv , Z 0 )

v∈Sf

M

C 00 =

• Ccont (Gv , Z 00 )

v∈Sf

for Z 0 = FΓ (Z) ⊗R R/J 2 ,

Z 00 = Z ⊗R ΓR

and the morphisms of complexes g, g 0 , g 00 = i+ S (−),

f, f 0 , f 00 = resSf , as ef• (Z), E=C

ef• (Z 0 ), E0 = C

ef• (Z 00 ) = C ef• (Z) ⊗R ΓR E 00 = C

in the notation of 1.3.11. As a result, the discussion in 1.3.11 yields the data (1.3.8.1-2) for the complexes • Aj = Ccont (GK,S , Zj ),

Bj = US+ (Zj ),

M

Cj =

• Ccont (Gv , Zj )

v∈Sf

from 6.3.1 (where j = 1, 2 and Z1 = X, Z2 = Y ). In fact, in the present situation we can make the formulas (1.3.11.1-2) very explicit, as follows. The canonical projection prΓ : GK,S = Gal(KS /K)  Gal(K∞ /K) = Γ and the map Γ −→ ΓR

(γ 7→ γ ⊗ 1)

define a tautological 1-cocycle z : GK,S  Γ ,→ ΓR ,

1 z ∈ Ccont (GK,S , ΓR ),

dz = 0.

For each v ∈ Sf , a fixed embedding K ,→ K v induces primes v∞ |v of K∞ and vα |v of Kα (for all finite subextensions Kα /K of K∞ /K). The localization of z at v, which is equal to z

resv (z) : Gv ,→ GK,S −−→ΓR , factors through zv : Gv  Γv −→ (Γv )R := Γv ⊗Zp R, where Γv = Gal((K∞ )v∞ /Kv ) = ← lim − Gal((Kα )vα /Kv ) ⊂ Γ α

is the decomposition group of v in K∞ /K. 281

(11.2.3) Lemma. The Bockstein map β in the exact triangle β

RΓcont (G, Z) ⊗R ΓR −−→RΓcont (G, (Z ⊗R R/J 2 ) < −1 >)−−→RΓcont (G, Z)−−→ β

−−→RΓcont (G, Z)[1] ⊗R ΓR

(Z = X, Y ; G = GK,S , Gv )

is induced by the morphism of complexes • • β : Ccont (GK,S , Z) −→ Ccont (GK,S , Z ⊗R ΓR )[1]

c 7→ −z ∪ c (and similarly for Gv , when z has to be replaced by resv (z)). Proof. It is enough to consider the case G = GK,S and Z = Z n being concentrated in degree n ∈ Z. For each i ≥ 0, the epimorphism i+n i+n i i Ccont (G, FΓ (Z) ⊗R R/J 2 ) = Ccont (G, (Z n ⊗R R/J 2 ) < −1 >) −→ Ccont (G, Z n ) = Ccont (G, Z)

has a canonical section s given by (s(c))(g1 , . . . , gi ) = c(g1 , . . . , gi ) ⊗ 1. A short calculation shows that (−1)n (d(s(c)) − s(d(c)))(g1 , . . . , gi+1 ) = g1 c(g2 , . . . , gi+1 ) ⊗ (prΓ (g1 )−1 − 1) (mod J 2 ) = = −g1 c(g2 , . . . , gi+1 ) ⊗ prΓ (g1 ) = −(−1)n (z ∪ c)(g1 , . . . , gi+1 ), where the sign (−1)n on the L.H.S. (resp., the R.H.S.) comes from the sign rule 3.4.1.3 (resp., 3.4.5.2). It follows that (d ◦ s − s ◦ d)(c) = −z ∪ c, which proves the result, by 1.1.4. (11.2.4) Lemma. For each v ∈ Σ0 , (i) the Bockstein map β in the exact triangle β

RΓur (Gv , Z) ⊗R ΓR −−→RΓur (Gv , (Z ⊗R R/J 2 ) < −1 >)−−→RΓur (Gv , Z)−−→ β

−−→RΓur (Gv , Z)[1] ⊗R ΓR

(Z = X, Y )

is induced by the morphism of complexes β : Uv+ (Z) −→ Uv+ (Z ⊗R ΓR )

β(mn , mn−1 , m0n−1 , mn−2 ) = (0, 0, fv (mn ) ⊗ z(fv ), θv (mn−1 ) ⊗ z(fv )), using the notation of 7.6.2. (ii) The morphism β is also equal to the restriction of the composite morphism µ

λ[1]

β

• • Uv (Z)−−→Ccont (Gv , Z)−−→Ccont (Gv , Z ⊗R ΓR )[1]−−→Uv (Z)[1] ⊗R ΓR

to the subcomplex Uv+ (Z) ⊂ Uv (Z), where β(c) = −z ∪ c is as in Lemma 11.2.3. Proof. The condition (U ) implies that, in the notation of 7.6.2, we have Uv+ ((Z ⊗R R/J 2 ) < −1 >) = (Uv+ (Z) ⊗R R/J 2 ) < −1 > 282

(and similarly for Uv (−) = U (−)). The epimorphism (Uv+ (Z) ⊗R R/J 2 ) < −1 >−→ Uv+ (Z) has a canonical section s given by s(c) = c ⊗ 1 (and similarly for Uv (−)). The explicit formula for the differential on Uv (−) from 7.6.2 gives, as in the proof of Lemma 11.2.3,

(d ◦ s − s ◦ d)(mn , mn−1 , m0n−1 , mn−2 ) = (0, 0, −fv (mn ) ⊗ prΓ (fv )−1 , −θv (mn−1 ) ⊗ prΓ (fv )−1 ) = = (0, 0, fv (mn ) ⊗ z(fv ), θv (mn−1 ) ⊗ z(fv )), as claimed in (i). A short calculation based on the explicit formulas for λ and µ from Proposition 7.2.3-4 shows that the composite map λ[1] ◦ β ◦ µ is given by the same formula, proving (ii). (11.2.5) To sum up, the construction from 1.3.11 yields morphisms of complexes βW : W −→ W [1] ⊗R ΓR ,

(W = A, B, C, E),

e• (Z) (Z = X, Y ) are as above and where A, B, C and E = C f βE (a, b, c) = (βA (a), βB (b), −βC (c) − u(a) + v(b)), with i a ∈ Ccont (GK,S , Z)

βA (a) = −z ∪ a, βC (c)v = −resv (z) ∪ cv , ( −resv (z) ∪ bv , βB (b)v = −λ(resv (z) ∪ µ(bv )),

i−1 cv ∈ Ccont (Gv , Z),

v ∈ Σ, v ∈ Σ0 ,

bv ∈

v ∈ Sf

i Ccont (Gv , Zv+ )

i bv ∈ Cur (Gv , Z).

As the section s in the proof of Lemma 11.2.3 is functorial in G, we have u = 0. For the same reason, the Σ-components of v(b) vanish, while the Σ0 -components can be described explicitly using the recipe from 1.3.11. We obtain, therefore, morphisms βj,W for W = A, B, C and homotopies uj = 0, vj (where j = 1 resp., j = 2 for Z = X resp., Z = Y ) as in (1.3.8.1-2). (11.2.6) The next step is to verify the conditions (1.3.8.3-5), with the products ∪A , ∪B = 0 and ∪C as in 6.3.1. In (1.3.8.3) we can take hB = 0, while hA and hC are constructed using the following diagram, in which • C • (−) stands for Ccont (G, −) with G = GK,S or Gv : 283

id

C • (X) ⊗R C • (Y )

−−−−→

C • (X) ⊗R C • (Y )

k

k id

C • (X) ⊗R R ⊗R C • (Y )  id⊗(−z)⊗id y

−−−−→

C • (X) ⊗R (C • (ΓR )[1]) ⊗R C • (Y )   y∪⊗id

12 −−−−→

C • (X ⊗R ΓR )[1] ⊗R C • (Y )  ∪ y

−−−−→

id

C • (X ⊗R ΓR )[1] ⊗R C • (Y )  ∪ y

C • (X ⊗R Y ⊗R ΓR )[1]

−−−−→

id

C • (X ⊗R Y ⊗R ΓR )[1]

s

R ⊗R C • (X) ⊗R C • (Y )  (−z)⊗id⊗id y

⊗id

C • (ΓR )[1] ⊗R C • (X) ⊗R C • (Y )   y∪⊗id

(11.2.6.1)

The composite map C • (X) ⊗R C • (Y ) −→ C • (X ⊗R Y ⊗R ΓR )[1] in the left (resp., right) column is equal to id ⊗ β2 (resp., β1 ⊗ id). The first, second and the fourth squares from the top are all commutative, while the commutative diagram s12 ◦(T ⊗T )

C • (X) ⊗R (C • (ΓR )[1])  ∪ y

−−−−−−→

C • (ΓR )[1] ⊗R C • (X)  ∪ y

C • (X ⊗R ΓR )[1]

←−

T

C • (X ⊗R ΓR )[1]

together with the bifunctorial homotopy a : id − in (11.2.6.1) is commutative up to the homotopy k ⊗ id : ∪ ⊗ id −

T from 3.4.5.5 show that the third square from the top (∪ ⊗ id) ◦ (s12 ⊗ id),

where k = −a ? (∪ ◦ s12 ◦ (T ⊗ T )) − (T ◦ ∪ ◦ s12 ) ? (a ⊗ a)1 . We define the homotopy hW : id ⊗ β2,W −

β1,W ⊗ id

(W = A, C)

by the formula hW = ∪ ? (k ⊗ id) ? (id ⊗ (−z) ⊗ id), with G = GK,S (resp., G = Gv , v ∈ Sf ) if W = A (resp., W = C). As hf = u 1 = u 2 = 0 and f3 [1] ? hA = hC ? (f1 ⊗ f2 ) (the last equality following from the bifunctoriality of the homotopy a : id − hA and hC ), the condition (1.3.8.4) is satisfied with Hf = 0. 284

(11.2.6.2) T used in the construction of

It remains to verify the condition (1.3.8.5), which boils down to the statement that, for each v ∈ Sf , the v-component mv of the homotopy

miv

m = ∪C [1] ? (v1 ⊗ g2 ) − ∪C [1] ? (g1 ⊗ v2 ) − hC ? (g1 ⊗ g2 )
i
i−1 II • • : Uv+ (X) ⊗R Uv+ (Y ) −→ τ≥2 Ccont (Gv , ωR (1)) ⊗R ΓR [1]

is 2-homotopic to zero. For v ∈ Σ we have mv = −hC ? (g1 ⊗ g2 ) = −hC ? (g1 ⊗ g2 ) + g3 [1] ? hB = 0, as in (11.2.6.2). For v ∈ Σ0 , the only possibly non-zero component of mv is 1 1 m2v : Hur (X) ⊗R Hur (Y ) −→ H 0 (ωR ) ⊗R ΓR .

It is enough to show that m2v = 0 (hence (1.3.8.5) holds with Hg = 0). Firstly, m2v factors through 1 1 Hur (X) ⊗R Hur (D(X)(1)),

(11.2.6.3)

so we can assume that Y = D(X)(1) and π = ev2 . The R-module (11.2.6.3) is annihilated by N v − 1, which proves that m2v = 0 in the case when R is flat over Zp . The general case follows by observing that the morphism 1 1 adj(m2v ) : Hur (X) −→ HomR (Hur (D(X)(1)), H 0 (ωR ) ⊗R ΓR )

is given by a “universal” formula which does not depend on R, as all homotopies hC , v1 , v2 are given by such universal formulas. This completes the verification of the conditions (1.3.8.3-5). (11.2.7) We are now ready to conclude the proof of Theorem 11.2.1(i). We apply Proposition 1.3.10 to e• (X), E2 = C e• (Y ) and E3 = ω • [−3] (as in 6.3.1). This is legitimate, since Greenberg’s local E1 = C R f f conditions admit the transposition data (1.3.5.1-7), by 6.7.8 and Proposition 7.7.3. The morphisms Tj for • j = 1, 2 (resp., for j = 3) are homotopic to the identity by Proposition 6.5.4(i) (resp., because ωR is a complex of injective R-modules). Finally, we use our earlier observation that the Bockstein map gf (Z) −→ RΓ gf (Z)[1] ⊗R ΓR β : RΓ

(Z = X, Y )

is represented by the morphism of complexes e • (Z) −→ C e• (Z)[1] ⊗R ΓR βE : C f f given by the formula from Proposition 1.3.9(i). The statement of Theorem 11.2.1(i) then follows from Proposition 1.3.10. gf (X) and (11.2.8) In the definition of the height pairings in 11.1.4 we let the Bockstein map β act on RΓ i e (X). If we chose instead the opposite convention and define H f

∪π,0 [1] id⊗β e gf (X)⊗R RΓ gf (Y )−−−−→ gf (X) ⊗R (RΓ gf (Y )[1]) ⊗R ΓR −−−−→ω h0π : RΓ RΓ R [−2] ⊗R ΓR , L

L

then the proof of Theorem 11.2.1 (namely the appeal to Proposition 1.3.9) shows that e h0π = e hπ (and 0 e hπ,i,j = e hπ,i,j ). (11.2.9) On the other hand, if we replace FΓ (X)/J 2 = (X ⊗R R/J 2 ) < −1 > in 11.1.3 by FΓ (X)ι /J 2 = (X ⊗R R/J 2 ) < 1 >, then the map β is replaced by −β and the height pairings e h, e hπ,i,j change sign. (11.3) Comparison with classical p-adic height pairings In contrast to various classical constructions of Qp -valued height pairings on Selmer groups ([Ne 2], [PR 4,5]), the definition in 11.1 requires no additional assumptions apart from the existence of orthogonal Greenberg’s 285

local conditions Xv+ ⊥π Yv+ (v ∈ Σ). In this (and the following) section we relate the pairing e hπ,1,1 for R = Zp to the height pairing constructed in terms of universal norms. ∼

(11.3.1) Assume that Γ = Gal(K∞ /K) −→ Zrp (r ≥ 1) and that we are given the following data: (11.3.1.1) Two Qp -representations X, Y of GK,S (i.e. finite-dimensional vector spaces over Qp with a continuous Qp -linear action of GK,S ). (11.3.1.2) A GK,S -map π : X ⊗Qp Y −→ Qp (1) such that

adj(π) : X −→ HomQp (Y, Qp (1)) = Y ∗ (1)

is an isomorphism. (11.3.1.3) For each v ∈ Sf , Qp -subrepresentations of Gv Xv+ ⊂ X, such that

(∀v|p) Xv+ ⊥⊥π Yv+ ,

Yv+ ⊂ Y (∀v - p) Xv+ = Yv+ = 0.

Putting Xv− = X/Xv+,

Yv− = Y /Yv+

(v ∈ Sf ),

it follows that adj(π) induces isomorphisms ∼

Xv± −→ HomQp (Yv∓ , Qp (1)) = (Yv∓ )∗ (1)

(v|p).

We also assume that the following conditions hold: (11.3.1.4) (∀v ∈ Sf ) H 0 (Gv , X) = H 0 (Gv , Y ) = 0. 0 (11.3.1.5) (∀v|p) H (Gv , Xv− ) = H 0 (Gv , Yv− ) = 0. Observe that (11.3.1.1-5) hold in the context of [Ne 2, 6.9] (where X, Y were denoted by V, V ∗ (1)). (11.3.2) There are two natural Greenberg’s local conditions for Z = X, Y : ∆(Z) with Σ = Sf (resp., ∆0 (Z) with Σ = {v|p}) and the subrepresentations Zv+ (v ∈ Σ) as in (11.3.1.3). The assumptions (11.3.1.2-4) imply that ∼

RΓcont (Gv , Z) = RΓur (Gv , Z) −→ 0

(v ∈ Sf , v - p; Z = X, Y )

in D(Qp Mod); it follows that the canonical maps gf (Z) := RΓ gf (GK,S , Z; ∆(Z)) −→ RΓ gf (GK,S , Z; ∆0 (Z)) RΓ are isomorphisms in

Dfb t (Qp Mod).

(Z = X, Y )

The condition (11.3.1.5) implies that, under the canonical map gf (Z) −→ RΓcont (GK,S , Z), RΓ

the cohomology group e 1 (Z) := H e 1 (GK,S , Z; ∆(Z)) H f f is identified with the Selmer group defined by Greenberg [Gre 2,3]   M 1 1 Sel(Z) = Ker Hcont (GK,S , Z) −→ Hcont (Gv , Zv− ) =  1 = Ker Hcont (GK,S , Z) −→

v∈Sf

M



1 Hcont (Gv , Zv− )

v|p

286

(Z = X, Y ).

(11.3.3) Under the assumptions (11.3.1.1-5) there is a canonical (up to a sign) height pairing ([Ne 2], [PR 4,5]) hnorm = hnorm : Sel(X) ⊗Qp Sel(Y ) −→ Γ ⊗Zp Qp , π the definition of which is recalled in 11.3.7 below. Fix a GK,S -invariant Zp -lattice T (X) ⊂ X. This determines a GK,S -invariant Zp -lattice T (Y ) ⊂ Y such that adj(π) defines an isomorphism ∼

adj(π) : T (X) −→ HomZp (T (Y ), Zp (1)) = T (Y )∗ (1), i.e. π : T (X) ⊗Zp T (Y ) −→ Zp (1) is a perfect duality over R = Zp . For each v ∈ Sf , put + T (Z)+ v = T (Z) ∩ Zv ,

+ T (Z)− v = T (Z)/T (Z)v

(Z = X, Y );

then adj(π) induces isomorphisms ∼

∓ ∓ ∗ T (X)± v −→ HomZp (T (Y )v , Zp (1)) = (T (Y )v ) (1)

(v|p).

+ Let ∆(T (Z)) (Z = X, Y ) be Greenberg’s local conditions with Σ = Sf and the above T (Z)+ v (i.e. T (Z)v = 0 if v - p). Then the Selmer complexes

gf (T (Z)) := RΓ gf (GK,S , T (Z); ∆(T (Z))) ∈ Dfb t (Zp Mod) RΓ

(Z = X, Y )

satisfy ∼ g gf (T (Z)) ⊗Z Qp −→ RΓ RΓf (Z) p

(Z = X, Y ),

hence the pairing from 11.1.4 e e f1 (T (X)) ⊗Zp H e f1 (T (Y )) −→ Γ hπ,1,1 : H induces a pairing e e 1 (X) ⊗Qp H e 1 (Y ) −→ Γ ⊗Zp Qp . hπ,1,1 : Sel(X) ⊗Qp Sel(Y ) = H f f The main goal of Sect. 11.3 is to show that e hπ,1,1 = −hnorm . π In order to simplify the arguments we work exclusively with the Qp -representations Z, Zv+ , not with the lattices T (Z), T (Z)+ v (Z = X, Y ). This means that we apply the constructions in 11.1.3-4 to the exact sequences e• (Z ⊗Zp Γ) −→ C e• (FΓ (Z) ⊗Λ Λ/J 2 ) −→ C e• (Z) −→ 0, 0 −→ C f f f obtained from e • (T (Z) ⊗Zp Γ) −→ C e• (FΓ (T (Z)) ⊗Λ Λ/J 2 ) −→ C e • (T (Z)) −→ 0 0 −→ C f f f by taking the tensor product − ⊗Zp Qp . Above, Λ = Zp [[Γ]] and FΓ (Z) := FΓ (T (Z)) ⊗Zp Qp . 287

(11.3.4) Universal norms. Let v be a non-archimedean prime of K, V a Qp -representation of Gv and T ⊂ V a Gv -stable Zp -lattice. Fixing an embedding K ,→ K v , we obtain a prime vα |v in each finite subextension Kα /K of K∞ /K. We define the universal norms with respect to K∞ /K at v as \
cor 1 1 1 N∞ Hcont (Gv , T ) = Im Hcont (Gvα , T )−−→Hcont (Gv , T ) = α


1 1 = Im Hcont (Gv , FΓv (T )) −→ Hcont (Gv , T ) =
1 1 = Im Hcont (Gv , FΓ (T )) −→ Hcont (Gv , T ) . We also define 1 1 1 N∞ Hcont (Gv , V ) = (N∞ Hcont (Gv , T )) ⊗Zp Qp ⊂ Hcont (Gv , V )

(this vector space over Qp depends only on V , not on T ). (11.3.5) Local reciprocity maps. Let v be a non-archimedean prime of K. The local reciprocity map recv : Kv∗ −→ Gab v is related to the invariant map ∼

invv : H 2 (Gv , Q/Z(1)) = Br(Kv ) −→ Q/Z by the formula χ(recv (a)) = invv (a ∪ δχ),

(11.3.5.1)

where a ∈ Kv∗ ,

χ ∈ H 1 (Gv , Q/Z) = Homcont (Gab v , Q/Z)

and δχ ∈ H 2 (Gv , Z) is the coboundary attached to the exact sequence 0 −→ Z −→ Q −→ Q/Z −→ 0 ([Se 1], Prop. XI.2). If we work with cohomology with finite coefficients, i.e. with the following exact sequences of (discrete) Gv -modules ∗ n

∗

0 −→ µn −→ K v −→K v −→ 0 n

0 −→ Z−→Z −→ Z/nZ −→ 0 and elements ∗

a ∈ Kv∗ = H 0 (Gv , K v ),

δa ∈ H 1 (Gv , µn )

χ ∈ H 1 (Gv , Z/nZ) = Homcont (Gab v , Z/nZ),

δχ ∈ H 2 (Gv , Z),

we obtain from (11.3.5.1) and the vanishing of 0 = invv (δ(a ∪ χ)) = invv (δa ∪ χ + a ∪ δχ) another useful formula χ(recv (a)) = −invv (δa ∪ χ) = invv (χ ∪ δa). Taking n = pr and passing to the projective limit, we obtain isomorphisms 288

(11.3.5.2)

∼

b p (= lim Kv∗ ⊗ Z/pr Z) −→ Gab b recv : Kv∗ ⊗Z v ⊗Zp ←r− ∼ 1 b p −→ δ : Kv∗ ⊗Z Hcont (Gv , Zp (1)).

Recall from 11.2 the tautological projection b zv : Gab v ⊗Zp  Γv ; denote by `v the composite map 1 `v = zv ◦ recv ◦ δ −1 : Hcont (Gv , Zp (1))  Γv .

The formula (11.3.5.2) implies that 1 ∀c ∈ Hcont (Gv , Zp (1)).

`v (c) = invv (zv ∪ c),

(11.3.5.3)

As 1 Ker(`v ) = N∞ Hcont (Gv , Zp (1)),

the map `v induces an isomorphism ∼

1 `v : Hcont (Gv , Zp (1))/N∞ −→ Γv .

(11.3.6) Lemma. For each v|p, 1 1 Hcont (Gv , Zv+ ) = N∞ Hcont (Gv , Zv+ )

(Z = X, Y ).

Proof. The statement is trivial if Γv = 0, so we can assume that Γv 6= 0. In this case H 0 (Gv , (Zv+ )∗ (1)) = 0 by (11.3.1.5), and we conclude by Corollary 8.11.8. (11.3.7) Definition of hnorm . Let 1 x ∈ Ccont (GK,S , X),

1 y ∈ Ccont (GK,S , Y ),

dx = 0,

dy = 0

be 1-cocycles with cohomology classes contained in 1 [x] ∈ Sel(X) ⊂ Hcont (GK,S , X),

1 [y] ∈ Sel(Y ) ⊂ Hcont (GK,S , Y ).

Then y defines an extension of GK,S -modules σ∗ (1)

0 −→ Y −−→Yy −→ Qp −→ 0 with extension class [Yy ] = [y]. Dualizing, we obtain an extension σ

0 −→ Qp (1) −→ Yy∗ (1)−→X −→ 0. For v ∈ Sf , put −1 Yy∗ (1)+ (Xv+ ), v = σ

∗ ∗ + Yy∗ (1)− v = Yy (1)/Yy (1)v .

The coboundary map δ in the exact cohomology sequence σ

δ

∗ 1 1 1 2 0 −→ Hcont (GK,S , Qp (1)) −→ Hcont (GK,S , Yy∗ (1))−−→H cont (GK,S , X)−−→Hcont (GK,S , Qp (1))

is equal, up to a sign, to the cup product with [y] under π : X ⊗Qp Y −→ Qp (1). As the map 289

M

2 resSf : Hcont (GK,S , Qp (1)) −→

2 Hcont (Gv , Qp (1))

v∈Sf

is injective and Xv+ ⊥π Yv+ for all v ∈ Sf , we have

δ([x]) = 0. Fix a lift of [x]

1 [b x] ∈ σ∗−1 ([x]) ⊂ Hcont (GK,S , Yy∗ (1)); 1 ∗ it is determined up to an element of Hcont (GK,S , Qp (1)) = OK,S ⊗ Z Qp .

For each v ∈ Sf , denote by 1 [b xv ] = resv ([b x]) ∈ Hcont (Gv , Yy∗ (1))

the localization of [b x] at v. The assumptions (11.3.1.3-5) imply that the following diagram has exact rows and columns:

0   y 0 −→

1 Hcont (Gv , Qp (1)) −→

k

1 Hcont (Gv , Yy∗ (1)+ v ) −→  + yiv 1 Hcont (Gv , Yy∗ (1))

1 Hcont (Gv , Xv+ ) −→  + yiv

−→

1 Hcont (Gv , X)

δv

2 Hcont (Gv , Qp (1))

k

2 Hcont (Gv , Qp (1)). (11.3.7.1) As δv is (again up to a sign) given by the cup product with [yv ] = resv ([y]) (hence δv ([xv ]) = 0) and + + 1 + [xv ] = i+ v ([xv ]) for some [xv ] ∈ Hcont (Gv , Xv ) by assumption, an easy diagram chase shows that there is a (unique) cohomology class

0 −→

1 Hcont (Gv , Qp (1)) −→

0   y

1 ∗ + [b x+ v ] ∈ Hcont (Gv , Yy (1)v )

satisfying

i+ x+ xv ]. v ([b v ]) = [b We claim that the following diagram also has exact rows and columns: 290

−→

0   y 1 Hcont (Gv , FΓv (Qp (1)))   y

−→

pr∗

1 Hcont (Gv , Qp (1))   y

1 Hcont (Gv , FΓv (Yy∗ (1)+ v ))  F (σ) y Γv ∗

−→

pr∗

1 Hcont (Gv , Yy∗ (1)+ v)  σ y ∗

1 Hcont (Gv , FΓv (Xv+ ))   y

−→

pr∗

1 Hcont (Gv , Xv+ )  δ yv

2 Hcont (Gv , Qp (1))

−→

id

2 Hcont (Gv , Qp (1))

k

`

v −→

Γv ⊗ Z p Q p

(11.3.7.2) −→

0

k id

Qp

−→

Qp

This follows from the fact that Im(pr∗ ) = N∞ and the description of the universal norms N∞ for Qp (1) (resp., for Xv+ ) given in 11.3.5 (resp., in Lemma 11.3.6). This implies that there is a cohomology class 

 1 ∗ + FΓv (x+ v ) ∈ Hcont (Gv , FΓv (Yy (1)v ))

which maps to 1 + [x+ x+ v ] = σ∗ ([b v ]) ∈ Hcont (Gv , Xv )

under the map pr∗ ◦ FΓv (σ)∗ . The difference uv = [b x+ v ] − pr∗

 
FΓv (x+ v)

is an element of 1 uv ∈ Hcont (Gv , Qp (1))

and the value `v (uv ) ∈ Γv ⊗Zp Qp does not depend on the choice of [FΓv (x+ v )]. If v - p, then Xv+ = 0, Yy∗ (1)+ v = Qp (1) and we have 1 ∗b uv = [b x+ v ] ∈ Hcont (Gv , Qp (1)) = Kv ⊗Qp ;

the latter group is isomorphic to Qp under the valuation map ∼ b p −→ v ⊗ 1 : Kv∗ ⊗Q Qp .

The height pairing hnorm is defined by the formula π X hnorm ([x], [y]) = `v (uv ) ∈ Γ ⊗Zp Qp . π v∈Sf 1 This is well-defined, since another choice of [b x] replaces each uv by uv + u for some u ∈ Hcont (GK,S , Qp (1)) and

291

X

`v (u) = 0

v∈Sf ∗ for such a global cohomology class u ∈ OK,S ⊗Z Qp . (11.3.8) If v ∈ Sf is unramified in K∞ /K, then the map

zv ◦ recv : Kv∗ −→ Γv factors through the valuation v : Kv∗ −→ Z. As the direct sum of the valuations (v⊗id) M ∗ OK,S ⊗Z Qp −−→ Qp v∈Sf

is surjective, it follows that there is always a lift [b x] of [x] with vanishing `v (uv ) = 0 for all v ∈ Sf unramified in K∞ /K (in particular, such that [b x+ ] = 0 for all v ∈ Sf , v - p). v (11.3.9) Theorem. Under the assumptions (11.3.1.1-5), the height pairings e 1 (X) ⊗Qp H e 1 (Y ) −→ Qp hnorm ,e hπ,1,1 : Sel(X) ⊗Qp Sel(Y ) = H π f f are related by

hnorm = −e hπ,1,1 . π

e 1 (X) and [yf ] ∈ H e 1 (Y ) be the cohomology classes of 1-cocycles Proof. Let [xf ] ∈ H f f e1 xf = (x, (x+ v ), (λv )) ∈ Cf (X),

e1 (Y ), yf = (y, (yv+ ), (µv )) ∈ C f

dxf = 0,

dyf = 0,

where 1 x ∈ Ccont (GK,S , X),

dx = 0

x+ v

dx+ v =0

1 ∈ Ccont (Gv , Xv+ ), 0 λv ∈ Ccont (Gv , X) =

X,

+ i+ v (xv )

(v ∈ Sf )

= xv + dλv

(v ∈ Sf )

+ and similarly for y, yv+ , µv . Recall that Xv+ = Yv+ = 0 if v - p, hence x+ v = yv = 0 for such v. It follows from Lemma 11.2.3 that

e 2 (X ⊗Zp Γ) β([xf ]) ∈ H f is represented by the 2-cocycle e2 β(xf ) = (−z ∪ x, (−zv ∪ x+ v ), (zv ∪ λv )) ∈ Cf (X ⊗Zp Γ), 1 where z ∈ Ccont (GK,S , Γ) is the tautological cocycle given by the projection GK,S  Γ. By definition, ∼ 3 e hπ,1,1 ([xf ], [yf ]) = [β(xf ) ∪π,0 yf ] ∈ Hc,cont (GK,S , Qp (1) ⊗Zp Γ) −→ Γ ⊗Zp Qp . 1 1 ∗ + (11.3.10) Lemma. Represent the cohomology classes [b x] ∈ Hcont (GK,S , Yy∗ (1)), [b x+ v ] ∈ Hcont (Gv , Yy (1)v ) norm 1 ∗ + 1 ∗ (v ∈ Sf ) from the construction of hπ by 1-cocycles x b ∈ Ccont (GK,S , Yy (1)), x bv ∈ Ccont (Gv , Yy (1)+ v) + + 0 ∗ ∗ b b (v ∈ Sf ). As iv (b xv ) = x bv + dλv for suitable λv ∈ Ccont (Gv , Yy (1)) = Yy (1) (v ∈ Sf ), we obtain a 1-cocycle bv )) ∈ C e 1 (Y ∗ (1)). Then (b x, (b x+ ), ( λ v y f

b β(b x, (b x+ a, (bbv ), (b cv )) ∈ Cf2 (Yy∗ (1)), v ), (λv )) = (b where

bbv = −zv ∪ uv ∈ C 2 (Gv , Qp (1) ⊗Z Γ) p cont 292

(v ∈ Sf )

and uv is as in 11.3.7. Proof. The diagram (11.3.7.2) has an analogue in which each FΓv is replaced by FΓ and Qp at the bottom of the left column is replaced by Zp [[Γ/Γv ]] ⊗Zp Qp . This implies that, for each v ∈ Sf , the element 1 ∗ + x b+ v − uv ∈ Ccont (Gv , Yy (1)v )

is equal to the image pr∗ (FΓ (x+ v )) of a 1-cocycle 1 ∗ + FΓ (x+ v ) ∈ Ccont (Gv , FΓ (Yy (1)v )),


d FΓ (x+ v ) = 0.

The definition of β and Lemma 11.2.3 then yield bbv = β(b x+ v ) = β(uv ) = −zv ∪ uv , as claimed. e 2 (X ⊗Zp Γ) is represented by a 2-cocycle of the form e = (11.3.11) Lemma. Assume that β([xf ]) ∈ H f e 2 (Y ∗ (1) ⊗Zp Γ) is also a 2-cocycle (i.e. db (a, (bv ), (cv )) = σ∗ (b e), where eb = (b a, (bbv ), (b cv )) ∈ C e = 0). Then y f [β(xf ) ∪ yf ] =

X

b invv (i+ bv + i+ v (bv ) ∪ y v (bv ) ∪ µv ),

v∈Sf 0 0 where yb ∈ Ccont (GK,S , Yy ) = Yy maps to 1 ∈ Ccont (GK,S , Qp ) = Qp and satisfies db y = (σ ∗ (1))∗ (y). If, in addition, bbv ∈ C 2 (Gv , Qp (1) ⊗Z Γ) ⊂ C 2 (Gv , Y ∗ (1)+ ⊗Z Γ) (∀v ∈ Sf ), p p cont cont y v

then [β(xf ) ∪ yf ] =

X

invv (bbv ) ∈ Γ ⊗Zp Qp .

v∈Sf

Proof. We have 3 e ∪1 yf = (a, (bv ), (cv )) ∪1 (y, (yv+ ), (µv )) = (a ∪ y, (cv ∪ yv + i+ v (bv ) ∪ µv )) ∈ Cc,cont (GK,S , Qp (1) ⊗Zp Γ),

where ∪1 denotes the product ∪r,h from Proposition 1.3.2(i) for r = 1 and h = 0. As 3 a ∪ y ∈ Ccont (GK,S , Qp (1) ⊗Zp Γ),

d(a ∪ y) = 0

3 and Hcont (GK,S , Qp (1) ⊗Zp Γ) = 0, we have

a ∪ y = dε,

2 ε ∈ Ccont (GK,S , Qp (1) ⊗Zp Γ).

Then 3 d(ε, 0) = (dε, (−εv )) ∈ Cc,cont (GK,S , Qp (1) ⊗Zp Γ)

e ∪1 yf − d(ε, 0) = (0, (cv ∪ yv + i+ v (bv ) ∪ µv + εv )), which implies that the cohomology class ∼

3 [e ∪1 yf ] ∈ Hc,cont (GK,S , Qp (1) ⊗Zp Γ) −→ Γ ⊗Zp Qp

is equal to [e ∪1 yf ] =

X

invv (cv ∪ yv + i+ v (bv ) ∪ µv + εv ).

v∈Sf

293

(11.3.11.1)

We are now going to use the fact that e = σ∗ (b e), db e = 0. For each v ∈ Sf , the commutative diagram 2 Ccont (Gv , Yy∗ (1) ⊗Qp Y ⊗Zp Γ)  id⊗(σ∗ (1)) ∗ y

σ ⊗id

∗ −−→

2 Ccont (Gv , X ⊗Qp Y ⊗Zp Γ)  π∗ y

2 Ccont (Gv , Yy∗ (1) ⊗Qp Yy ⊗Zp Γ) −−→

2 Ccont (Gv , Qp (1) ⊗Zp Γ)

shows that 2 cv ∪ y v = b cv ∪ (db y )v ∈ Ccont (Gv , Qp (1) ⊗Zp Γ).

(11.3.11.2)

Combining (11.3.11.1-2) and using the formulas i+ (bbv ) = b av + db cv

0 = invv (d(b cv ∪ ybv )) = invv (db cv ∪ ybv − b cv ∪ db yv ),

(the second one following from db e = 0), we obtain X [e ∪1 yf ] = invv (i+ (bbv ) ∪ ybv + i+ (bv ) ∪ µv + (ε − b a ∪ yb)v ) v∈Sf

=

X

invv (i+ (bbv ) ∪ ybv + i+ (bv ) ∪ µv ),

v∈Sf

as the sum of the local invariants of the global 2-cocycle 2 ε−b a ∪ yb ∈ Ccont (GK,S , Qp (1) ⊗Zp Γ),

d(ε − b a ∪ yb) = 0,

vanishes. This proves the first statement of the Lemma. In the special case when bbv ∈ C 2 (Gv , Qp (1) ⊗Z Γ) p cont

(∀v ∈ Sf ),

we have i+ (bbv ) ∪ ybv = i+ (bbv ) ∪ 1 = i+ (bbv ).

i+ (bv ) ∪ µv = 0,

(11.3.12) We are now ready to conclude the proof of Theorem 11.3.9: combining the statements of Lemma 11.3.10-11, we obtain (using (11.3.5.3)) e hπ,1,1 ([xf ], [yf ]) = [β(xf ) ∪ yf ] =

X

invv (−zv ∪ uv ) = −

v∈Sf

X

`v (uv ) = −hnorm ([x], [y]). π

v∈Sf

(11.3.13) Plater [Pl] used universal norms to construct Frac(R)-valued height pairings on Selmer groups attached to certain two-dimensional Galois representations GK,S −→ GL2 (R), ∼

where R is an integral domain, finite and flat over Zp [[Γ]], Γ −→ Zp . It is very likely that the proof of Theorem 11.3.9 applies to the situation considered in [Pl]; we have not checked the details, though. (11.3.14) Rubin’s formula. Assume that e1 [xf ] = [(x, (x+ v ), (λv ))] ∈ Hf (X),

e f1 (Y ) [yf ] = [(y, (yv+ ), (µv ))] ∈ H

are as in the proof of Theorem 11.3.9 and that there exists a cohomology class 1 1 [xIw ] ∈ HIw (K∞ /K, X) = Hcont (GK,S , FΓ (X))

such that 294

1 pr∗ ([xIw ]) = [x] ∈ Hcont (GK,S , X).

(11.3.14.1)

The commutative diagram with exact rows and columns L v

L v

0= L v

L v

H 1 (GK,S , FΓ (X)/J 2 )  pr∗ y H 0 (Gv , Xv− )  −β=0 y

∂=0

−→

e 1 (X) H f  β y

H 1 (GK,S , X)  β y

−→

∂ e 2 (X) ⊗ ΓR −→ H 1 (Gv , Xv− ) ⊗ ΓR −→ H f

resSf

−→

resSf

−→

L v

H 0 (Gv , Xv− ) = 0  β=0 y

H 1 (Gv , Xv− ) ⊗ ΓR   yi

H 1 (Gv , FΓ (Xv− )/J 2 )  pr∗ y L 1 − v H (Gv , Xv )

H 2 (GK,S , X) ⊗ ΓR

(in which the direct sums are taken over v ∈ Sf and the subscript “cont” is suppressed in all cohomology groups) shows that there is a unique collection of cohomology classes [DxIw ] = [(DxIw )v ],

1 (DxIw )v ∈ Hcont (Gv , Xv− ) ⊗R ΓR

(v ∈ Sf )

satisfying i([DxIw ]) = resSf ([xIw ] (mod J 2 )). The pair ([xIw ], [DxIw ]) is an analogue of (z, Derρ (z)) in the situation considered by Rubin [Ru]. The following result is a variant of Rubin’s theorem ([Ru], Thm. 3.2(ii)) in our context. (11.3.15) Proposition. In the situation of 11.3.14 (still assuming (11.3.1.1-5), we have e hπ,1,1 ([xf ], [yf ]) = −

X

invv ((DxIw )v ∪ yv+ ).

v∈Sf

Proof. The assumption (11.3.14.1) implies that the 2-cocycle a in β(xf ) = (a, (bv ), (cv )) is a coboundary, a = dA,

1 A ∈ Ccont (GK,S , X).

For each v ∈ Sf , we have i+ v (bv ) = d(cv + Av ), hence − β([xf ]) = ∂([(c− v + Av )v ]),

where (−)− v denotes the image of (−)v under the map 1 1 Ccont (Gv , X) −→ Ccont (Gv , Xv− ).

295

In the notation used in the proof of Lemma 11.3.11, we can take ε = A ∪ y, hence X

e hπ,1,1 ([xf ], [yf ]) = =

X

invv (cv ∪ yv + i+ v (bv ) ∪ µv + εv ) =

v∈Sf

invv ((cv + Av ) ∪ (yv + dµv )) =

v∈Sf

X

X

invv ((cv + Av ) ∪ yv + d(cv + Av ) ∪ µv ) =

v∈Sf + invv ((cv + Av ) ∪ i+ v (yv )) =

v∈Sf

X

− + invv ((c− v + Av ) ∪ yv ).

v∈Sf

On the other hand, Lemma 1.2.19 implies that β([xf ]) = −∂([DxIw ]), hence there is 1 B ∈ Ccont (GK,S , X),

dB = 0

satisfying − − c− v + Av = −(DxIw )v + Bv

(∀v ∈ Sf )

(where DxIw is a 1-cocycle representing [DxIw ]). Thus X e hπ,1,1 ([xf ], [yf ]) = invv ((−(DxIw )v + Bv− ) ∪ yv+ ), v∈Sf

but the sum X

invv (Bv− ∪yv+ ) =

v∈Sf

X

+ invv (Bv ∪i+ v (yv )) =

v∈Sf

X

X

invv (Bv ∪(yv +dµv )) =

v∈Sf

invv ((B∪y)v −d(Bv ∪µv )) = 0

v∈Sf

vanishes. Proposition follows. (11.4) Comparison with classical p-adic height pairings (bis) e 1 (Z) no In this section we slightly relax the conditions on the Galois representations X, Y from 11.3, so that H f longer coincides with Sel(Z) (Z = X, Y ). The difference between the two groups is an algebraic counterpart of trivial zeros of p-adic L-functions (cf. [M-T-T]). In this case there are two ‘natural’ height pairings Sel(X) ⊗Qp Sel(Y ) −→ Γ ⊗Zp Qp , namely the one constructed via universal norms, and the “canonical” height pairing. A relation between them is proved in ([Ne 2], Thm. 7.13), in the context of p-adic Hodge theory. ∼

(11.4.1) Assume that Γ = Gal(K∞ /K) −→ Zrp (r ≥ 1) and that we are given the data (11.4.1.n) = (11.3.1.n)

(n = 1,2,3)

satisfying (11.4.1.4) = (11.3.1.4). For each prime v|p in K, put Av = H 0 (Gv , Xv− ) = H 0 (Gv , (Yv+ )∗ (1)) Bv = H 0 (Gv , Yv− ) = H 0 (Gv , (Xv+ )∗ (1)). Taking the pull-back (resp., push-forward) of the tautological exact sequence 0 −→ Xv+ −→ X −→ Xv− −→ 0 296

with respect to the inclusion Av ,→ Xv− (resp., the projection ρv : Xv+  Bv∗ (1)), we obtain a short exact sequence of Gv -modules 0 −→ Bv∗ (1) −→ Wv −→ Av −→ 0. Similarly, applying Bv ,→ Yv− and ρ0v : Yv+  A∗v (1) to 0 −→ Yv+ −→ Y −→ Yv− −→ 0, we obtain an exact sequence 0 −→ A∗v (1) −→ Wv0 −→ Bv −→ 0. We denote the corresponding extension classes by 1 qv (X) := [Wv ] ∈ Ext1Qp [Gv ] (Av , Bv∗ (1)) = HomQp (Av , Bv∗ ) ⊗Qp Hcont (Gv , Qp (1)) 1 qv (Y ) := [Wv0 ] ∈ Ext1Qp [Gv ] (Bv , A∗v (1)) = HomQp (Bv , A∗v ) ⊗Qp Hcont (Gv , Qp (1)).

Equivalently, they are equal to the maps (ρv )∗

∂

v 1 + 1 ∗ ∗ 1 qv (X) : Av = H 0 (Gv , Xv− )−−→H cont (Gv , Xv )−−→Hcont (Gv , Bv (1)) = Bv ⊗Qp Hcont (Gv , Qp (1))

∂v0

(ρ0 )∗

v 1 1 ∗ ∗ 1 qv (Y ) : Bv = H 0 (Gv , Yv− )−−→Hcont (Gv , Yv+ )−−→H cont (Gv , Av (1)) = Av ⊗Qp Hcont (Gv , Qp (1)).

1 b p = Hcont If q ∈ Kv∗ ⊗Q (Gv , Qp (1)) is the extension class of an exact sequence of Gv -modules β

α

0 −→ Qp (1)−→E −→Qp −→ 0, then q −1 (we write the group law multiplicatively) is the extension class of β ∗ (1)

α∗ (1)

0 −→ Qp (1) −→ E ∗ (1) −→ Qp −→ 0. Applying this remark to Wv and Wv0 = Wv∗ (1), we see that qv (Y ) = qv (X)−1 under the canonical isomorphism ∼

∼

∼

HomQp (Av , Bv∗ ) −→ A∗v ⊗Qp Bv∗ −→ Bv∗ ⊗Qp A∗v −→ HomQp (Bv , A∗v ). Any homomorphism of vector spaces over Qp 1 F : Hcont (Gv , Qp (1)) −→ U

induces Qp -linear maps F (qv (X)) : Av −→ Bv∗ ⊗Qp U,

F (qv (Y )) : Bv −→ A∗v ⊗Qp U

satisfying F (qv (Y )) = −F (qv (X)) ∈ A∗v ⊗Qp Bv∗ ⊗Qp U. This applies, in particular, to the valuation 1 b p −→ Qp v = v ⊗ 1 : Hcont (Gv , Qp (1)) = Kv∗ ⊗Q

and the map 1 `v : Hcont (Gv , Qp (1)) −→ Γv ⊗Zp Qp .

297

We assume that the following conditions (11.4.1.5-7) are satisfied: (11.4.1.5) (∀v|p) v(qv (X)) : Av −→ Bv∗ is an isomorphism (equivalently, v(qv (Y )) : Bv −→ A∗v is an isomorphism). We denote by Ssp = {v|p; Av 6= 0} = {v|p; Bv 6= 0} the set of “special” primes of K; new phenomena arise at such primes. (11.4.1.6) (∀v ∈ Ssp ) either Γv = 0 or H 0 (Gv∞ , Xv− /Av ) = H 0 (Gv∞ , Yv− /Bv ) = 0 (where we put Gv∞ = Gal(K v /(K∞ )v∞ ) for a fixed prime v∞ of K∞ above v). (11.4.1.7) (∀v ∈ Ssp ) `v (qv (X)) : Av −→ Bv∗ ⊗Zp Γv is surjective (equivalently, `v (qv (Y )) : Bv −→ A∗v ⊗Zp Γv is surjective. (11.4.2) We now list the most important consequences of (11.4.1.1-7). For each v ∈ Ssp , the boundary maps 1 ∂v : Av ,→ Hcont (Gv , Xv+ ),

1 ∂v0 : Bv ,→ Hcont (Gv , Yv+ )

(v ∈ Ssp )

are injective and admit canonical splittings (ρv )∗

v

v(qv (X))−1

(ρ0 )∗

v

v(qv (Y ))−1

1 1 splv : Hcont (Gv , Xv+ )−−→Hcont (Gv , Bv∗ (1))−−→Bv −−−−−−→Av v 1 1 ∗ splv0 : Hcont (Gv , Yv+ )−−→H cont (Gv , Av (1))−−→Av −−−−−−→Bv .

As in 11.3.2, we have ∼

RΓcont (Gv , Z) = RΓur (Gv , Z) −→ 0

(v ∈ Sf , v - p; Z = X, Y )

in D(Qp Mod); hence the Selmer complexes gf (Z) := RΓ gf (GK,S , Z; ∆(Z)) −→ RΓ gf (GK,S , Z; ∆0 (Z)) RΓ

(Z = X, Y )

(with the local conditions defined as in 11.3.2) are again isomorphic in Dfb t (Qp Mod). The image of the canonical map 1 e f1 (Z) −→ Hcont H (GK,S , Z)

(Z = X, Y )

is again equal to the Selmer group Sel(Z) from 11.3.2, but the map can have a non-trivial kernel. Indeed, the third of the exact sequences in 6.1.3 yields exact sequences M e f1 (X) −→ Sel(X) −→ 0 0 −→ Av −→ H v∈Ssp

0 −→

M

e 1 (Y ) −→ Sel(Y ) −→ 0. Bv −→ H f

v∈Ssp

These sequences have canonical splittings M e 1 (X) −→ spl : H Av , f

+ [(x, (x+ v ), (λv ))] 7→ (splv ([xv ]))

v∈Ssp

spl

0

e 1 (Y :H f

) −→

M

Bv ,

[(y, (yv+ ), (µv ))] 7→ (splv0 ([yv+ ])).

v∈Ssp

The induced decompositions ∼ e 1 (X) −→ H Sel(X) ⊕ f

M

∼ e 1 (Y ) −→ H Sel(Y ) ⊕ f

Av ,

v∈Ssp

M v∈Ssp

can be characterized as follows: the lift 298

Bv

e f1 (X), Sel(X) −→ H

[x] 7→ [(x, (x+ v ), (λv ))]

satisfies ∗b ∗ ∗b ∗ ρv ([x+ v ]) ∈ (Ov ⊗Qp ) ⊗Qp Bv ⊂ (Kv ⊗Qp ) ⊗Qp Bv

(∀v ∈ Ssp )

(and similarly for Y ). The conditions (11.4.1.5-7) also imply that ∼

0 1 Ssp = Ssp ∪ Ssp ,

i Ssp = {v ∈ Ssp | Γv −→ Zip }.

(11.4.3) Lemma. For each v|p, the canonical maps (ρv )∗

`

(ρ0v )∗

`

v 1 1 ∗ Hcont (Gv , Xv+ )/N∞ −−→Hcont (Gv , Bv∗ (1))/N∞ −−→B v ⊗ Z p Γv v 1 1 ∗ Hcont (Gv , Yv+ )/N∞ −−→Hcont (Gv , A∗v (1))/N∞ −−→A v ⊗ Z p Γv

are isomorphisms. Proof. This is trivial for Γv = 0, so we can assume that Γv 6= 0. If v 6∈ Ssp , then Av = Bv = 0 and the statement follows from Lemma 11.3.6. If v ∈ Ssp , then (11.4.1.6) implies that ρv

1 Hcont (Gv , Ker(Xv+ −→Bv∗ (1)))/N∞ = 0,

by the same argument as in the proof of Lemma 11.3.6. The fact that the map 1 1 (ρv )∗ : Hcont (Gv , Xv+ )/N∞ −→ Hcont (Gv , Bv∗ (1))/N∞

is an isomorphism then follows from the Snake Lemma applied to the following commutative diagram with exact rows: ρv

1 Hcont (Gv , FΓv (Ker(Xv+ −→Bv∗ (1))))  pr∗ y ρv

1 Hcont (Gv , Ker(Xv+ −→Bv∗ (1)))

−→

−→

1 Hcont (Gv , FΓv (Xv+ )) −→  pr∗ y 1 Hcont (Gv , Xv+ )

1 Hcont (Gv , FΓv (Bv∗ (1)))  pr∗ y

−→ 0

1 Hcont (Gv , Bv∗ (1))

−→ 0

−→

(note that the zeros on the right come from the assumption (11.4.1.6) and the local Tate duality). (11.4.4) Definition of hnorm . Under the assumptions (11.4.1.1-7) one can define a height pairing hnorm : Sel(X) ⊗Qp Sel(Y ) −→ Γ ⊗Zp Qp π by modifying the construction from 11.3.7 at the special primes v ∈ Ssp as follows. Let x, y, Yy and [b x] be as in 11.3.7. The top of the diagram (11.3.7.1) has to be replaced by 0   y H 0 (Gv , Yy∗ (1)− v)  ∂ y v

0   y ∼

−→

1 Hcont (Gv , Yy∗ (1)+ v ) −→

H 0 (Gv , Xv− )  ∂ y v 1 Hcont (Gv , Xv+ ).

This implies that δv ◦ i+ v ◦ ∂v = 0, and as before we obtain a uniquelly determined cohomology class 1 ∗ + [b x+ v ] ∈ Hcont (Gv , Yy (1)v ),

299

i+ x+ xv ]. v ([b v ]) = [b

The diagram (11.3.7.2) also has to be modified by adding ∂

v 1 Hcont (Gv , Yy∗ (1)+ v ) ←−  σ y ∗

H 0 (Gv , Yy∗ (1)− v)

∂

1 Hcont (Gv , Xv+ )

v ←−

There exist

k H 0 (Gv , Xv− ).

  1 ∗ + FΓv (x+ v ) ∈ Hcont (Gv , FΓv (Yy (1)v )),

tv ∈ H 0 (Gv , Xv− ) = Av

such that [x+ v ] + ∂v (tv ) = pr∗ ◦ FΓv (σ)∗




FΓv (x+ v) .

Then uv = [b x+ v ] − pr∗




FΓv (x+ + ∂v (tv ) v)

is an element of 1 uv ∈ Hcont (Gv , Qp (1))

and the height pairing hnorm is defined by the formula π X hnorm ([x], [y]) = `v (uv ) ∈ Γ ⊗Zp Qp . π v∈Sf

(11.4.5) Definition of hcan . One defines the “canonical height” hcan : Sel(X) ⊗Qp Sel(Y ) −→ Qp π as follows. For each v ∈ Ssp , the subgroup of units 1 b p ) ⊗Qp Bv∗ ⊂ (Kv∗ ⊗Q b p ) ⊗Qp Bv∗ = Hcont Hf1 (Gv , Bv∗ (1)) := (Ov∗ ⊗Q (Gv , Bv∗ (1)) 1 (resp., Hf1 (Gv , A∗v (1)) ⊂ Hcont (Gv , A∗v (1))) is complementary to Im(qv (X)) (resp., Im(qv (Y ))), hence the localizations at v define morphisms



(ρv )∗ resv 1 1 1 ev : Sel(X)−−→Ker Hcont (Gv , X) −→ Hcont (Gv , Xv− ) = Coker ∂v : H 0 (Gv , Xv− ) −→ Hcont (Gv , Xv+ ) −−→
∼ (ρv )∗ 1 −−→Coker qv (X) : Av −→ Hcont (Gv , Bv∗ (1)) ←− Hf1 (Gv , Bv∗ (1)), ev : [x] 7→ [e xv ] ∈ Hf1 (Gv , Bv∗ (1)) and Sel(Y ) −→ Hf1 (Gv , A∗v (1)),

[y] 7→ [e yv ].

We define norm hcan ([x], [y]) + π ([x], [y]) := hπ

X

`v (qv (X))−1 `v ([e xv ])`v ([e yv ]) ∈ Γ ⊗Zp Qp ,

(11.4.5.1)

1 v∈Ssp

where the term on the R.H.S. should be interpreted as follows: we have `v ([e xv ]) ∈ Bv∗ ⊗Zp Γv ,

`v (qv (X))−1 `v ([e xv ]) ∈ Av

and the latter element of Av is evaluated against `v ([e yv ]) ∈ A∗v ⊗Zp Γv . Although we have used (11.4.5.1) as a definition, in the situation of ([Ne 2], Sect. 7) one defines hcan directly π and (11.4.5.1) then becomes the statement of ([Ne 2], Thm. 7.13(2)). 300

e 1 (X), [yf ] ∈ H e 1 (Y ) be the cohomology (11.4.6) Theorem. Under the assumptions (11.4.1.1-7), let [xf ] ∈ H f f + 1 + 1 e (X), yf = (y, (y ), (µv )) ∈ C e (Y ) (using the same notation as classes of 1-cocycles xf = (x, (xv ), (λv )) ∈ C v f f in the proof of Theorem 11.3.9). Then X

e hπ,1,1 ([xf ], [yf ]) = −hnorm ([x], [y]) − π

0 + `v (qv (X))−1 `v (ρv (x+ v ))`v (ρv (yv )).

1 v∈Ssp

Proof. We indicate only the necessary modifications to the proof of Theorem 11.3.9 at v ∈ Ssp . Firstly, the statement of Lemma 11.3.10 has to be replaced by bbv = zv ∪ (∂v (tv ) − uv ), where 0 − tv ∈ H 0 (Gv , Yy∗ (1)− v ) = H (Gv , Xv ) = Av

is as in 11.4.4 (the proof of the Lemma remains the same). Secondly, we have e i+ v (∂v (tv )) = dtv ,

0 e tv ∈ Ccont (Gv , Yy∗ (1)),

e tv 7→ tv .

It follows that b e i+ v (bv ) = −zv ∪ uv + zv ∪ dtv ,

e i+ v (bv ) ∪ µv = zv ∪ σ∗ (dtv ) ∪ µv ,

hence b e bv + zv ∪ dσ∗ (e `v (uv ) + invv (i+ bv + i+ tv ) ∪ µv ) = v (bv ) ∪ y v (bv ) ∪ µv ) = invv (zv ∪ dtv ∪ y ∗ = −invv (zv ∪ e tv ∪ db yv + zv ∪ σ∗ (e tv ) ∪ dµv ) = −invv (zv ∪ (e tv ∪ (σ (1))∗ (yv ) + σ∗ (e tv ) ∪ dµv )) = + + + + e = −invv (zv ∪ tv ∪ iv (yv )) = −invv (zv ∪ tv ∪ yv ) = −`v (tv ∪ yv ) = −`v (tv ∪ ρ0v (yv+ )). The first formula in Lemma 11.3.11 (which still holds in our case) yields e hπ,1,1 ([xf ], [yf ]) = [β(xf )∪yf ] = −

X v∈Sf

`v (uv )−

X

`v (tv ∪ρ0v (yv+ )) = −hnorm ([x], [y])− π

v∈Ssp

1 v∈Ssp

0 1 (as `v = 0 for v ∈ Ssp ). For each v ∈ Ssp , the composite map ∂

ρv

`

v v 1 + 1 ∗ ∗ Av −→H cont (Gv , Xv )/N∞ −→Hcont (Gv , Bv (1))/N∞ −→Bv ⊗Zp Γv

is equal to `v (qv (X)), which means that we can take tv = `v (qv (X))−1 `v (ρv (x+ v )), hence the correction term in the above formula is equal to 0 + `v (tv ∪ ρ0v (yv+ )) = `v (qv (X))−1 `v (ρv (x+ v ))`v (ρv (yv )).

Theorem is proved. 301

X

`v (tv ∪ρ0v (yv+ ))

(11.4.7) Corollary. With respect to the canonical decompositions defined in 11.4.2: ∼ e f1 (X) −→ H Sel(X) ⊕

M

M

∼ e f1 (Y ) −→ H Sel(Y ) ⊕

Av ,

v∈Ssp

Bv ,

v∈Ssp

0 e 1 (Y ) (resp., H e 1 (X) ⊗ Bv ) for all v ∈ Ssp the pairing e hπ,1,1 vanishes on Av ⊗ H , and is given by the formulas f f

e hπ,1,1 ([x], [y]) = −hcan π ([x], [y]),

e hπ,1,1 ([av ], [y]) = −`v ([e yv ])(av ), ( −`v (qv (X))(av )(bw ), e hπ,1,1 ([av ], [bw ]) = 0,

e hπ,1,1 ([x], [bw ]) = −`w ([e xw ])(bw ) v = w, v 6= w,

1 where [x] ∈ Sel(X), [y] ∈ Sel(Y ), v, w ∈ Ssp , av ∈ Av , bw ∈ Bw .

(11.4.8) In matrix terms, Corollary 11.4.7 says that the restriction of e hπ,1,1 to the subspaces e 1 (X)(1) = Sel(X) ⊕ H f

M

e 1 (Y )(1) = Sel(Y ) ⊕ H f

e 1 (X), Av ⊂ H f

1 Ssp

M

e 1 (Y ) Bv ⊂ H f

1 Ssp

has matrix e hπ,1,1 = −

hcan π

`v ◦ev

`v ◦ev

`v (qv (X))

!

(11.4.9) Proposition. Under the assumptions (11.4.1.1-7), the pairing e e f1 (X)(1) ⊗Qp H e f1 (Y )(1) −→ Γ ⊗Zp Qp hπ,1,1 : H is non-degenerate if and only if hnorm : Sel(X) ⊗Qp Sel(Y ) −→ Γ ⊗Zp Qp π is. If true, then

det(−e hπ,1,1 ) = det(hnorm ) π

Y

det(`v (qv (X))

1 v∈Ssp

in

det(Sel(X))∗ ⊗ det(Sel(Y ))∗ ⊗

O

(det(Av )∗ ⊗ det(Bv )∗ ).

1 Ssp

Proof. As in ([Ne 2], 7.13(3)), this follows from Corollary 11.4.7 and the matrix identity ! ! ! ! A B 1 0 A − BD−1 C B 1 0 = . C D 0 D 0 1 D−1 C 1 (11.4.10) If K = Q and X is the one-dimensional Galois representation attached to a non-trivial Dirichlet character χ 6= 1 whose restriction to Gp is trivial, then the formulas from 11.4.7 (for the local conditions Xv+ = 0, Yv+ = Y = X ∗ (1)) yield the statement of ([Bu-Gr], Prop. 9.3(iii)). (11.5) Higher height pairings In this section we define analogues of the “derived heights” of Bertolini-Darmon [B-D 1,2] in our setting. (11.5.1) Let K∞ /K, X, Y be as in 11.1. Fix an integer k ≥ 1 and assume that (P ), (U ) and the following strengthening of (F l(Γ)) hold: (F lk (Γ)) For all i = 0, . . . , k, J i /J i+1 is a flat (hence free) R-module. 302

The condition (F lk (Γ)) implies that, for all 0 ≤ i ≤ j ≤ k + 1, the exact sequence 0 −→ J j /J k+1 −→ J i /J k+1 −→ J i /J j −→ 0

(11.5.1.1)

consists of free R-modules, hence splits. If ∆ = 0, then (F lk (Γ)) holds for all k ≥ 1. If ∆ = Z/pm Z and R is an Z/pm Z-algebra, then (F lp−1 (Γ)) holds. (11.5.2) The condition (F lk (Γ)) implies that the J-adic filtrations on FΓ (X)/J k+1 = (X ⊗R R/J k+1 ) < −1 >,

FΓ (Y )ι /J k+1 = (Y ⊗R R/J k+1 ) < 1 >

(11.5.2.1)

have graded quotients grJi (FΓ (X)/J k+1 ) = (X ⊗R J i /J i+1 ) < −1 >= X ⊗R J i /J i+1 grJi (FΓ (Y )ι /J k+1 ) = (Y ⊗R J i /J i+1 ) < 1 >= Y ⊗R J i /J i+1

(0 ≤ i ≤ k)

(grJi = 0 for i > k). As in 11.1, the given Greenberg’s local conditions for X, Y induce similar local conditions for all (X ⊗R J i /J j ) < −1 >,

(Y ⊗R J i /J j ) < 1 >

e • (FΓ (X)/J k+1 ) (resp., C e • (FΓ (Y )ι /J k+1 )) The assumptions (U ), (F lk (Γ)) imply that the filtrations FJ• on C f f induced by the J-adic filtrations on the coefficient modules (11.5.2.1) satisfy ( •   e (X) ⊗R J i /J i+1 , C (0 ≤ i ≤ k) f • i k+1 e (FΓ (X)/J grFJ C ) = f 0, otherwise, and similarly for FΓ (Y )ι /J k+1 (one can show that the filtrations FJ• are the J-adic ones, but we do not need this fact). It follows that the corresponding spectral sequences e • (FΓ (X)/J k+1 ), F • ) Er (X) = Er (C f J 0

e • (FΓ (Y )ι /J k+1 ), F • ) Er (Y ) = Er (C f J

(11.5.2.2)

have initial terms e i+j (X) ⊗R J i /J i+1 , E1i,j (X) = H f

0

e i+j (Y ) ⊗R J i /J i+1 E1i,j (Y ) = H f

(0 ≤ i ≤ k)

(and E1i,j = 0 E1i,j = 0 for i > k). The differential ej e j+1 (X) ⊗R J/J 2 = H e j+1 (X) ⊗R ΓR d0,j 1 : Hf (X) −→ Hf f coincides with the coboundary map induced by the morphism gf (X) −→ RΓ gf (X)[1] ⊗R J/J 2 β : RΓ from 11.3. Other differentials di,j 1 are also induced by β: d0,i+j ⊗id

id⊗mult

i,j 0,i+j 0,i+j+1 1 di,j ⊗R J i /J i+1 −−−−→E ⊗R J/J 2 ⊗R J i /J i+1 −−−−→E10,i+j+1 ⊗R J i+1 /J i+2 = E1i+1,j . 1 : E1 = E1 1

More generally, multiplication induces maps Eri,j ⊗R J l /J l+1 −→ Eri+l,j−l 303

(i, l ≥ 0, r ≥ 1)

compatible with the differentials. Almost the same remarks apply to the differentials 0 dr in the spectral sequence 0 Er (Y ), the only difference being that (0 Er (Y ), 0 dr ) = (Er (Y ), (−1)r dr ). For r = 1, the equality ej e j+1 (Y ) ⊗R J/J 2 = −d0,j 1 : Hf (Y ) −→ Hf

0 0,j d1

follows from the description of β in 11.2.5, as the presence of the involution ι changes z into −z. If k ≥ k 0 ≥ 1, then the canonical morphism of spectral sequences e• (FΓ (X)/J k+1 )) −→ E∗ (C e• (FΓ (X)/J k +1 )) E∗ (C f f 0

induces isomorphisms in the region ∼

Eri,j (−) −→ Eri,j (−0 )

(0 ≤ i ≤ k 0 − r + 1).

(11.5.3) A slightly modified construction from 5.4.1 can be applied to the J-adic truncation • F (π)k+1 : (FΓ (X)/J k+1 ) ⊗R (FΓ (Y )ι /J k+1 ) −→ ωR ⊗R (R/J k+1 )(1)

of the pairing • F (π) : FΓ (X) ⊗R FΓ (Y )ι −→ ωR ⊗R R(1),

yielding cup products (depending on various choices, as in 5.4.1 and 6.3.1) • • ι k+1 II e• (FΓ (X)/J k+1 ) ⊗ C e• ∪:C ) −→ τ≥3 Cc,cont (GK,S , ωR ⊗R R/J k+1 (1)) = f R f (FΓ (Y ) /J

II • • • = τ≥3 Cc,cont (GK,S , ωR (1)) ⊗R R/J k+1 −→ (ωR ⊗R R/J k+1 )[−3],

(11.5.3.1)

the homotopy classes of which are independent of any choices. These cup products are compatible with the filtrations in the sense that 0

0

• FJi ∪ FJi ⊆ (ωR ⊗R J i+i /J k+1 )[−3],

hence they induce products 0

0

0

0

• ∪r : Eri,j (X) ⊗R 0 Eri ,j (Y ) −→ Eri+i ,j+j ((ωR ⊗R R/J k+1 )[−3], J • )

satisfying dr (x ∪r y) = (dr x) ∪r y + (−1)x x ∪r (0 dr y) = (dr x) ∪r y + (−1)i+j x ∪r (0 dr y). • As the sequences (11.5.1.1) split, the spectral sequence for the J-adic filtration on ωR ⊗R R/J k+1 has vanishing differentials, hence degenerates into

( 0 • Ern,n ((ωR

⊗R R/J

k+1

)[−3], J ) = •

0 E1n,n

=

0

H n+n −3 (ωR ) ⊗R J n /J n+1 ,

(0 ≤ n ≤ k)

0,

otherwise.

To sum up, we have constructed products (for each r ≥ 1) ( ∪r :

Eri,j (X)

0 0 ⊗R Eri ,j (Y

0

) −→

0

H i+i +j+j

0

−3

0

0

(ωR ) ⊗R J i+i /J i+i +1 ,

0,

(0 ≤ i, i0 , i + i0 ≤ k) otherwise

304

satisfying (dr x) ∪r y + (−1)i+j x ∪r (0 dr y) = 0.

(11.5.3.2)

For r = 1, the cup product 0

0

e j (X) ⊗R H e j (Y ) −→ H j+j 0 −3 (ωR ) ∪1 : E10,j (X) ⊗R 0 E10,j (Y ) = H f f is induced by the cup products e• (X) ⊗R C e • (Y ) −→ ω • [−3]. C f f R (11.5.4) The spectral sequences (11.5.2.1) induce decreasing filtrations e fi (X) := Er0,i (X), F ilΓr H

0

e fi (Y ) := 0 Er0,i (Y ) F ilΓr H

(1 ≤ r ≤ k + 1)

on e i (X) = F il 1 ⊃ F il 2 ⊃ · · · ⊃ F il k+1 H f Γ Γ Γ e i (Y ) = 0 F il 1 ⊃ 0 F il 2 ⊃ · · · ⊃ 0 F il k+1 . H f Γ Γ Γ As 0 dr = (−1)r dr , we have 0 F ilΓr = F ilΓr . If we replace k by k 0 ≤ k, then the terms F ilΓr for r ≤ k 0 + 1 do not change. (11.5.5) Definition of the higher order height pairings. The pairings 0,i

dr ⊗id (r) e e i (X)) ⊗R F il r (H e j (Y )) = E 0,i (X) ⊗R 0 E 0,j (Y )−−−−→ hπ,i,j : F ilΓr (H f Γ r r f ∪

r i+j−2 −→ Err,i−r+1 (X) ⊗R 0 Er0,j (Y )−−→H (ωR ) ⊗R J r /J r+1

(1 ≤ r ≤ k)

do not depend on k ≥ r (assuming that (F lk (Γ)) holds), and for r = 1 coincide with the pairings defined in 11.1.4: (1) e hπ,i,j = e hπ,i,j .

As before, it is the case i + j = 2 which is interesting, when we obtain pairings with values in H 0 (ωR ) ⊗R J r /J r+1 . We denote by e (r) L Ker(hπ,i,j )

(r) e j (Y )) KerR (e hπ,i,j ) ⊂ F ilΓr (H f

e i (X)), ⊂ F ilΓr (H f

(r) the left (resp., right) kernel of e hπ,i,j .

(11.5.6) Proposition. (i) The pairings (r) e e i (X)) ⊗R F il r (H e j (Y )) −→ H i+j−2 (ωR ) ⊗R J r /J r+1 hπ,i,j : F ilΓr (H f Γ f

satisfy

(1 ≤ r ≤ k)

(r) (r) e hπ◦s12 ,j,i (y, x) = (−1)ij+r−1 e hπ,i,j (x, y).

(ii) In particular, if X = Y , Xv+ = Yv+ (v ∈ Σ), π ◦ s12 = c · π, c = ±1, then the pairing (r) e e 1 (X)) ⊗R F il r (H e 1 (X)) −→ H 0 (ωR ) ⊗R J r /J r+1 hπ,1,1 : F ilΓr (H f Γ f

(1 ≤ r ≤ k)

is symmetric (resp., skew-symmetric) if c = (−1)r (resp., c = (−1)r−1 ). (iii) For each r ≤ k, (r) e fi (X)) ⊆ L Ker(e F ilΓr+1 (H hπ,i,j ),

(r) e j (Y )) ⊆ KerR (e F ilΓr+1 (H hπ,i,j ). f

305

(iv) For each r ≤ k and i + j = 2, (r) e fi (X)))R−tors ⊆ L Ker(e (F ilΓr (H hπ,i,j ),

(r) e j (Y )))R−tors ⊆ KerR (e (F ilΓr (H hπ,i,j ). f

Proof. The formula (11.5.3.2) yields (r) e hπ◦s12 ,j,i (y, x) = (dr y) ∪r x = −(−1)j y ∪r (0 dr x) = −(−1)j y ∪r (−1)r dr x =

= (−1)j+r−1 (−1)(i+1)j (dr x) ∪r y = (−1)ij+r−1 e hπ,i,j (x, y), (r)

proving (i) (and its special case (ii)). If x ∈ F ilΓr+1 , then dr x = 0 and e h(r) (x, y) = (dr x) ∪r y = 0 vanishes r+1 (r) (r) (r) e e e by the definition of h . If y ∈ F ilΓ , then h (x, y) = ±h (y, x) = 0 by (i); this proves (iii). Finally, (iv) follows from the fact that H 0 (ωR ) ⊗R J r /J r+1 is torsion-free over R, by Lemma 2.8.8 and the assumption (F lk (Γ)). (11.5.7) Proposition. Assume that (F l1 (Γ)) holds, p ∈ Spec(R), Rp has no embedded primes, and that • for each q ∈ Q = {q ∈ Spec(R) | q ⊂ p, ht(q) = 0}, the localized pairing πq : Xq ⊗Rq Yq −→ (ωR )q (1) is a + + perfect duality (over Rq ) and (∀v ∈ Σ) (Xv )q ⊥⊥πq (Yv )q . Then  e (1) L Ker(hπ,i,2−i )p

e i (X)p −→ = Ker H f

M

 e i (X)q /(F il 2 (H e i (X)))q  H f Γ f

(∀i).

q∈Q

Proof. For every Rp -module M of finite type,  MRp −tors = Ker M −→

M

 Mq  .

q∈Q

As H 0 (ωR )p is torsion-free over Rp (by Lemma 2.10.5(v)), the right vertical arrow of the following commutative diagram is injective: (1) (e hπ,i,2−i )p :

e i (X)p ⊗Rp H e 2−i (Y )p H f f   y

L (1) ei e 2−i ((e hπ,i,2−i )q ) : q∈Q Hf (X)q ⊗Rq Hf (Y )q

−−−−→

−−−−→

L

(H 0 (ωR ) ⊗R ΓR )p   y

q∈Q (H

0

(ωR ) ⊗R ΓR )q .

This implies that it is sufficient to prove the result for p = q. The exact triangle in D(R Mod) gf (X) −→ RHomR (RΓ gf (Y ), ωR [−3]) −→ Err(∆(X), ∆(Y ), π) RΓ gives, after localizing at q, isomorphisms ∼ e i+1 (X)q −→ e 2−i (Y )q , H 0 (ωR )q ). adj(∪1 )q : H HomRq (H f f

As the height pairing is defined by the formula (1) e hπ,i,2−i (x, y) = (d0,i 1 x) ∪1 y,

it follows that e (1) L Ker(hπ,i,2−i )q

2 ei = Ker(d0,i 1 )q = (F ilΓ (Hf (X)))q ,

as claimed. ∼ r(Γ) (11.5.8) If ∆ = 0, i.e. Γ −→ Zp (r(Γ) ≥ 1), then (F lk (Γ)) holds for k ≥ 1. The cup products in (11.5.3.1) are compatible as k varies and induce a cup product 306

ι • ef• (FΓ (X)) ⊗ C e• ∪:C R f (FΓ (Y ) ) −→ (ωR ⊗R R)[−3],

the homotopy class of which does not depend on any choices. The corresponding spectral sequences e • (FΓ (X)), F • ), Er (X) = Er (C f J

0

e• (FΓ (Y )ι ), F • ) Er (Y ) = Er (C f J

e i+j (X) ⊗R J i /J i+1 , E1i,j (X) = H f

0

then satisfy e i+j (Y ) ⊗R J i /J i+1 E1i,j (Y ) = H f

e i (X) (Z = X, Y ) and the higher height pairings for all i ≥ 0. Similarly, the decreasing filtrations F ilΓr on H f (r) e h are defined for all r ≥ 1, and the intersection π,i,j

e i (Z)) := F ilΓ∞ (H f

\

e i (Z)) F ilΓr (H f

(Z = X, Y )

r≥1

is equal to the submodule of universal norms   i e fi (Z)) = Im H e f,Iw e fi (Z) F ilΓ∞ (H (K∞ /K, Z) −→ H

(Z = X, Y ).

(11.5.9) Proposition (Dihedral case). Assume that (F lk (Γ)) holds, K∞ /K + is as in 10.3.5.1 and the conditions (10.3.5.1.1-5) are satisfied. Then e i (X) (i) Fix τ ∈ Γ+ − Γ and its lift τe ∈ G(KS /K + ). Then the action of Ad(e τ )f defines an involution τ on H f e j (Y ), which does not depend on any choices. and H f e i (X) and H e j (Y ) are τ -stable. (ii) The filtrations F il r on H Γ

f

f

(r) (iii) The pairings e hπ,i,j satisfy

(r) (r) e hπ,i,j (τ x, τ y) = (−1)r e hπ,i,j (x, y)

(1 ≤ r ≤ k).

Proof. (i) This follows from Lemma 8.6.4.4(i). e • (G+ , FΓ+ (X)) (ii) The spectral sequence Er•,• (X) (resp., 0 Er•,• (Y )) can be defined using the complex C f e• (G+ , FΓ+ (Y )ι )) – equipped with the filtration induced by the J-adic filtration on the coefficient (resp., C f

modules – and the filtered quasi-isomorphism sh+ f from 10.3.5.5. This implies that the differentials commute with τ , proving the claim. (iii) Lemma 10.3.5.6 implies that the products 0

0

0

∪r : Eri,j (X) ⊗R 0 Eri ,j (Y ) −→ H i+i +j+j

0

−3

0

0

(ωR ) ⊗R J i+i /J i+i +1

(0 ≤ i, i0 , i + i0 ≤ k)

satisfy 0

τ x ∪r τ y = ι(x ∪r y) = (−1)i+i x ∪r y, hence (r) 0,i e hπ,i,j (τ x, τ y) = d0,i r (τ x) ∪r τ y = τ (dr (x)) ∪r τ y = re = (−1)r d0,i r (x) ∪r y = (−1) hπ,i,j (x, y), (r)

for all x ∈ Er0,i (X), y ∈ 0 Er0,j (Y ). (11.5.10) Rubin’s formula revisited. Howard ([Ho 3], Thm. 3.4) generalized Rubin’s result ([Ru], Thm. 3.2(ii)) to higher height pairings for Zp -extensions (defined as in 11.8.4 below). This result holds in 307

our context, too: assume that we are in the situation of 11.5.1 with Σ = Sf and that (F lk (Γ)) holds. Each e 1 (X) = E 0,1 (X) (r ≤ k) can be represented by a triple element [xf ] ∈ F ilΓr H r f   e• (FΓ (X)), J • ), xf,Iw = xIw , (x+ ), (α ) ∈ Zr0,1 (C v f Iw,v where 1 + x+ Iw,v ∈ Ccont (Gv , FΓ (Xv )),

1 xIw ∈ Ccont (GK,S , FΓ (X)),

0 αv ∈ Ccont (Gv , FΓ (X))

(v ∈ Sf )

and   + + e2 (J r FΓ (X)). eb := dxf,Iw = dxIw , (dx+ a, (bbv ), (b cv )) ∈ C f Iw,v ), (−(xIw )v + iv (xIw,v ) − dαv ) = (b Denote by e = (a, (bv ), (cv )) the image of eb in e 2 (J r FΓ (X)/J r+1 FΓ (X)) = C e 2 (X) ⊗R J r /J r+1 . C f f e 1 (Y ) can be represented by a 1-cocycle yf = (y, (y + ), (µv )), where Similarly, an element [yf ] ∈ H v f 1 y ∈ Ccont (GK,S , Y ),

1 yv+ ∈ Ccont (Gv , Yv+ ),

0 µv ∈ Ccont (Gv , Y )

(v ∈ Sf )

and + dyf = (dy, (dyv+ ), (−yv + i+ v (yv ) − dµv )) = 0. 1 r − (11.5.11) Proposition. Assume that dxIw = 0. For each v ∈ Sf , i− v ((xIw )v ) ∈ Ccont (Gv , J FΓ (Xv )) is a 1-cocycle; denote by 1 1 [(Dr xIw )v ] ∈ Hcont (Gv , FΓ (Xv− ) ⊗R J r /J r+1 ) = Hcont (Gv , Xv− ) ⊗R J r /J r+1

the cohomology class of its reduction modulo J r+1 FΓ (Xv− ). Then (j) e (i) For 1 ≤ j < r ≤ k, hπ,1,1 ([xf ], [yf ]) = 0. P (r) (ii) e h ([xf ], [yf ]) = − invv ([(Dr xIw )v ] ∪ [y + ]) π,1,1

v

v∈Sf

(r ≤ k).

Proof. (i) This follows from Proposition 11.5.6(iii). (ii) By definition, d0,1 r ([xf ]) is represented by e = (0, (bv ), (cv )); the same argument as in the proof of Proposition 11.3.15 (this time with A = ε = 0) shows that X X  

(r) + + e hπ,1,1 ([xf ], [yf ]) = d0,1 invv [i− invv [i− r (xf ) ∪r yf = v (cv ) ∪ yv ] = v (cv )] ∪ [yv ] . v∈Sf

v∈Sf

However, for each v ∈ Sf , [i− v (cv )] = −



i− v (xIw )v




 (mod J r+1 FΓ (Xv− )) = − [(Dr xIw )v ] .

(11.5.12) Localized height pairings. If S ⊂ R is a multiplicative subset, let S ⊂ R be its inverse image under the augmentation map. The constructions in 11.1-2,5 all work if we replace (R, R, J) by (RS , RS , JS ). (11.5.13) As in 11.1.7, one can generalize the constructions from 11.5.2-5, by considering an intermediate field K ⊂ L ⊂ K∞ and replacing the ideal J by JL and the assumption (F lk (Γ)) by (F lk (ΓL )). The analogues of the spectral sequences (11.5.2.2) have initial terms e i+j (L/K, X) ⊗R (J L )i /(J L )i+1 , E1i,j (X) = H f,Iw

0

e i+j (L/K, Y )ι ⊗R (J L )i /(J L )i+1 (0 ≤ i ≤ k) E1i,j (Y ) = H f,Iw 308

As in 11.5.4, they define decreasing filtrations e i (L/K, X) := E 0,i (X), F ilΓr H f,Iw r

0

e i (L/K, Y )ι := 0 E 0,i (Y ) F ilΓr H f,Iw r

(1 ≤ r ≤ k + 1)

on i e f,Iw H (L/K, X) = E10,i (X),

i e f,Iw H (L/K, Y )ι = 0 E10,i (Y )

and higher height pairings (r) ι i+j−2 e e i (L/K, X)⊗R[[Γ ]] 0 F il r H ei hπ,L/K,i,j : F ilΓr H (ωR )⊗R R[[ΓL ]]⊗R (J L )r /(J L )r+1 , f,Iw Γ f,Iw (L/K, Y ) −→ H L (1 ≤ r ≤ k) which generalize the pairings (11.1.7.5).

(11.6) Bockstein spectral sequence and descent ∼

∼

In the case when Γ −→ Zp (γ 7→ 1) we have R = R[[Γ]] −→ R[[T ]] (T = γ − 1) and the height pairings e hπ,i,j are defined in terms of the Bockstein maps βi : H i (M • /T M •) −→ H i+1 (M • /T M • ), i.e. the coboundary maps attached to the exact sequence of complexes T

0 −→ M • /T M •−→M • /T 2 M • −→ M • /T M • −→ 0, where e • (FΓ (X)) = C e • ((X ⊗R R[[T ]]) < −1 >). M• = C f f As observed in 11.5.2, the maps βi are equal to the differentials d1 in the spectral sequence Er (X) defined by the filtration FJ• induced on M • by the T -adic filtration on FΓ (X). In our case, FJ• coincides with the T -adic filtration on M • , hence Er (X) becomes, essentially, the “Bockstein spectral sequence”. In this section we list some elementary properties of this spectral sequence; they play an important role in a descent formalism gf,Iw (K∞ /K, X))” to certain cohomological invariants relating the “leading term of detR (M • ) = detR (RΓ over K. (11.6.1) Bockstein spectral sequence (11.6.1.1) We shall work with triples (A, Λ, T ), where A is a (non-zero) noetherian ring, Λ is an A-algebra, T ∈ Λ is an element of Λ not dividing zero and such that the composite map A −→ Λ −→ Λ/T Λ is an isomorphism (we use this map to identify A with Λ/T Λ). A typical example is provided by (R, R[[T ]], T ). (11.6.1.2) If (A, Λ, T ) are as in 11.6.1.1 and p ∈ Spec(A), denote by p ∈ Spec(Λ) its inverse image under the projection Λ −→ Λ/T Λ = A. Then the triple (Ap , Λp , T ) is again of the type considered in 11.6.1.1. (11.6.1.3) We say that a Λ-module M is T -flat if the multiplication by T on M is injective. This is equivalent to the vanishing of TorΛ 1 (M, A) = 0, by the exact sequence of Tor’s attached to T

0 −→ Λ−→Λ −→ A −→ 0. (11.6.1.4) From now on (until the end of 11.6), let M • be a cohomologically bounded complex of T -flat Λ-modules. It is equipped with the T -adic filtration F i M • = T i M • (i.e. F i M • = M • for i ≤ 0), which has ∼ graded quotients isomorphic to grFi (M • ) −→ M • /T M • (i ≥ 0). By definition, the corresponding spectral sequence Er = Er (M • , F • ) satisfies Eri,j = Zri,j /Bri,j , where Zri,j = {a ∈ F i M i+j | da ∈ F i+r M i+j+1 }
Bri,j = d(F i−r+1 M i+j−1 ) + F i+1 M i+j ∩ Zri,j 309

(r ≥ 1),

hence multiplication by T induces isomorphisms ∼

∼

T∗ : Zri,j −→ Zri+1,j−1

T∗ : Bri,j −→ Bri+1,j−1

(i ≥ 0),

(i ≥ r − 1),

∼

T∗ : Eri,j −→ Eri+1,j−1

monomorphisms T∗ : Bri,j ,→ Bri+1,j−1

(0 ≤ i < r − 1)

T∗ : Eri,j  Eri+1,j−1

(0 ≤ i < r − 1).

and epimorphisms

In particular, multiplication by T −i induces isomorphisms ∼

(T −i )∗ : Zri,j −→ Zri+j := M i+j ∩ d−1 (T r M i+j+1 ) ∼

(T −i )∗ : Bri,j −→ Bri+j := (T −i )∗ Bri,j

(i ≥ 0)

(i ≥ r − 1).

This means that, in the “stable range” i ≥ r − 1, the terms Wri,j (W = Z, B, E) depend only on i + j and r, as ∼

(T −i )∗ : (Zri,j , Bri,j , Eri,j ) −→ (Zri+j , Bri+j , Eri+j )

(i ≥ r − 1)

(11.6.1.4.1)

(where Ern := Zrn /Brn ) is an isomorphism. The differentials di,j r define, under the isomorphisms (11.6.1.4.1), “stable” differentials dnr : Ern −→ Ern+1 , which depend on T . If we replace T by uT (u ∈ Λ∗ ), then the isomorphism (11.6.1.4.1) is multiplied by u−i ; this implies that dnr is replaced by (u)−r dnr , where u = u (mod T Λ) ∈ A∗ . The simply graded “stable” spectral sequence (Ern , dnr ) is usually referred to as the “Bockstein spectral sequence” It arises from the exact couple T

H ∗ (M • )−→H ∗ (M • ) −→ H ∗ (M • /T M •) (cf. [We], 5.9.9-12). (11.6.1.5) In the special case when (∀j 6= j0 , j0 + 1)

E10,j = 0,

we have (∀i + j 6= j0 , j0 + 1, ∀r ≥ 1)

Eri,j = 0

and the maps T∗ : Eri,j −→ Eri+1,j−1

(r ≥ 1)

are isomorphisms for all pairs (i, j) 6= (0, j0 + 1). (11.6.2) In concrete terms, the spectral sequence Ern can be described as follows: E1n = H n (M • /T M •) and dn1 : H n (M • /T M •) −→ H n+1 (M • /T M •) is the coboundary map attached to 310

(11.6.1.4.2)

T

0 −→ M • /T M •−→M • /T 2 M • −→ M • /T M • −→ 0. Put H n = H n (M • ) and let cnr : T r−1H n ∩ H n [T ] −→ H n /(T H n + H n [T r−1 ])

(r ≥ 1)

be the map sending T r−1 a to the class of a. For r = 0, let T

cn0 : H n −→H n be the multiplication by T . The description of Ern in terms of the exact couple (11.6.1.4.2) shows that, for each r ≥ 1, there are exact sequences 0

0

H n /(T H n + H n [T r−1])   y0

−→

−→ H n+1 /(T H n+1 + H n+1 [T r−1])

−→ in+1 r

−→

jn

r Ern −→  dn y r

Ern+1

−→

T r−1 H n+1 ∩ H n+1 [T ] −→   y0

0

T r−1 H n+2 ∩ H n+2 [T ] −→

0,

(11.6.2.1)

in which dnr = in+1 ◦ cn+1 ◦ jrn . Indeed, for r = 1, these are nothing but the exact sequences r r 0 −→ H n /T H n −→ H n (M • /T M •) −→ H n+1 [T ] −→ 0 coming from the cohomology sequence of T

0 −→ M • −→M • −→ M • /T M • −→ 0. Given (11.6.2.1) for r ≥ 1, it can be viewed as an exact sequence of complexes 0 −→

M

(H n /(T H n + H n [T r−1 ]))[−n] −→ (Er• , dr ) −→

n

M

(T r−1 H n ∩ H n [T ])[−n + 1] −→ 0,

n

for which the corresponding boundary map in D(A Mod) is equal to M M (cnr [−n + 1])n : (T r−1 H n ∩ H n [T ])[−n + 1] −→ (H n /(T H n + H n [T r−1 ]))[−n + 1]. n

n

Taking cohomology gives exact sequences 0 −→ Coker(cnr ) −→ H n (Er• , dr ) −→ Ker(cn+1 ) −→ 0, r

(11.6.2.2)

i.e. (11.6.2.1) for r + 1. Noting that, for each r ≥ 1, multiplication by T r−1 induces an isomorphism ∼

T r−1 : H n /(T H n + H n [T r−1 ]) −→ T r−1 H n /T r H n , we see that (11.6.2.1) can be written as 0 −→ T r−1 H n /T r H n −→ Ern −→ T r−1 (H n+1 [T r ]) −→ 0, and cnr : T r−1 (H n [T r ]) ,→ T r−1 H n  T r−1 H n /T r H n . 311

(11.6.3) Lemma. If each H n = H n (M • ) is a Λ-module of finite type, then ∼

∼

Ern = Ern0 −→ T r0 −1 (H n [T r0 ]) −→ T r−1 (H n [T r ]).

(∃r0 ) (∀r ≥ r0 ) (∀n) n We denote this stationary value by E∞ .

Proof. As M • is cohomologically bounded and each H n is a noetherian Λ-module, there exists r0 ≥ 1 such that H n [T ∞ ] = H n [T r0 −1 ], for all n. Then Ern = H n /(T H n + H n [T r−1 ]) = H n /(T H n + H n [T r0 −1 ]) = Ern0 for all r ≥ r0 . (11.6.4) Lemma 11.6.3 states that, if H is a Λ-module of finite type, then the inductive system of Amodules of finite type T r−1H/T r H (with transition maps given by T ) becomes stationary. Its limit value is isomorphic to ∼

lim T r−1 H/T r H −→ H/(T H + H[T ∞ ]). −→ r (11.6.5) Lemma. Assume that A is a local ring, dim(A) = 0 and each H n = H n (M • ) is a Λ-module of finite type. Then: (i) The following conditions are equivalent: n (∀n) `Λ (H n ) < ∞ ⇐⇒ (∃r0 )(∀n) Ern0 = 0 ⇐⇒ (∀n) E∞ = 0.

(ii) If the equivalent conditions of (i) are satisfied, then (∀r ≥ 1)

0=

X

(−1)n `A (Ern )

n

X XX (−1)n `Λ (H n ) = (−1)n n `A (Ern ). n

r≥1 n

In particular, if the complex (E1• , d1 ) is acyclic, then X

(−1)n `Λ (H n (M • )) =

n

X

(−1)n n `A (H n (M • /T M •)).

n

Proof. (i) If each H n has finite length over Λ, then there is r0 ≥ 1 such that T r0 −1 H n = 0 for all n; then n Ern = 0 for all r ≥ r0 . Conversely, if E∞ = 0, then T r H n = T r+1 H n for some r; Nakayama’s Lemma then r n n implies T H = 0, hence `Λ (H ) < ∞. (ii) By (i), Ern = 0 for r ≥ r0 (and all n). The following exact sequences of Λ-modules of finite length 0 −→ T r−1 H n /T r H n −→ Ern −→ T r−1 (H n+1 [T r ]) −→ 0 T r−1 0 −→ H n+1 [T r−1 ] −→ H n+1 [T r ]−−→H n+1 [T ] −→ H n+1 [T ]/(T r−1H n+1 [T r ]) −→ 0 Tr

0 −→ H n+1 [T r ] −→ H n+1 −−→T r H n+1 −→ 0 imply that `A (Ern ) = `Λ (Ern ) = `Λ (T r−1 H n /T r H n ) + `Λ (T r−1 H n+1 /T r H n+1 ), hence X

(−1)n `A (Ern ) = 0

n

312

(r ≥ 1)

and X

(−1)n n `A (Ern ) =

n

XX

X

(−1)n `Λ (T r−1 H n /T r H n )

n

(−1)

n

n `A (Ern )

=

X

r≥1 n

(−1)n `Λ (H n ).

n

(11.6.6) Lemma-Definition. Assume that A is a local ring, dim(A) = 1 and Λ is catenary. Then the following properties of a Λ-module of finite type H are equivalent:   r−1 r `A lim H/T H < ∞ ⇐⇒ codimΛ (supp(H)) ≥ 1. −→ T r

If they are satisfied, we put

  r−1 r aA (H) := `A lim T H/T H . −→ r

If K • is a cohomologically bounded complex of Λ-modules such that each cohomology module H n (K • ) is of finite type over Λ and satisfies (i), we put X aA (H(K • )) := (−1)n aA (H n (K • )). n

Proof. Set Q = {q ∈ Spec(A) | ht(q) = 0},

Q = {q ∈ Spec(Λ) | q ∈ Q}

(using the notation of 11.6.1.2). For each q ∈ Q we have dim(Λq ) = 1; Lemma 11.6.5(i) applied to (Aq , Λq , Hq ) shows that the following conditions are equivalent: (∀q ∈ Q) `Λq (Hq ) < ∞ ⇐⇒ (∀q ∈ Q)

    r−1 r r−1 r lim T H/T H = 0 ⇐⇒ ` lim T H/T H < ∞. A − −→ → r

q

r

In order to conclude the proof of the Lemma, it remains to show that each minimal prime ideal qΛ ⊂ Λ is contained in some q ∈ Q. As Λ is catenary, dimΛ (Λ/qΛ ) = 2 and dimΛ (Λ/qΛ + T Λ) ≥ 1, hence there is P ∈ Spec(Λ) with P 6= mΛ and P ⊇ qΛ + T Λ. This implies that P = q for some q ∈ Spec(A), q 6= mA , hence q ∈ Q. (11.6.7) If A is a discrete valuation ring, then Λ is a regular local ring (hence a unique factorization domain) of dimension dim(Λ) = 2 and T is an irreducible element of Λ. If H is a finitely generated torsion Λ-module, then its “characteristic power series” charΛ (H) ∈ (Λ − {0})/Λ∗ ⊂ Frac(Λ)∗ /Λ∗ (defined as in 9.1.2) is of the form charΛ (H) = T r(H) char∗Λ (H), where
r(H) = `Λ(T ) H(T ) ,

char∗Λ (H) 6∈ T Λ/Λ∗.

We define char∗Λ (H)(0) := char∗Λ (H) (mod T Λ) ∈ A/A∗ = (Λ/T Λ)/Λ∗ ⊂ Frac(A)∗ /A∗ . The integer r(H) ≥ 0 is “the order of vanishing” of charΛ (H) at T = 0, while char∗Λ (H)(0) is its “leading term”. These invariants (which do not change if we multiply T by a unit of Λ) are usually studied in Iwasawa theory for A = Zp , Λ = Zp [[T ]]. 313

(11.6.8) Lemma. If A is a discrete valuation ring and H a finitely generated torsion Λ-module, then aA (H) = ordA (char∗Λ (H)(0)) . Proof. If H is pseudo-null over Λ, then T r H = T r+1 H for some r, hence aA (H) = 0. As aA (−) is additive in exact sequences, the structure theory for torsion Λ-modules implies that it is enough to consider the case of a cyclic module: H = Λ/T n f Λ, where f ∈ Λ, f (0) = 6 0 (where f (0) := f (mod T ) ∈ Λ/T Λ = A). As ∼ T n H −→ Λ/f Λ and aA (H) = aA (T n H), we can assume that n = 0. Then the map T : H −→ H is injective, hence aA (H) = `A (H/T H) = `A (Λ/(f, T )Λ) = `A (A/f (0)A) = ordA (f (0)). (11.6.9) Euler-Poincar´ e characteristic - notation. Let B be a ring and K • a bounded (resp., cohomologically bounded) complex of B-modules. If each K n (resp., each H n (K • )) is a B-module of finite length, put χB (K • ) :=

X

(−1)n `B (K n )

n

resp., χB (H(K • )) :=

X (−1)n `B (H n (K • )). n

If χB (K ) is defined, so is χB (H(K )), and the two integers coincide. •

•

(11.6.10) Proposition. Assume that A is a local ring, dim(A) = 1, Λ is catenary and each H n = H n (M • ) is a Λ-module of finite type with codimΛ (supp(H n )) ≥ 1. Then: (i) (∃r0 ≥ 1) (∀n) `A (Ern0 ) < ∞. Fix such r0 . (ii) (∀r ≥ r0 ) (∀n) `A (Ern ) < ∞. • (iii) (∀r ≥ r0 ) χA (Er• , dr ) = χA (Er+1 , dr+1 ).P
• (iv) (∀r ≥ r0 ) aA (H(M )) = χA (Er• , dr ) = n (−1)n `A (Coker(cnr−1 )) − `A (Ker(cnr−1 )) . (v) If (∀n) `A (E1n ) < ∞, then aA (H(M • )) =

X

(−1)n (`A (H n /T H n) − `A (H n [T ])) .

n

(vi) If (E1• , d1 )q is acyclic for all minimal prime ideals q ⊂ A, then aA (H(M • )) =

X

(−1)n (`A (Coker(H n [T ] −→ H n /T H n)) − `A (Ker(H n [T ] −→ H n /T H n ))) .

n

Proof. The statement (i) follows from Lemma 11.6.5(i) applied to (Aq , Λq , Mq• ) instead of (A, Λ, M • ), for all minimal prime ideals q ⊂ A. The statements (ii) and (iii) are immediate consequences of (i), as H ∗ (Er• ) = ∗ n Er+1 . For (iv), note that there exists s ≥ r0 such that Esn = E∞ (for all n). Then aA (H(M • )) =

X n (−1)n `A (E∞ ) = χA (Es• ) n

(by Lemma 11.6.3) and we have χA (Es• ) = χA (Er• ) =

X


(−1)n `A (Coker(cnr−1 )) − `A (Ker(cnr−1 ))

n

for all r ≥ r0 , by (iii) and (11.6.2.2). The statements (v) and (vi) are special cases of (iv), for r = r0 = 1 and r = r0 = 2, respectively. 314

(11.6.11) In the special case when A = Zp , Λ = Zp [[T ]] and H n = 0 for n 6= 0, then the statements (v) and (vi) in Proposition 11.6.10 boil down to the following well-known facts about the finitely generated torsion Λ-module H = H 0 : If charΛ (H)(0) 6= 0, then
ordp (charΛ (H)(0)) = ordp |HΓ |/|H Γ | . If the canonical map c : H Γ −→ HΓ has finite kernel and cokernel, then ordp (char∗Λ (H)(0)) = ordp (|Coker(c)|/|Ker(c)|) . (11.6.12) If A is a discrete valuation ring, then aA (H(M • )) = ordA (char∗Λ (H(M • ))(0)) , where char∗Λ (H(M • ))(0) :=

Y

(−1)n

(char∗Λ (H n (M • ))(0))

(11.6.12.1)

n

(by Lemma 11.6.8). The R.H.S. of (11.6.12.1) can be reinterpreted as follows: as Λ is regular, M • is a perfect complex over Λ, with Λ-torsion cohomology. This implies that there is a canonical (up to a sign) isomorphism ∼

detΛ (M • ) ⊗Λ Frac(Λ) −→ Frac(Λ); it identifies detΛ (M • ) with the invertible Λ-module Λ · charΛ (H(M • ))−1 ⊂ Frac(Λ), where charΛ (H(M • )) is defined analogously to (11.6.12.1). Similarly, in the situation of Proposition 11.6.10, for each r ≥ r0 − 1 there is a canonical (up to a sign) isomorphism ∼

detA (Er• , dr ) ⊗A Frac(A) −→ Frac(A); it identifies detA (Er• , dr ) with A · charA (H(Er• , dr ))−1 ⊂ Frac(A), where charA (H(Er• , dr )) =

Y


(−1)n n charA (Er+1 ) .

n

This means that we can reformulate Proposition 11.6.10(iv) in more suggestive terms as follows: det∗Λ (M • )(0) = detA (Er• , dr )

(∀r ≥ r0 − 1).

(11.6.12.2)

This formulation is used, in a special case, in [Bu-Gr] (the idea of using determinants in this context is due to Kato [Ka 1]). It is not clear whether one can make sense of (11.6.12.2) for more general one-dimensional local rings A; the problem is (even for r = 1) to find appropriate conditions guaranteeing that the complex (Er• , dr ) is perfect over A. (11.7) Formulas of the Birch and Swinnerton-Dyer type Perrin-Riou [PR 2-4] and Schneider [Sch 2,3] computed the leading term of the characteristic power series of a Selmer group arising in Iwasawa theory of abelian varieties. Their results (further generalized in [PR 5] and [Pl]) involved p-adic variants of various terms appearing in the conjecture of Birch and Swinnerton-Dyer. In this section we deduce similar formulas in our context, as an application of the formalism from 11.6. 315

(11.7.1) Assume that we are in the situation of 11.1.3, with Γ isomorphic to Zp (hence the condition ∼ (F lk (Γ)) from 11.5.1 holds for all k ≥ 1). Fixing an isomorphism Γ −→ Zp , we identify R = R[[Γ]] with R[[T ]] in the usual way (T = γ − 1). For a prime ideal p0 ∈ Spec(R) consider the following conditions: • (11.7.1.1) πp0 : Xp0 ⊗Rp0 Yp0 −→ (ωR )p0 (1) is a perfect duality (over Rp0 ). + (11.7.1.2) (∀v ∈ Σ) (Xv )p0 ⊥⊥πp0 (Yv+ )p0 . ∼ ∼ gf (X)p0 −→ gf (Y )p0 −→ (11.7.1.3) τ≤0 RΓ τ≤0 RΓ 0 in D(Rp Mod). 0

(11.7.2) The spectral sequence Eri,j (X) (resp., 0 Eri,j (Y )) from 11.5.8 is equal to the spectral sequence Eri,j (M • , T • ) (resp., Eri,j ((N • )ι , T • )) from 11.6.1.4 for A = R, Λ = R[[T ]], where e• (FΓ (X)) = C e• ((X ⊗R R[[T ]]) < −1 >) M• = C f f e• (FΓ (Y )ι ) = C e• ((Y ⊗R R[[T ]]) < 1 >). (N • )ι = C f f As in 11.6.1.4, we have the corresponding stable (simply graded) spectral sequences Ern (X), 0 Ern (Y ) and the surjective maps (T −i )∗ : Eri,j (X)  Eri+j (X),

(T −i )∗ : 0 Eri,j (Y )  0 Eri+j (Y ),

(i ≥ 0).

As ι(T ) = (1 + T )−1 − 1 ≡ −T (mod T 2 ), we have (0 Ern (Y ), 0 dnr ) = (Ern (Y ), (−1)r dnr ). The cup products ∪r from 11.5.3 are compatible with T∗ , hence induce “stable” cup products ∪st r fitting to the following commutative diagram: 0

0

Eri,j (X) ⊗R 0 Eri ,j (Y ) −−−−→   −i 0 y(T )∗ ⊗(T −i )∗

∪r :

∪st r :

0

Ern (X) ⊗R 0 Ern (Y )

0

H i+i +j+j

0

−3

0

(ωR ) ⊗R R · T i+i   −i−i0 y(T )∗

(11.7.2.1)

0

H n+n −3 (ωR )

−−−−→

(in which n = i + j, n0 = i0 + j 0 , i, i0 ≥ 0, r ≥ 1). The formula (11.5.3.2) holds also for ∪st r , hence the adjoint map adj(∪st ) induces, for each q ∈ Z, a morphism of complexes r • • 0 • 0 q adjq (∪st r ) : (Er (X), dr ) −→ HomR (( E (Y ), dr ), H (ωR )[−3 + q]).

The most interesting case is that of q = 0: • 0 • • 0 adj0 (∪st r ) : Er (X) −→ HomR ( E (Y ), H (ωR )[−3])

These maps determine inductively each other, as the components of

(r ≥ 1). adj0 (∪st r+1 )

are equal to

adj (∪st )∗

0 r • 0 n n Er+1 (X) = H n (Ern (X))−−−−−−→H (HomR ( E • (Y ), H 0 (ωR )[−3])) −→ HomR (0 E 3−n (Y ), H 0 (ωR )). (11.7.2.2) For each r ≥ 1, we define “stable” higher height pairings by the formula i

st

dr ⊗id ∪r (r),st i+1 0 j i+j−2 e hπ,i,j : Eri (X) ⊗R 0 Erj (Y )−−−−→E (ωR ). r (X) ⊗R Er (Y )−−→H

These pairings satisfy the formula in Proposition 11.5.6(ii) and depend on the choice of γ ∈ Γ; if we replace γ by γ c (c ∈ Z∗p ), then T is replced by (1 + T )c − 1 ≡ cT (mod T 2 ), 316

(r),st hence dir and e hπ,i,j are multiplied by c−r . The commutative diagram (r) e hπ,i,j :

Er0,i (X) ⊗R 0 Er0,j (Y )  1 ⊗1 y∗ ∗

−−−−→

H i+j−2 (ωR ) ⊗R R · T r   −r y(T )∗

(r),st e hπ,i,j :

Eri (X) ⊗R 0 Erj (Y )

−−−−→

H i+j−2 (ωR )

(r) (induced by (11.7.2.1)) shows that e hπ,i,j factors through the canonical surjective maps

e i (X)) = E 0,i (X)  E i (X), F ilΓr (H f r r

e j (Y )) = 0 E 0,j (Y )  0 E j (Y ). F ilΓr (H r r f

(11.7.3) Proposition. Assume that q ∈ Spec(R), ht(q) = 0 and the conditions (11.7.1.1-2) are satisfied for p0 = q. Then: (i) The localized maps ∼

n 0 3−n adj0 (∪st (Y )q , H 0 (ωR )q ) r )q : Er (X)q −→ HomRq ( Er

(r ≥ 1)

are all isomorphisms.  
e (r),st (ii) (∀i) = Ker (dir )q : Eri (X)q −→ Eri+1 (X)q . L Ker hπ,i,2−i q

Proof. (i) For r = 1 the statement follows from the duality Theorem 6.3.4, as observed in the proof of Proposition 11.5.7. Assume that the claim is proved for r ≥ 1; then the localization at q of the first (resp., the second) arrow in (11.7.2.2) is an isomorphism by the induction hypothesis (resp., because H 0 (ωR )q = IRq is an injective Rq -module), proving the statement for r + 1. The part (ii) follows from (i) and the definition of e h(r),st . (11.7.4) Proposition. Assume that p ∈ Spec(R), Rp has no embedded primes and the conditions (11.7.1.13) are satisfied for all p0 ∈ Q = {q ∈ Spec(R) | q ⊂ p, ht(q) = 0}. Then  e (r) L Ker(hπ,1,1 )p

e 1 (X)p −→ = Ker H f

M

 e 1 (X)q /(F il r+1(H e 1 (X)))q  H f f Γ

q∈Q



   M (r) e 1 (Y )p −→ e 1 (Y )q /(F il r+1 (H e 1 (Y )))q  KerR e hπ,1,1 = Ker H H f f f Γ p

(r ≥ 1).

q∈Q

Proof. The symmetry property from Proposition 11.5.6(i) (which also holds for the stable height pairings) shows that it is sufficient to consider only the left kernel. The same argument as in the proof of Proposition 11.5.7 reduces to the case p = q, ht(q) = 0. The assumption (11.7.1.3) and the duality isomorphisms ∼ e i (X)q −→ e 3−i (Y )q , H 0 (ωR )q ) H HomRq (H f f

imply that (∀j 6= 1, 2)

e j (Z)q = 0 H f

(Z = X, Y ).

This means that Eri,j (X)q , 0 Eri,j (Y )q satisfy (11.6.1.5) for j0 = 1, hence all the maps ∼

1∗ : Er0,1 (X)q −→ Er1 (X)q ,

∼

1∗ : 0 Er0,1 (Y )q −→ 0 Er1 (Y )q ,

are isomorphisms. It follows that     (r) (r),st e hπ,1,1 = e hπ,1,1 q

317

q

(r ≥ 1)

(11.7.4.1)

and  
∼ (r) e hπ,1,1 = Ker (d1r )q : Er1 (X)q −→ Er2 (X)q −→ q
∼ 0,1 e 1 (X))q , −→ Ker (dr )q : Er0,1 (X)q −→ E r,2−r (X)q = F ilΓr (H f L Ker

by Proposition 11.7.3(ii) and (11.7.4.1). (11.7.5) We now fix p ∈ Spec(R) with ht(p) = 1 and put Q = {q ∈ Spec(R) | q ⊂ p, q 6= p} (i.e. ht(q) = 0 for all q ∈ Q). We are going to apply the discussion from 11.6 to the localized objects A = Rp , Λ = Rp (using e• (FΓ (X))p (resp., (N • )ι = C e • (FΓ (Y )ι )p ). The reader should keep the notation from 11.6.1.2) and M • = C f f in mind the following simplest case: R = Zp , R = Zp [[T ]], p = (p), p = (p, T ), A = Zp , Λ = Zp [[T ]], q = (0), q = (T ). (11.7.6) Proposition. Let q ∈ Q and assume that the conditions (11.7.1.1-3) hold for p0 = q. Then: e j (Z)q = 0 (i) (∀j 6= 1, 2) H (Z = X, Y ). f (ii) The following conditions are equivalent: (∀n) `R

q

 en H

f,Iw (X)q



< ∞ ⇐⇒ (∀n) `R

q

 en H

f,Iw (Y )q



< ∞ ⇐⇒

e 1 (X))q = F il r+1 (H e 1 (Y ))q = 0 ⇐⇒ ⇐⇒ (∃r ≥ 1) F ilΓr+1 (H f f Γ   (r) r 1 r 1 0 e e f (X))q × F ilΓ (H e f (Y ))q −→ H (ωR )q has trivial left and right kernels. hπ,1,1 : F ilΓ (H

⇐⇒ (∃r ≥ 1)

q

If these equivalent conditions hold, then:     r e1 e 1 (Y ))q , (iii) (∀r ≥ 1) `Rq F ilΓ (Hf (X))q = `Rq F ilΓr (H f     1 2 e e `Rq Hf (Z)q = `Rq Hf (Z)q e n (Z)q = 0 (iv) (∀n 6= 2) H (Z = X, Y ).

(Z = X, Y ).

f,Iw

e 1 (X))q = 0, then (v) If r0 ≥ 1 is the smallest positive integer for which F ilΓr0 +1 (H f `R

q

r0   X   2 e f,Iw e f1 (X))q H (Z)q = `Rq F ilΓr (H

(Z = X, Y ).

r=1

(vi) r0 ≥ 1 from (v) is the smallest positive integer for which   (r0 ) e e 1 (X))q × F il r0 (H e 1 (Y ))q −→ H 0 (ωR )q hπ,1,1 : F ilΓr0 (H f f Γ q

has trivial leftand right kernels.    e 2 (Z)q ≥ `Rq H e 1 (X)q (vii) `R H f,Iw f q with equality occurring if and only if

(Z = X, Y ),

  e e 1 (X)q × H e 1 (Y )q −→ H 0 (ωR )q hπ,1,1 : H f f q

has trivial left and right kernels. Proof. The statement (i) was proved in 11.7.4. As regards (ii), Lemma 11.6.5(ii) applied to Aq , Λq and Mq• (resp., (N • )ιq ) shows (if we take into account (i)) equivalences (∀n) `R

q



 e n (Z)q < ∞ ⇐⇒ (∃r ≥ 1) E 1 (Z)q = E 2 (Z)q = 0 H f,Iw r r 318

(Z = X, Y ).

(11.7.6.1)

The duality result in Proposition 11.7.3(i) gives

`Rq Eri (X)q = `Rq Er3−i (Y )q ,

(11.7.6.2)

and (11.7.4.1) yields ∼ e 1 (Z))q Er1 (Z)q −→ F ilΓr (H f

(Z = X, Y ).

(11.7.6.3)

Combining the statements (11.7.6.1-3) we obtain (ii). Assuming that the equivalent conditions in (ii) are satisfied, Lemma 11.6.5(ii) gives

0 = `Rq Er2 (Z)q − `Rq Er1 (Z)q , which proves (iii), if we take into account (11.7.6.2-3). For Z = X, Y , the exact sequences e n (Z)q /T H e n (Z)q −→ H e n (Z)q −→ H e n+1 (Z)q [T ] −→ 0 0 −→ H f,Iw f,Iw f f,Iw together with (i) and the Nakayama Lemma show that (∀n 6= 1, 2)

e n (Z)q = 0, H f,Iw

e 1 (Z)q [T ] = 0, H f,Iw

which proves (iv), since `R

q

  e 1 (Z)q < ∞. H f,Iw

The formula (v) follows from (iii), (iv), Lemma 11.6.5(ii) and (11.7.6.2-3):

`R

q



 X  


X
X 2 1 1 r e1 e 2 (Z)q = H 2` E (Z) − ` E (Z) = ` E (X) = ` F il ( H (X)) Rq q Rq q Rq q Rq q . f,Iw r r r Γ f r≥1

r≥1

r≥1

Finally, the statement (vi) follows from Proposition 11.7.4 for p = q, and (vii) is a direct consequence of (v) and (vi). (11.7.7) Lemma. Assume that, for each q ∈ Q, the conditions (11.7.1.1-3) (for p0 = q) and 11.7.6(ii) hold. Then, for each Z = X, Y : n n (i) (∀n) codimR (supp(Hf,Iw (Z)p )) ≥ 1, hence the integer aRp (Hf,Iw (Z)p ) is defined. p r0 +1 e 1 (ii) If r0 ≥ 1 and codimRp (supp(F il (H (X))p )) ≥ 1, then Γ

X

f

n (−1)n aRp (Hf,Iw (Z)p ) = χRp (H(Er•0 (Z)p )) = χRp (Er•0 +1 (Z)p ).

n

Proof. The condition 11.7.6(ii) for any q ∈ Q implies (i), while (ii) follows from Proposition 11.6.10(iv) (for e• (FΓ (X))p ) and Proposition 11.7.6(iv). A = Rp , Λ = R p , M • = C f (11.7.8) Recall that, for each r ≥ 1, we have a morphism of complexes • 0 0 • • adj0 (∪st r ) : Er −→ HomR ( Er (Y ), H (ωR )[−3])

such that the composite map i 0 2−i adj0 (∪st (Y ), H 0 (ωR )) r )i+1 ◦ dr : Er (X) −→ HomR ( Er (r),st is equal to adj(e hπ,i,2−i ):

319

···

···

dr

dr dr / E i (X) / E i+1 (X) UUUU r r UUUU UUUUadj(eh) UUUU UUUU UUUU  *  / HomR (0 E 3−i (Y ), H 0 (ωR )) / HomR (0 E 2−i (Y ), H 0 (ωR )) r r

/ ···

(11.7.8.1)

/ ···

We form a new complex Er• (X, Y, π) by traversing the diagonal arrow with i = 1 in (11.7.8.1): Er• (X, Y, π) : · · · −→ Er0 (X) −→ Er1 (X) −→ HomR (0 Er1 (Y ), H 0 (ωR )) −→ HomR (0 Er0 (Y ), H 0 (ωR )) −→ · · · For r = 1, the complex E1• (X, Y, π) is equal to d01 Hom(d01 ,id) adj(e hπ,1,1 ) 0 0 e f0 (X)−→ e f1 (X)−−−−−−→Hom e1 e0 · · · −→ H H R (Hf (Y ), H (ωR ))−−−−−−→HomR (Hf (Y ), H (ωR )) −→ · · ·

(11.7.9) Lemma. Assume that the conditions (11.7.1.1-3) hold for p0 = p and depth(Rp ) = 1(= dim(Rp )). Then: (i) (∀r ≥ 1) (∀i 6= 1, 2, 3) Eri (Z)p = 0, Er3 (Z)p is Rp -torsion (Z = X, Y ). (ii) The only possibly non-zero components of the localization of (11.7.8.1) at p are 0

0

/ E 1 (X)p / E 2 (X)p r r VVVV VVVV VVVV VVVV VVVV VVVV  *  0 2 0 / HomRp ( Er (Y )p , H (ωR )p ) / HomRp (0 Er1 (Y )p , H 0 (ωR )p )

/ E 3 (X)p r  /0

e i (Z). The assumptions (11.7.1.1-2) Proof. It is sufficient to prove the Lemma for r = 1, when E1i (Z) = H f yield the exact triangle gf (X)p −→ RHomRp (RΓ gf (Y )p , ωRp ) −→ Err(∆(X), ∆(Y ), π)p RΓ in Df t (Rp Mod), which induces the following two exact sequences (by Lemma 2.10.11(ii)): 0   y e 4−i (Y )p )Rp −tors ) DRp ((H f   y · · · −→ H i−1 (Err)p

−→

e i (X)p H f

−→

gf (Y )p , ωRp )) −→ H i (RHomRp (RΓ   y

H i (Err)p

−→ · · ·

e 3−i (Y )p , H 0 (ωR )p ) HomRp (H f   y 0, (11.7.9.1) e i (X)p = where the error terms H i (Err)p vanish for i 6= 1, 2 and are torsion over Rp for i = 1, 2. As H f e i (Y )p = 0 for i ≤ 0 by the assumption (11.7.1.3), it follows from (11.7.9.1) that H e i (X)p = 0 for i > 3 H f f 320

e 3 (X)p is Rp -torsion. Interchanging X, Y and replacing π by π ◦ s12 gives the same result for Y , and that H f proving (i). The statement (ii) follows from (i) and the fact that H 0 (ωR )p is torsion-free over Rp . (11.7.10) Under the assumptions of Lemma 11.7.9, the diagram (11.7.9.1) yields a complex 0 −→

e 1 (Xp )tors H f

−→

e 3 (Yp )tors ) DRp (H f

−→

H 1 (Errp )

−→

−→

e 2 (Xp )tors H f

−→

e 2 (Yp )tors ) DRp (H f

−→

H 2 (Errp )

−→

−→

e 3 (Xp )tors H f

−→

e 1 (Yp )tors ) DRp (H f

−→

0,

(11.7.10.1)

in which we write e i (Zp )tors = (H e i (Z)p )Rp −tors , H f f

Err = Err(∆(X), ∆(Y ), π).

The complex (11.7.10.1) is acyclic everywhere except possibly at the terms H i (Errp ); denote its cohomology at these two terms by H(H i (Errp )) = H(H i (Err(∆(X), ∆(Y ), π)p ))

(i = 1, 2).

According to 7.6.10.7 we have X
`Rp H i (Errp ) = Tamv (X, p)

(i = 1, 2),

v∈Σ0

hence X
0 ≤ `Rp H(H i (Errp )) ≤ Tamv (X, p).

(11.7.10.2)

v∈Σ0

The generalized Cassels-Tate pairings 10.5.3 and the isomorphisms (10.2.8.3) show that the DRp -dual of (11.7.10.1) is isomorphic to the same sequence in which (X, Y, π) are replaced by (Y, X, π ◦ s12 ). This implies that

`Rp H(H i (Err(∆(X), ∆(Y ), π)p )) = `Rp H(H 3−i (Err(∆(Y ), ∆(X), π ◦ s12 )p ))

(i = 1, 2)

and 3 X i=1 3 X

 
e fi (Xp )tors ) + `Rp H(H 2 (Err(∆(X), ∆(Y ), π)p )) = (−1)i `Rp H  
e fi (Yp )tors ) + `Rp H(H 2 (Err(∆(Y ), ∆(X), π ◦ s12 )p )) . (−1) `Rp H

(11.7.10.3)

i

i=1

(11.7.11) Theorem. Assume that depth(Rp ) = dim(Rp ) = 1 and that the conditions (11.7.1.1-3) hold for p0 = p. If, for all q ∈ Q, the pairings e f1 (X)q × H e f1 (Y )q −→ H 0 (ωR )q (e hπ,1,1 )q : H have trivial left and right kernels, then (for Z = X, Y ):  we have   1    `R H e (X) , n=2 q q f e n (Z)q = (i) (∀q ∈ Q) `R H f,Iw q  0, n 6= 2. e n (Z)p is an (Rp )-module of finite type and codim (supp(H e n (Z)p )) ≥ 1. (ii) (∀n) H f,Iw f,Iw Rp (iii) (The formula of the Birch and Swinnerton-Dyer type) X n

3     X   n e f,Iw e fi (X)p )Rp −tors + (−1)n aRp H (X)p = `Rp det((e hπ,1,1 )p ) + (−1)i `Rp (H i=1


+ `Rp H(H 2 (Err(∆(X), ∆(Y ), π)p ) , 321

where the first term on the right is defined as in 2.10.21, i.e. is equal to    e f1 (X)p −→ HomRp (H e f1 (Y )p , H 0 (ωR )p ) , `Rp Coker adj((e hπ,1,1 )p ) : H and

X
0 ≤ `Rp H(H 2 (Err(∆(X), ∆(Y ), π)p )) ≤ Tamv (X, p). v∈Σ0

Proof. As the conditions (11.7.1.1-3) hold for p0 = p, they are also satisfied by all p0 = q ∈ Q. Thus (i) follows from Proposition 11.7.6(vii), and (ii) is a consequence of (i). In order to prove (iii), Lemma 11.7.7(ii) for r0 = 1 together with Lemma 11.7.9 give X

  e n (X)q = χRp (H(E • (X)p )) = χRp (H(σ≤2 E • (X)p )) − `Rp ((H e 3 (X)p )Rp −tors ). (−1)n aRp H f,Iw 1 1 f

n

The exact triangle σ≤2 E1• (X)p −→ E1• (X, Y, π)p −→ Cone((λ2r )p )[−2],

(11.7.11.1)

where λnr is a shorthand for n 0 3−n λnr = adj0 (∪st (Y ), H 0 (ωR )), r )n : Er (X) −→ HomR ( Er

then yields χRp (H(σ≤2 E1• (X)p )) = χRp (H(E1• (X, Y, π)p )) + `Rp (Ker((λ21 )p )) − `Rp (Coker((λ21 )p )). As the complex E1• (X, Y, π)p ) coincides with e 1 (X)p −→ HomRp (H e 1 (Y )p , H 0 (ωR )p )] [adj((e hπ,1,1 )p ) : H f f in degrees 1, 2 (by Lemma 11.7.9(ii)), it follows that e f1 (X)p )Rp −tors ). χRp (H(E1• (X, Y, π)p )) = `Rp (det((e hπ,1,1 )p )) − `Rp ((H A diagram chase on (11.7.9.1) for i = 2 shows that ∼

e 2 (X)p )Rp −tors , Ker((λ21 )p ) = (H f

Coker((λ21 )p ) −→ H 2 (H 2 (Errp )).

Putting everything together, we obtain (iii). (11.7.12) By (11.7.10.3), Lemma 2.10.20(iv) and Theorem 11.7.11(iii),   X   X e n (X)p = e n (Y )p . (−1)n aRp H (−1)n aRp H f,Iw f,Iw n

n

This can also be deduced from the exact triangle gf,Iw (X)p −→ D (RΓ gf,Iw (Y )p )ι [−3] −→ Err(FΓ (X), FΓ (X)ι , FΓ (π))p , RΓ R p

as the contributions from the error term Err cancel each other. (11.7.13) Attentive readers will have noticed that the statements and proofs in 11.7.5-12 depend only on Zp , (Zv+ )p , πp (Z = X, Y ). This implies that everything in these sections works in the context of 10.5.1. ∼

(11.7.14) Many results of Sect. 11.7 also hold in the case when Γ −→ Z/pm Z (m ≥ 1), pm · R = 0, ∼ dim(R) = 0, r ≤ p − 1 (as R/J p −→ R[T ]/(T p), where T = γ − 1 for a fixed generator γ ∈ Γ). (11.7.15) A variant of Theorem 11.7.11 in non-commutative Iwasawa theory is proved in [Bu-Ve], Thm. 6.7. 322

(11.8) Higher height pairings and the Cassels-Tate pairing ∼ In this section we assume that Γ −→ Zp . We show that, in this case, the J-adic graded quotients of the generalized Cassels-Tate pairing over R coincide (up to a sign) with higher height pairings. (11.8.1) Consider first the abstract situation: let (A, Λ, T ) be as in 11.6.1.1 and denote the complex −i [Λ−−→ΛT ] in degrees 0, 1 (where i denotes the canonical map) by C • . For a complex X • of Λ-modules denote by H!i (X • ) the cohomology groups of the complex RΓ! (X • ) = X • ⊗Λ C • . As in 2.10.7-9, there is an exact sequence of T -primary torsion Λ-modules 0 −→ H i−1 (X • ) ⊗Λ (ΛT /Λ) −→ H!i (X • ) −→ H i (X • )[T ∞ ] −→ 0 (the first term of which is T -divisible), quasi-isomorphisms u

v

C • −−→C • ⊗Λ C • −−→C • ,

vu = id,

uv −

id

and products s

id⊗v

23 • (X • ⊗Λ C • ) ⊗Λ (Y • ⊗Λ C • )−−→(X ⊗Λ Y • ) ⊗Λ (C • ⊗Λ C • )−−−−→(X • ⊗Λ Y • ) ⊗Λ C •

(note that both id ⊗ v and id ⊗ u are quasi-isomorphisms), which induce products H!i (X • ) ⊗Λ H!j (Y • ) −→ H!i+j (X • ⊗Λ Y • ) factoring through ∪ij : H i (X • )[T ∞ ] ⊗Λ H j (Y • )[T ∞ ] −→ H!i+j (X • ⊗Λ Y • ). If X • is a complex of T -flat Λ-modules, then the canonical projection C • −→ (ΛT /Λ)[−1] induces isomorphisms ∼

H!i (X • ) −→ H i−1 (X • ⊗Λ (ΛT /Λ)). (11.8.2) Assume that we are given the following data: (11.8.2.1) An involution ι : Λ −→ Λ satisfying ι(T ) ≡ εT (mod T 2 ), where ε = ±1. The induced involution on A = Λ/T Λ will also be denoted by ι. (11.8.2.2) Complexes of T -flat Λ-modules M • , N • , P • . (11.8.2.3) A morphism of complexes π : M • ⊗Λ (N • )ι −→ P • [−3], where (N • )ι = N • ⊗Λ,ι Λ. The construction recalled in 11.8.1 then yields products ∪π,i,j : H i (M • )[T ∞ ] ⊗Λ H j (N • )[T ∞ ]ι −→ H i+j−4 (P • ⊗Λ (ΛT /Λ)), which induces for each r ≥ 1 pairings H i (M • )[T r ] ⊗Λ H j (N • )[T r ]ι −→ H i+j−4 (P • ⊗Λ (T −r Λ/Λ)) and (r)

h , iπ,i,j :

H i (M • )[T r ] H j (N • )[T r ]ι ⊗ −→ H i+j−4 (P • ⊗Λ (T −r Λ/T −r+1Λ)). A H i (M • )[T r−1 ] H j (N • )[T r−1 ]ι

(11.8.3) For each of the complexes Z • = M • , N • , P • [−3] the T -adic filtration on Z • (resp., on (Z • )ι ) induces a spectral sequence (Eri,j (Z • ), dr ) (resp., (0 Eri,j (Z • ), 0 dr ) = (Eri,j (Z • ), εr dr )) and the corresponding stable spectral sequence (Ern (Z • ), dr ) (resp., (0 Ern (Z • ), 0 dr ) = (Ern (Z • ), εr dr )). As in 11.5.3 we obtain from 11.8.2.3 products 0

0

0

0

∪r : Eri,j (M • ) ⊗A 0 Eri ,j (N • )ι −→ Eri+i ,j+j (P • [−3]) satisfying 323

dr (x ∪r y) = (dr x) ∪r y + (−1)i+j x ∪r (0 dr y). We assume, from now on, that •,• E1•,• (P • [−3]) = E∞ (P • [−3]).

(11.8.3.1)

This allows us to define abstract height pairings by the same recipe as in 11.5.5: d0,i ⊗id

(r)

∪

r r r,i+1−r r,i+j+1−r hπ,i,j : Er0,i (M • ) ⊗A 0 Er0,j (N • )ι −−−−→E (M • ) ⊗A 0 Er0,j (N • )ι −−→E (P • [−3]) = r r

= E1r,i+j+1−r (P • [−3]) = H i+j−2 (P • ⊗Λ (T r Λ/T r+1Λ)) 0

0

0

0

(note that in 11.5.3 the involution ι acts trivially on R, hence 0 Eri ,j (Y • ) = 0 Eri ,j (Y • )ι then). Recall from 11.6.2 that the differentials (depending on T ) in Ern (Z • ) are given by the following formula: jn

T 1−r

r r−1 dnr : Ern (Z • )−−→T (H n+1 (Z • )[T r ])−−→

H n+1 (Z • )[T r ] in+1 r n+1 −−→E (Z • ). r H n+1 (Z • )[T r−1 ]

(11.8.4) Proposition. The pairing (T 1−r j i ⊗T 1−r j j )

1 ⊗1

∗ ∗ r r i 0 j • • ι h : Er0,i (M • ) ⊗A 0 Er0,j (N • )ι −−−−→E r (M ) ⊗A Er (N ) −−−−−−−−−−→ H i+1 (M • )[T r ] H j+1 (N • )[T r ]ι f T 2r −→ i+1 ⊗ −→H i+j−2 (P • ⊗Λ (T −r Λ/T −r+1Λ))−−→ A • • r−1 j+1 r−1 ι H (M )[T ] H (N )[T ]

−→ H i+j−2 (P • ⊗Λ (T r Λ/T r+1Λ)) (r)

(r)

(where the arrow f is equal to f = h , iπ,i+1,j+1 ) is equal to h = (−1)j+1 hπ,i,j . Proof. An element a ∈ Er0,i (M • ) (resp., b ∈ 0 Er0,j (N • )ι ) is represented by α0 ∈ M i (resp., β 0 ∈ (N j )ι ) (r) satisfying dα0 = T r α (resp., dβ 0 = T r β) for some α ∈ M i+1 (resp., β ∈ (N j+1 )ι ). Then hπ,i,j (a ⊗ b) = r,i+j+1−r d0,i (P • [−3]) is represented by dα0 ∪ β 0 = T r α ∪ β 0 . On the other hand, the same r (a) ∪r b ∈ Er calculation as in the proof of Lemma 10.1.6 shows that the value of the pairing h(a ⊗ b) is represented by e where T 2r v ◦ s23 (e α ∪ β), βe = β ⊗ 1 + (−1)j+1 β 0 ⊗ T −r ,

α e = α ⊗ 1 + (−1)i+1 α0 ⊗ T −r , i.e. by

T 2r (α ∪ β) ⊗ 1 + (−1)j+1 (α ∪ β 0 ) ⊗ T −r




(mod P • [−3] ⊗Λ T r+1 Λ),

which proves the claim. ∼

(11.8.5) Assume that we are in the situation of 11.1.3 with Γ −→ Zp (γ 7→ 1) and q ∈ Spec(R) is a prime ideal with ht(q) = 0 such that Rq is an integral domain ( ⇐⇒ Rq is a field). Denote by q ⊂ R = R[[Γ]] = R[[γ − 1]] the inverse image of q under the augmentation map; the localization Λ = Rq is a discrete valuation ring with involution ι (induced by the standard involution on R), T = γ − 1 is a prime element of Λ and ι(T ) ≡ −T (mod T 2 ). The discussion in 11.8.1-4 applies to the complexes e• (FΓ (X)) , M• = C f q

e • (FΓ (Y )) , N• = C f q

• P • = (ωR ⊗R R)q

and the cup product ∪ : M • ⊗Λ (N • )ι −→ P • [−3] from (11.5.3.1). (r)

If i + j = 2, then the pairing h , iπ,i+1,j+1 defined in 11.8.2 is equal to the r-th graded quotient (with respect to the T -adic = q-adic filtration) of the localized generalized Cassels-Tate pairing 324

  e i+1 (K∞ /K, X) H f,Iw q

(Rq )−tors

⊗R

 q

e j+1 (K∞ /K, Y )ι H f,Iw q

 (Rq )−tors

−→ Frac(Rq )/Rq

(i + j = 2).

According to Proposition 11.8.4, it is equal (up to a sign) to the localization at q of the higher height pairing (r) e hπ,i,j from 11.5.5. In the special case R = Zp , Howard [Ho 3] uses this description as a definition of higher height pairings. (11.9) Higher height pairings and parity results ∼ In this section we continue to assume that Γ −→ Zp , γ 7→ 1. See [Ho 3] for similar results in the special case R = Zp . (11.9.1) Proposition. Assume that we are in the situation of 11.7.1 and the conditions (11.7.1.1-3) hold for p0 = q ∈ Spec(R), ht(q) = 0. Let q ⊂ R = R[[T ]] = R[[γ − 1]] be the inverse image of q under the augmentation map. Then, for each Z = X, Y and n ∈ Z, (i) There is an exact sequence n n e f,Iw e f,Iw e fn (Z)q −→ H e n+1 (Z)q [T ] −→ 0. 0 −→ H (Z)q /T H (Z)q −→ H f,Iw

e n (Z)q = 0; H e 1 (Z)q [T ] = 0. (ii) (∀n 6= 1, 2) H f,Iw f,Iw   e 1 (Z) is equal to the inverse image of T r−1 H e 2 (Z)q [T r ] ; in particular, (iii) For each r ≥ 1, F ilΓr H f f,Iw q   e 1 (Z) coincides with the submodule of universal norms Im H e 1 (Z)q /T ,→ H e 1 (Z) . F ilΓ∞ H f f,Iw f q q (r)

(iv) For each r ≥ 1, hπ,1,1 induces a non-degenerate paring on the graded quotients e 1 (X) × grr H e 1 (Y ) −→ H 0 (ωR )q . ( , )r : grr H f f q q (v) If X = Y , Xv+ = Yv+ (v ∈ Σ) and π ◦ s12 = −π, then the pairing ( , )r is symmetric (resp, skewsymmetric) for r odd (resp., r even). Proof. The statements (i) and (ii) were proved in the course of the proof of Proposition 11.7.6; (iii) follows from (11.6.2.1) and 11.6.1.5 (which applies with j0 = 1, by the proof of Proposition 11.7.4). Finally, (iv) (resp., (v)) follows from Proposition 11.7.4 (resp., Proposition 11.5.6(ii)). (11.9.2) Proposition (Dihedral case). In the situation of Proposition 11.9.1, assume that we are given 0 K∞ /K + as in 10.3.5.1 satisfying (10.3.5.1.1-5). For ε, ε0 = ±, denote by ( , )εε the restriction of ( , )r to r 0 ε ε r e1 r e1 ± τ =±1 gr H (X) × gr H (Y ) , where (−) = (−) . Assume that 2 is invertible in Rq . Then f

q

f

q

(i) For r odd, ( , )++ = ( , )−− = 0 and the pairings r r ±

∓

e 1 (X) × grr H e 1 (Y ) −→ H 0 (ωR )q ( , )±∓ : grr H r f f q q are non-degenerate. In particular,

    e 1 (X)± = `Rq grr H e 1 (Y )∓ . `Rq grr H f f q q

(ii) For r even, ( , )+− = ( , )−+ = 0 and the pairings r r ±

±

e f1 (X) × grr H e f1 (Y ) −→ H 0 (ωR )q ( , )±± : grr H r q q are non-degenerate. Proof. By Proposition 11.5.9, (τ x, τ y)r = (−1)r (x, y)r , hence 0

0

r 0 εε ( , )εε r = (−1) εε ( , )r . 0

r 0 As 2 is invertible in Rq , it follows that ( , )εε r = 0 if (−1) εε = −1. The rest follows from the non-degeneracy of ( , )r .

325

(11.9.3) Proposition (Self-dual dihedral case). In the situation of Proposition 11.9.2, assume that X = Y , Xv+ = Yv+ (v ∈ Σ) and π ◦ s12 = −π. Then (i) For r odd,     e f1 (X) = 2`Rq grr H e f1 (X)± ≡ 0 (mod 2). `Rq grr H q

q

e 1 (X)± and (ii) For r even, ( , )±± is a symplectic (= non-degenerate alternating) pairing on grr H r f q   e f1 (X)± ≡ 0 (mod 2). `Rq grr H q   e 1 (X) /(F il ∞ )q ≡ 0 (mod 2). (iii) `Rq H Γ f q (iv) Let q0 = qR ∈ Spec(R). Then     1 e f1 (X) ≡ ` e f,Iw 0 `Rq H H (X) (mod 2). q R 0 q q

e 1 (X)± are equal, by Proposition 11.9.2(i). Proof. (i) The lengths of grr H f q (ii) This follows from Proposition 11.9.2(ii) and Lemma 10.7.4(ii). (iii) Combining (i) and (ii), we obtain   X   r e1 e f1 (X) /(F ilΓ∞ )q = ` Rq H ` gr H (X) ≡ 0 (mod 2). R q f q q r≥1

e 1 (X)q [T ] = 0, we have (iv) As H f,Iw

    1 1 e f,Iw e f,Iw `Rq ((F ilΓ∞ )q ) = `Rq H (X)q /T = `R 0 H (X)q0 . q

(11.9.4) In the special case when Rq is an integral domain ( ⇐⇒ Rq is a field), we have H 0 (ωR )q = Rq and Rq is a discrete valuation ring with prime element T = γ − 1. The statement (resp., the proof) of Proposition 11.9.3 then boils down to that of Lemma 10.6.8, as the pairing h , ir,T from loc. cit. coincides, up to a sign, with the pairing ( , )r , by 11.8.5.

326

12. Parity of ranks of Selmer groups In this chapter we generalize the parity results of [Ne 3] to elliptic curves over totally real number fields and to Selmer groups attached to Hilbert modular forms of parallel (even) weight. (12.1) The general setup (12.1.1) Let F and L be number fields (contained in a fixed closure Q of Q) and M a “motive” Palgebraic ∞ −s over F with coefficients in L. The L-function L(M, s) = a n (assuming it is well-defined) is a n n=1 P∞ Dirichlet series with coefficients an ∈ L. For each embedding ι : Q ,→ C, define L(ιM, s) = n=1 ι(an )n−s . Conjecturally, L(ιM, s) admits a meromorphic continuation to C and a functional equation of the form ?

L∞ (ιM, s) L(ιM, s) = ε(ιM, s) L∞ (ιM ∗ (1), −s) L(ιM ∗ (1), −s), where L∞ (ιM, s) =

Y

Lv (ιM, s)

v|∞

is a suitable product of Γ-factors (independent on ι) and Y ε(ιM, s) = εv (ιM, s) = ι(ε(M, 0)) · f (M )−s , v ∗

where ε(M, 0) ∈ Q is the global ε-factor of M and f (M ) ∈ N is the conductor of M . (12.1.2) If M is a pure motive of weight w = 6 −2, the Bloch-Kato Conjectures ([B-K], [Fo-PR], [Fo 1]) predict that ?

1 ords=0 L(ιM, s) = dimL Hf,M (F, M ∗ (1)),

(CBK )

1 where Hf,M is a suitable motivic cohomology group (in particular, the L.H.S. should not depend on ι). For each prime p | p of L, let Mp be the p-adic realization of M ; this is a continuous Lp -representation of GF,S , for a suitable finite set of primes S. Bloch and Kato conjectured that the “p-adic realization map” induces an isomorphism

(Cp )

∼

1 Hf,M (F, M ∗ (1)) ⊗L Lp −→ Hf1 (F, Mp∗ (1)),

where the R.H.S. is the Bloch-Kato generalized Selmer group. Combining (CBK ) with (Cp ) leads to a much weaker (and more accessible) version of the conjecture (CBK ), namely ?

ords=0 L(ιM, s) = dimLp Hf1 (F, Mp∗ (1)).

(CBK,p )

∼

(12.1.3) We shall concentrate on the case of a self-dual (pure) motive M −→ M ∗ (1). In this case w = −1 and s = 0 is the centre of symmetry of the conjectural functional equation (CF E )

?

L∞ (ιM, s) L(ιM, s) = ε(ιM, s) L∞ (ιM, −s) L(ιM, −s),

ε(ιM, s) = ι(ε(M, 0)) · f (M )−s . ∼

For each prime p | p of L, the induced isomorphism of Galois representations Mp −→ Mp∗ (1) should be skew-symmetric. The conjecture (CBK,p ) then reads as ?

ords=0 L(ιM, s) = h1f (F, Mp ) := dimLp Hf1 (F, Mp ). 327

(12.1.3.1)

(12.1.4) Examples: (i) An archetypal example of a self-dual pure motive is given by M = h1 (E)(1), where E is an elliptic curve over F . In this case we have L = Q, L(ιM, s) = L(E/F, s + 1), p = p, ∼ 1 Mp = Tp (E) ⊗ Q = Vp (E), the isomorphism Mp −→ Mp∗ (1) is induced by the Weil pairing, Hf,M (F, M ) = E(F ) ⊗ Q and there is an exact sequence 0 −→ E(F ) ⊗ Qp −→ Hf1 (F, Vp (E)) −→ (Tp X(E/F )) ⊗ Q −→ 0. This means that ords=0 L(ιM, s) = ords=1 L(E/F, s),

h1f (F, Vp (E)) = rkZ E(F ) + corkZp X(E/F )[p∞ ],

(12.1.4.1)

hence the conjecture (CBK ) (resp., (Cp )) is equivalent to the Conjecture of Birch and Swinnerton-Dyer (resp., to the finiteness of the p-primary part of X(E/F )). (ii) More generally, let A be an abelian variety over F and L a totally real number field with OL ,→ EndF (A), dim(A) = [L : Q]. Then M = h1 (A)(1) is self-dual as a rank 2 motive with coefficients in L, Mp = Vp (A), 1 L(ιM, s) = L(ιA/F, s + 1), Hf,M (F, M ) = A(F ) ⊗ Q, there is an exact sequence 0 −→ A(F ) ⊗OL Lp −→ Hf1 (F, Vp (A)) −→ (Tp X(A/F )) ⊗ Q −→ 0 and ords=0 L(ιM, s) = ords=1 L(ιA/F, s),

h1f (F, Vp (A)) = rkOL A(F ) + corkOL,p X(A/F )[p∞ ].

(12.1.4.2)

∼

(12.1.5) Returning to the case of an arbitrary self-dual pure motive M −→ M ∗ (1), the general recipe [Se 4, §3] implies that ords=0 L∞ (ιM, s) = 0, i.e. the central point s = 0 is “critical” in the sense of Deligne [De 3]. The functional equation (CF E ) (if available) then yields (−1)ords=0 L(ιM,s) = ε(M, 0) ∈ {±1}.

(12.1.5.1)

We shall be interested in the conjecture (12.1.3.1) modulo 2, i.e. in ?

ords=0 L(ιM, s) ≡ h1f (F, Mp ) (mod 2).

(12.1.5.2)

Thanks to (12.1.5.1), this conjectural congruence can be reformulated (assuming the validity of (CF E )) as Y 1 ? (−1)hf (F,Mp ) = ε(M, 0) = εv (M, 0). (12.1.5.3) v

The local ε-factors εv (M, 0) ∈ {±1} depend only on the Galois representation Mp (assuming the standard compatibilities between Mp and other realizations of M ): If v | ∞, then εv (M, 0) depends only on the Hodge numbers of the de Rham realization MdR of M (and on the action of the complex conjugation at v on Mp if v is a real prime). These Hodge numbers can be ∼ read off from the comparison isomorphism MdR ⊗ BdR −→ Mp ⊗ BdR (at a fixed prime of F above p). If v - ∞, then the action of Gv on Mp gives rise to a representation W Dv (Mp ) of the Weil-Deligne group of Fv (for v - p, this is a classical fact ([De 2], 8.4); for v | p one uses the potential semistability of Mp at v and applies the recipe in [Fo 2] (cf. [Sa], remarks before Thm. 1)). One defines, using the notation of [De 2, (5.5.2), 8.12], εv (Mp , 0) = εv (W Dv (Mp ), ψv , dxψv , 0), 328

where ψv is any non-trivial additive character of Fv and dxψv the corresponding self-dual Haar measure on Fv . The value of εv (Mp , 0) is equal to ±1 ([De 2], (5.7.1), 8.12) and does not depend on ψv ([De 2], (5.3), (5.4), 8.12). It is expected that the isomorphism class of W Dv (Mp ) is defined over L and does not depend on p; if true, then the local ε-factor εv (M, 0) := εv (Mp , 0) ∈ {±1} is well-defined. Assuming this independence on p, one can reformulate (12.1.5.3) as (Csgn (Mp ))

1

?

(−1)hf (F,Mp ) = ε(Mp , 0) =

Y

εv (Mp , 0).

v

(12.1.6) The conjectural equality (Csgn (Mp )) does not involve the motive M , only its p-adic realization Mp . Moreover, it makes sense to consider (Csgn (V )) for an arbitrary self-dual p-adic Galois representation ∼ V −→ V ∗ (1) of GF which is geometric in the sense of ([Fo-PR], II.3.1.1; [Fo-Ma], I.1), i.e. unramified outside a finite set S of primes of F and potentially semistable at all primes above p. We propose to study the variation of the conjecture (Csgn (V )) in p-adic families of Galois representations. In vague terms, assume that (a) X is an irreducible p-adic analytic space. (b) V (λ) is a family of Lp -representations of GF,S depending analytically on a parameter λ ∈ X (Lp ). ∼ (c) Each representation V (λ) is equipped with a skew-symmetric isomorphism V (λ) −→ (V (λ) )∗ (1), varying analytically in λ. (d) There is a Zariski dense subset X mot ⊂ X (Lp ) of “motivic” values of the parameter for which V (λ) is the p-adic realization of a (pure) motive M (λ) . The first observation is that neither side of the conjecture (Csgn (V (λ) )) is, in general, constant on X mot . The point is that both terms are related to the complex-valued L-function L(ιM (λ) , s), while a good p-adic behaviour can only be expected from a suitable p-adic L-function Lp (λ, s) (which depends on fixed embeddings ι : Q ,→ C and ιp : Q ,→ Qp ). In favourable circumstances, such a p-adic L-function satisfies, for critical values (λ, n) ∈ X mot × Z, an interpolation property of the form Lp (λ, n) = Eu(λ, n)

L(ιM (λ) , n) , Ω(ι, λ, n)

where Ω(ι, λ, n) is a suitable period and Eu(λ, n) =

Y

Euv (λ, n)

v|p

a product of appropriate Euler factors at primes of F above p. Whenever one of the Euler factors Euv (λ, 0) vanishes, it forces the function s 7→ Lp (λ, s) to have a “trivial” (or “exceptional”) zero at s = 0. In a self-dual situation, this phenomenon was first studied by MazurTate-Teitelbaum [M-T-T], who observed that the presence of a trivial zero at λ ∈ X mot may cause a mismatch between the signs in the functional equations of L(ιM (λ) , s) and s 7→ Lp (λ, s). One would expect, in general, that an appropriate functional equation for the “two-variable” p-adic L-function Lp (λ, s) will induce functional equations for each “one-variable” p-adic L-function s 7→ Lp (λ, s) (λ ∈ X mot ) with a common sign εe(V (λ) , 0) = εe = ±1, while 329

ε(M (λ) , 0) = ε(V (λ) , 0) = εe(V (λ) , 0)

Y (contribution from a possible trivial zero of Euv (λ, 0)) v|p

(i.e. the presence of a trivial zero at λ ∈ X mot causes a jump in the value of ε(M (λ) ) = ε(V (λ) )). This happens, for example, in the case of the two-variable p-adic L-function attached to a Hida family of p-ordinary modular forms [G-S]. (12.1.7) As proposed first in a special case in [M-T-T], a p-adic variant of the conjecture (12.1.3.1) for M = M (λ) (λ ∈ X mot ) should read as ? e f1 (F, V (λ) ), ords=0 Lp (λ, s) = e h1f (F, V (λ) ) := dimLp H

(12.1.7.1)

e 1 (F, V (λ) ) is an appropriate “extended Selmer group”, which incorporates H 1 (F, V (λ) ), as well as where H f f the trivial zeros of Eu(λ, 0). If M (λ) = h1 (E)(1) for an elliptic curve E over F (say, with semistable reduction at all primes above p), then the Euler factor Euv (λ, 0) at a prime v | p vanishes ⇐⇒ E has split multiplicative reduction at v. The appropriate extended Selmer group was defined in this case in [M-T-T]. (12.1.8) The theory developed in the previous chapters leads naturally to extended Selmer groups, at least in the quasi-ordinary case. More precisely, assume that V is a (continuous) Lp -representation of GF,S (with S containing all archimedean primes and all primes above p in F ), which is equipped with the following data: ∼

(12.1.8.1) A skew-symmetric isomorphism of Lp [GF,S ]-modules V −→ V ∗ (1). (12.1.8.2) For each prime v | p of F , an exact sequence of Lp [Gv ]-modules 0 −→ Vv+ −→ V −→ Vv− −→ 0, ∼

for which the self-duality isomorphism V −→ V ∗ (1) from (a) induces isomorphisms of Lp [Gv ]-modules ∼ ∼ Vv+ −→ (Vv− )∗ (1), Vv− −→ (Vv+ )∗ (1). These data define Greenberg’s local conditions 7.8.2 ( • Ccont (Gv , Vv+ ), + Uv (V ) = • Ccont (Gv /Iv , V Iv ),

v|p v ∈ Sf , v - p

e • (GF,S , V ; ∆(V )) for V (with Σ = {v | p}). The cohomology groups of the corresponding Selmer complex C f e j (F, V ). The third triangle in (6.1.3.2) do not depend on S (by Proposition 7.8.8(ii)); we denote them by H f gives rise to an exact sequence 0 −→

M

1 e f1 (F, V ) −→ Hcont H 0 (Gv , Vv− ) −→ H (GF,S , V ) −→

v|p

M v|p

M

H 1 (Gv , Vv− ) ⊕

H 1 (Iv , V )Gv /Iv ,

v∈Sf ,v-p

which boils down, under suitable additional assumptions (cf. 12.5.9.2 below), to M e f1 (F, V ) −→ Hf1 (F, V ) −→ 0. 0 −→ H 0 (Gv , Vv− ) −→ H

(12.1.8.3)

v|p

e 1 (F, V ) as an “extended Selmer group” and each term H 0 (Gv , V − ) (if non-trivial) as a We interpret H v f contribution from a trivial zero. For example, if E is an elliptic curve over F with ordinary reduction (= multiplicative or good ordinary reduction) at each prime above p, then V = Vp (E) is naturally equiped with the data (12.1.8.1-2) (see 9.6.7.2) and the sequence (12.1.8.3) is exact, with 330

( 0

dimQp H (Gv , Vp (E)) =

1,

E has split multiplicative reduction at v

0,

otherwise,

e 1 (F, V ) in this special by Lemma 9.6.3, 9.6.7.3 and 9.6.7.6(ii). See [M-T-T] for a more explicit definition of H f case. (12.1.9) Assume that the representation V from (12.1.8.1-2) is potentially semistable at all primes above p and that the sequence (12.1.8.3) is exact. For each prime v of F , we define ( εev (V, 0) = εv (V, 0) ·

(−1)dimLp H

0

(Gv ,Vv− )

,

1,

v|p v-p

and we set εe(V, 0) =

Y v

εev (V, 0) = ε(V, 0)

Y 0 − (−1)dimLp H (Gv ,Vv ) ∈ {±1}. v|p

With this notation, the conjecture (Csgn (V )) is equivalent to esgn (V )) (C

Y 1 ? (−1)ehf (F,V ) = εe(V, 0) = εev (V, 0). v

If the above discussion applies to the representation V = V (λ) (λ ∈ X mot ) from 12.1.6, then εe(V, 0) should esgn (V )) would be equal to the sign in the functional equation of the p-adic L-function s 7→ Lp (λ, s), hence (C be just the conjecture (12.1.7.1) modulo 2. (12.1.10) Assume that each member V = V (λ) (λ ∈ X (Lp )) of the family of Galois representations considered in 12.1.6 is equipped with the structure (12.1.8.1-2), depending analytically on λ. It seems very likely that the following principles hold under very general circumstances: e h1f (F, V ) (mod 2) does not depend on λ ∈ X mot . (∀v - p∞) εv (V (λ) , 0) does not depend on λ ∈ X mot . Q (IIb) ev (V (λ) , 0) does not depend on λ ∈ X mot . v|p∞ ε (III) The presence of a non-trivial Euler system for V (λ) implies the conjectural equality (12.1.3.1) (resp., (12.1.7.1)), provided the L.H.S. is equal to 0 or 1. (I) (IIa)

esgn (V (λ) )) ( ⇐⇒ Once the principles (I) and (II) are established, then the validity of the conjecture (C (λ) mot (λ) esgn (V )) for all λ ∈ X mot , it then (Csgn (V ))) does not depend on λ ∈ X . In order to prove (C remains to find one motivic value λ to which the principle (III) would apply. (12.1.11) In [N-P] (resp., in [Ne 3]), this strategy was applied to a Hida family of classical p-ordinary modular forms (resp., to anticyclotomic Iwasawa theory of an elliptic curve with good ordinary reduction at p). In both cases, the principles (I) and (a global version of) (II) were established. The results of [N-P] were conditional, as the principle (III) applied only half-way. The relevant Euler system arguments were available ([Ka 2], [Ne 1]); what was missing was the proof that the Euler systems in question were non-trivial. In the anticyclotomic situation considered in [Ne 3], the principle (III) was available, thanks to the proof of Mazur’s conjecture on Heegner points ([Cor],[Va]). This chapter offers a simultaneous generalization of [N-P] and [Ne 3] to (twists of p-ordinary) Hilbert modular forms of parallel weight. Our main results are stated in 12.2.3,6 below. In the proof, one applies the principles (I) and (II) in three different contexts: (a) level raising congruences; (b) twisted Hida family; (c) dihedral Iwasawa theory. 331

In each case, the principle (I) follows from the existence of a suitable symplectic form, which is provided by the generalized Cassels-Tate pairing from Chapter 10. The principle (IIa) appears to be quite general (cf. Corollary 12.7.14.3 and Proposition 12.8.1.4 below); by contrast, we prove (IIb) in each case by an explicit calculation. The principle (III) applies in the context of dihedral Iwasawa theory, namely to CM points on Shimura curves. The Euler system argument is provided by a generalization of Kolyvagin’s original method ([Ne 4], Thm. 3.2), while the non-triviality of the Euler system follows from the proof of the generalized Mazur conjecture ([Cor-Va], Thm. 4.1). Finally, a descent argument due to R. Taylor ([Tay 4], §6) allows us to treat twodimensional self-dual Galois representations over totally real number fields that are only potentially modular (of parallel weight) and potentially ordinary. This applies, in particular, to elliptic curves and abelian varieties of GL(2)-type with a totally real coefficient field (modulo potential modularity results that seem to be well-known to the experts). The final result is stated in 12.2.8. (12.1.12) The ad hoc approach to the cases (IIb) and (IIc) in the present chapter is superseded by the subsequent work [Ne 5], in which the principle (II) is established in a rather general context, namely for families of Galois representations which are pure and satisfy the Panˇciˇskin condition at all primes above p. (12.2) The parity results In this section we state our main parity results and describe the main steps of the proofs. The proofs themselves will occupy the rest of Chapter 12. Let F be a totally real number field. Fix a prime number p and embeddings ιp : Q ,→ Qp , ι∞ : Q ,→ C. (12.2.1) Let f ∈ Sk (n, ϕ) be a (non-zero) cuspidal Hilbert modular form of parallel weight (k, . . . , k), level n ⊂ OF and character ϕ : A∗F /F ∗ −→ C∗ (see 12.3 below for a more detailed discussion). We assume that f is a newform, which implies that f is an eigenform for all Hecke operators T (a)f = λf (a)f

(a ⊂ OF ).

Let L be a sufficiently large number field (considered as a subfield of Q via ι∞ ) containing all Hecke eigenvalues λf (a) and the values of ϕ. We shall be free to replace L by a finite extension, if necessary. The embedding ιp induces a prime ideal p|p of L. There is a continuous two-dimensional representation V (f ) of GF,S over Lp attached to f (where S = {v | pn∞}). It is characterized by the fact that, for each prime v 6∈ S of F , the Euler factor Lv (f, s) = 1 − λf (v)(N v)−s + ϕ(v)(N v)k−1−2s


−1

of the standard L-function of f L(f, s) =

Y

Lv (f, s) =

v-∞

X λf (a) (N a)s a

coincides with Lv (V (f ), s) = det 1 − Fr(v)geom (N v)−s | V (f )Iv


−1

.

(12.2.2) Assume that 2|k and that there exists a character χ : A∗F /F ∗ −→ C∗ satisfying χ−2 = ϕ. Fix such a χ and denote by g = f ⊗ χ ∈ Sk (n(g), 1) the unique newform satisfying V (g) = V (f ) ⊗ χ (above, n(g) denotes the level of g). The Galois representation V := V (g)(k/2) = V (f )(k/2) ⊗ χ ∼

satisfies Λ2 V −→ Lp (1), hence it is self-dual in the sense of (12.1.8.1). For each prime v - p∞ of F we have Lv (V, s) = Lv (g, s + k/2). The L-function L(g, s + k/2) satisfies a functional equation of the form (CF E ). We are interested in the following variant of the conjectural equality (12.1.5.2): 332

?

ran (F, g) := ords=k/2 L(g, s) ≡ h1f (F, V ) (mod 2).

(12.2.2.1)

If F 0 /F is a finite solvable Galois extension (not necessarily totally real), then there is an automorphic form g 0 = BCF 0 /F (g) on GL2 (AF 0 ) satisfying, for each prime w of F 0 above a prime v - pn(g)∞ of F , the equality Lw (g 0 , s) = det 1 − Fr(w)geom (N w)−s | V (g)Iw


−1

.

If F 0 is totally real or if g does not have CM, then the form g 0 is cuspidal. One can generalize (12.2.2.1) to the base change form g 0 , namely ?

ran (F 0 , g) := ords=k/2 L(g 0 , s) ≡ h1f (F 0 , V ) (mod 2).

(12.2.2.2)

Our first result in this direction is the following Theorem. (12.2.3) Theorem. Let g = f ⊗ χ ∈ Sk (n(g), 1), where f is p-ordinary, i.e. (∀v | p) ordp (λf (v)) = 0. Let F 0 /F be a finite 2-abelian extension. Assume that at least one of the following assumptions (1)–(4) holds. (1) 2 - [F : Q]. (2) There exists a character µ : A∗F /F ∗ −→ {±1} such that the level n(g ⊗ µ) of the newform g ⊗ µ is not a square. (3) The following conditions (i)-(iii) are satisfied: (i) p is prime to k! n(g)dF 0 /Q w2 (L(g)), where L(g) is the (totally real) number field generated by the Hecke eigenvalues of g and w2 (L(g)) denotes the order of H 0 (GL(g) , Q/Z(2)). (ii) The residual representation ρp : GF −→ Aut(T /pT ) (where T ⊂ V is a GF -stable OL,p -lattice) is irreducible. (iii) If g has no CM, then there exists a basis of T /pT in which Im(ρp ) ⊇ SL2 (Fp ). (4) For each character α : Gal(F 0 /F ) −→ {±1} there exists a congruence f ⊗ α ≡ fα (mod pM(f ⊗α,p) ) in the sense of (12.10.3.1), where fα ∈ Sk (n(f ⊗ α)Qα , χ−2 ) is a p-ordinary newform, Qα = qα,1 · · · qα,rα , qα,j pn(f ⊗ α)cond(χ) are distinct prime ideals satisfying (∀j = 1, . . . , rα ) λfα (qα,j ) = −χ−1 (qα,j )(N qα,j )k/2−1 and M (f ⊗α, p) is a certain constant (more precisely, M (f ⊗α, p) is any value of M satisfying the assumptions of Proposition 12.8.4.14 for the form f ⊗ α instead of f ). Then: for each finite Galois extension F 00 /F 0 of odd degree, ran (F 00 , g) ≡ h1f (F 00 , V ) (mod 2). Proof. See 12.10. (12.2.4) Remarks. (i) It is likely that our methods can be used to prove a similar result for quasi-ordinary Hilbert modular forms of non-parallel weight. (ii) The condition 12.2.3(2) is satisfied if there exists a prime v of F for which π(g)v (the local component of the automorphic representation π(g) attached to g) is a twisted Steinberg representation: in this case there exists µ : A∗F /F ∗ −→ {±1} such that ordv (n(g ⊗ µ)) = 1. (iii) The condition 12.2.3(3i) implies that p > 3 (as w2 (Q) = 24) and that g is p-ordinary (as p is prime to n(g)). (iv) The condition 12.2.3(3) rules out only finitely many prime ideals p of L ([Dim], Prop. 3.1(i)). (v) In 12.2.3(4), instead of requiring the existence of congruences f ⊗ α ≡ fα (mod pM(f ⊗α,p) ) separately for each α : Gal(F 0 /F ) −→ {±1}, it would be sufficient to assume that, for each M ≥ 1, there is a congruence f ≡ f(M) (mod pM ) with an appropriate p-ordinary newform f(M) ∈ Sk (n(f ⊗ α)Q(M) , χ−2 ). One could then take fα = f(M) ⊗ α, for any M ≥ max{M (f ⊗ α, p) | α : Gal(F 0 /F ) −→ {±1}}. (vi) It is possible that the existence of such congruences f ≡ f(M) (mod pM ) could be established (under suitable additional assumptions) using the techniques of [K-R]. 0 (vii) In fact, it would be sufficient to assume the existence of congruences f 0 ≡ f(M) (mod pM ) for a suitable 0 fixed element f of the Hida family containing f . (viii) In the original version of 12.2.3 it was assumed that F 00 /F is an abelian extension. The current formulation was suggested by a reading of [M-R 1]. 333

(ix) In the case when F 00 = F = Q, χ = 1 and g = f ∈ Sk (N, 1) is p-ordinary, Skinner and Urban [S-U] proved that 2 - ran (Q, g) =⇒ h1f (Q, V ) ≥ 1. (x) Combining Theorem 12.2.3 (and Theorem 12.2.8 below) with Theorem 10.7.17(v), we obtain a generalization of ([M-R 3], Cor. 3.6-7). See Sect. 12.12 for more details. (12.2.5) One can combine Theorem 12.2.3 with base change [Lan] (cf. also [A-C], [J-PS-S]), obtaining the following result. (12.2.6) Corollary. Let g = f ⊗ χ ∈ Sk (n(g), 1), where f is p-ordinary. Let F1 /F be a finite solvable extension of totally real number fields and F10 /F1 a finite 2-abelian extension. Assume that at least one of the following conditions (1)–(3) holds: (1) 2 - [F1 : Q]. (2) There exists a prime v of F such that π(g)v is a twisted Steinberg representation. (3) The conditon 12.2.3(3) is satisfied and p is unramified in F10 /Q. Then: for each finite Galois extension of odd degree F100 /F10 , ran (F100 , g) ≡ h1f (F100 , V ) (mod 2). Proof. This follows from Theorem 12.2.3 applied to the base change form g1 = BCF1 /F (g). In view of 12.2.4(ii), the condition 12.2.6(2) implies that 12.2.3(2) holds for g1 . If the condition 12.2.6(3) holds, then 12.2.3(3iii) holds for g1 over F1 , since Gal(F1 /F ) is solvable and SL2 (Fp ) has no non-trivial solvable quotients (as p > 3, by 12.2.3(3i) for g). (12.2.7) Furthermore, combining Corollary 12.2.6 with the descent arguments of [Tay 3,4], we deduce the following theorem. (12.2.8) Theorem. Let F and L be totally real number fields, F0 /F an abelian 2-extension and A an abelian variety over F satisfying OL ⊂ EndF (A) and dim(A) = [L : Q]; assume that A is potentially modular in the sense of 12.11.3(1) below (1) . Fix an embedding ι : L ,→ R. Let p be a prime number such that A has potentially ordinary (= potentially good ordinary or potentially totally multiplicative) reduction at each prime of F above p; let p a prime of L above p. Assume that at least one of the following conditions holds: (1) A is modular over F in the sense of 12.11.3(1)(ii) (with F 0 = F ) and 2 - [F : Q]. (2) A does not have potentially good reduction everywhere. (3) A has potentially good reduction everywhere, A has good ordinary reduction at each prime of F above p, the prime number p is unramified in F0 /Q and does not divide w2 (L) (=⇒ p > 3). If A does not have CM, assume, in addition, that there exists a basis of A[p] in which Im(GF −→ Aut(A[p])) ⊇ SL2 (Fp ). Then: for each finite Galois extension of odd degree F1 /F0 , rkOL A(F1 ) + corkOL,p X(A/F1 )[p∞ ] ≡ ords=1 L(ιA/F1 , s) (mod 2). Proof. See 12.11. (12.2.9) If A does not have CM, then Im(GF −→ Aut(A[p])) contains SL2 (Fp ) for all but finitely many p ([Dim, Prop. 3.8] + potential modularity; see also [Ri], §5). (12.2.10) Corollary. Let F be a totally real number field, F0 /F an abelian 2-extension, E an elliptic curve over F which is potentially modular in the sense of 12.11.3(1) below (2) and p a prime number such that E has potentially ordinary (= potentially good ordinary or potentially multiplicative) reduction at each prime of F above p. Assume that at least one of the following conditions holds: (1) E is modular over F and 2 - [F : Q]. (2) j(E) 6∈ OF . (1)

Potential modularity of A seems to be well-known to the experts [Tay 5]; a proof is expected to be written down in a forthcoming thesis of a student of R. Taylor. (2) See the previous footnote. 334

(3) j(E) ∈ OF , E has good ordinary reduction at each prime of F above p, the prime number p is unramified in F0 /Q and p > 3. If E does not have CM, assume, in addition, that Im(GF −→ Aut(E[p])) ⊇ SL2 (Fp ). Then: for each finite Galois extension of odd degree F1 /F0 , rkZ E(F1 ) + corkZp X(E/F1 )[p∞ ] ≡ ords=1 L(E/F1 , s) (mod 2). (12.2.11) Example (cf. [M-R 3], 5.3). Let a(x) ∈ Q[X] be an irreducible polynomial of degree deg(a) = 4 with non-real roots α, α, β, β; denote by b(x) ∈ Q[X] its cubic resolvent with (real) roots γ1 = αα + ββ, γ2 = αβ + αβ, γ3 = αβ + αβ. Assume that the splitting field K = Q(α, α, β, β) of a(X) has maximal degree [K : Q] = 24, in other words that Gal(K/Q) = S4 . Let Q ⊂ k ⊂ K be an intermediate field such that 3 | [k : Q] and k is not equal to the fixed field of a transposition in S4 . This implies that exactly one of the following two cases occurs: (0) k is an abelian extension of degree 2 or 4 of the splitting field K0 = Q(γ1 , γ2 , γ3 ) of b(X) (which is a totally real S3 -extension of Q); (j) (j = 1, 2, 3) k is an extension of degree 1 or 2 of at least one of the (totally real, non-normal) cubic fields Kj = Q(γj ). Let E be an elliptic curve over Q with potentially ordinary reduction at a prime p. Our results imply that the congruence rkZ E(k) + corkZp X(E/k)[p∞ ] ≡ ords=1 L(E/k, s) (mod 2) holds in the following situations: (a) j(E) 6∈ Z [by 12.2.10(1), for F = Kj (j = 0, 1, 2, 3), F1 = F0 = k]. (b) j(E) ∈ Z, [k : Kj ] ≤ 2 for some j = 1, 2, 3 [by 12.2.3(1) and 12.11.5(iv) (or by 12.2.10(2)) applied to F = Kj , F 00 = F 0 = k and the cubic base change g = BCF/Q (gE ) (see [J-PS-S] for the non-Galois cubic base change) of the classical modular form gE ∈ S2 (NE , 1) attached to E]. (c) j(E) ∈ Z, [k : K0 ] = 2 or 4, p > 3 is unramified in k/Q, E has good ordinary reduction at p, E[p] is an irreducible Fp [GQ ]-module; if E has no CM, then Im(GQ −→ Aut(E[p])) = Aut(E[p]) [by 12.2.6(3) for F1 = K0 , F100 = F10 = k, g = gE ]. (12.2.12) A general outline of the proof of Theorem 12.2.3 was given in 12.1.11. We now describe the main steps in more detail. (12.2.12.1) Reduction to the case F 00 = F 0 = F for twisted forms g ⊗ α (α : Gal(F 0 /F ) −→ {±1}) There is a tower of fields F = F0 ⊂ F1 = F 0 ⊂ · · · ⊂ Fn = F 00 such that each Galois group Gi = Gal(Fi+1 /Fi ) is abelian (and of odd order for i > 0). For each i = 0, . . . , n − 1, we have X X ran (Fi+1 , g) = ran (Fi , g ⊗ α), h1f (Fi+1 , V ) = h1f (Fi , V ⊗ α), α

α

where α runs through all characters of Gi . If α2 6= 1, then the contributions of α and α−1 to both terms are the same; this implies that, for each i = 1, . . . , n − 1, h1f (Fi+1 , V ) ≡ h1f (Fi , V ) (mod 2),

ran (Fi+1 , g) ≡ ran (Fi , g) (mod 2), which reduces to the case F 00 = F 0 = F1 , and ran (F 0 , g) ≡

X

h1f (F 0 , V ) ≡

ran (F, g ⊗ α) (mod 2),

α:G0 −→{±1}

X

h1f (F, V ⊗ α) (mod 2),

α:G0 −→{±1}

which further reduces to the case F 00 = F 0 = F for the twisted forms g ⊗ α (α : Gal(F 0 /F ) −→ {±1}). (12.2.12.2) Reduction to the assumptions 12.2.3(1)-(2) One establishes the principles 335

(I) e h1f (F, V ⊗ α) ≡ e h1f (F, V1 ) (mod 2)

(V1 = V (fα )(k/2) ⊗ χ)

(II) εe(V ⊗ α, 0) = εe(V1 , 0) for the “level raising congruences” f ⊗ α ≡ fα (mod pM ), whose existence is guaranteed by the condition 12.2.3(4) (resp., to analogous congruences (mod p), whose existence is deduced from the condition 12.2.3(3)). The key point in the proof of (II) is the fact that, for each prime v - p∞, the local ε0,v -constants preserve congruences modulo pM ([De 2], Thm. 6.5 for M = 1; [Ya], Thm. 5.1 for M > 1). (12.2.12.3) Reduction to the case [2 - [F : Q] or (∃P | p) (∃q 6= P ) 2 - ordq (n(g))] This follows from an argument analogous to that from 12.2.12.1 for a suitable quadratic (resp., cyclic cubic) extension of F . (12.2.12.4) Further reduction to the case k = 2 One embeds (the p-stabilization of) f to a Hida family; such a family contains a form f 0 of weight (2, . . . , 2). We show that the principles (I) and (II) hold in this context: e h1f (F, V ) ≡ e h1f (F, V 0 ) (mod 2),

εe(V, 0) = εe(V 0 , 0),

where g 0 = f 0 ⊗ χ0 ∈ S2 (n(g 0 ), 1) and V 0 = V (g 0 )(1). (12.2.12.5) Switch to a Shimura curve Thanks to the previous reduction steps, one can assume that g ∈ S2 (n(g), 1) is attached by the Jacquet∗ Langlands correspondence to an automorphic form gB (with trivial central character) on BA , where B is the quaternion algebra over F ramified at all but one archimedean primes (and at the prime q from 12.2.12.3 if 2 | [F : Q]). This implies that there exists a Shimura curve X over F and a surjection Vp (J(X))⊗Qp Lp  V . (12.2.12.6) Dihedral Iwasawa theory By construction, there exists a prime P | p of F at which B does not ramify. We fix such a prime and a suitable totally imaginary quadratic extension K/F which embeds into B. For example, we can take any K in which all primes dividing n(g)P split (resp., in which q is inert and all remaining primes dividing n(g)P split) if 2 - [F : Q] (resp., if 2 | [F : Q]). One considers the tower of ring class fields K ⊂ K[1] ⊂ K[P ] ⊂ · · · ⊂ K[P ∞ ] =

∞ [

K[P n ];

n=1 ∞

the Galois group G = Gal(K[P ]/K) is abelian, its torsion subgroup is finite and the quotient group G/Gtors is isomorphic to ZrpP , where rP = [FP : Qp ]. The extension K[P ∞ ]/F is dihedral: any lift τ of the non-trivial element of Gal(K/F ) satisfies τ gτ −1 = g −1 , for all g ∈ G. ∼ Set K∞ = K[P ∞ ]Gtors ; then Γ = Gal(K∞ /K) −→ ZrpP . If β : Γ −→ Lp (β)∗ is a character of finite order Ker(β)

(where Lp (β) is a finite extension of Lp ), put Kβ = K∞

and

h1f (K, V ⊗ β) := dimLp (β) Hf1 (K, V ⊗ β) = dimLp (β) Hf1 (Kβ , V ) ⊗ β


Gal(Kβ /K)

.

For K chosen as above we have principle (II) ran (K, g) ≡ ran (K, g ⊗ β) ≡ 1 (mod 2), where ran (K, g ⊗ β) = ords=1 L(g × θ(β), s) denotes the order of vanishing of the Rankin-Selberg L-function attached to g and to the theta series of β. We also have a version of the principle (I), namely h1f (K, V ) ≡ h1f (K, V ⊗ β) (mod 2). (12.2.12.7) CM points 336

The Shimura curve X contains a large supply of CM points defined over the ring class fields K[P n ]; they give rise to cohomology classes in Hf1 (K[P n ], V ). A fundamental result of Cornut-Vatsal [Cor-Va, Thm. 4.1] (“generalized Mazur’s conjecture”) states that, for n >> 0, there exists a CM point defined over K[P n ] such that the corestriction from K[P n ] to K[P n ] ∩ K∞ of the corresponding cohomology class yields a non-zero ∗ element of (Hf1 (Kβ , V ) ⊗ β)Gal(Kβ /K) , for a suitable character of finite order β : Γ −→ Lp . An Euler system argument ([Ne 4], Thm. 3.2) then shows that h1f (K, V ⊗ β) = 1. Combined with 12.2.12.6, this implies that ran (K, g) ≡ h1f (K, V ) (mod 2). (12.2.12.8) Descent to F (varying K) Denoting by η the quadratic character attached to K/F , the result established in 12.2.12.7 can be written as ran (F, g) + ran (F, g ⊗ η) ≡ h1f (F, V ) + h1f (F, V ⊗ η) (mod 2). In order to eliminate one of the contributions on each side, we vary K as in [Ne 3]; applying the non-vanishing results of [Wa 2] and [F-H], the generalized Gross-Zagier formula [Zh 1,2] and an Euler system argument [Ne 4], we obtain in the end the desired congruence ran (F, g) ≡ h1f (F, V ) (mod 2). (12.2.13) The contents of this chapter is the following: in 12.3 (resp., 12.4) we recall basic properties of Hilbert modular forms of parallel weight and the corresponding automorphic representations (resp., of the corresponding Galois representations). In 12.5 we investigate p-ordinary forms and their twists. In 12.6, 12.7 and 12.8 we establish, respectively, the principles (I) and (II) in the context of dihedral Iwasawa theory, Hida families and level raising congruences. In 12.9 we prove a parity result over ring class fields, from which we deduce in 12.10 Theorem 12.2.3. In 12.11, we prove Theorem 12.2.8, by combining 12.2.3 with potential modularity results and R. Taylor’s descent arguments [Tay 4]. The results on the Euler system of Heegner points used in the steps 12.2.12.7-8 are proved in [Ne 4]. (12.3) Hilbert modular forms Let F be a totally real number field with ring of integers OF . We shall consider only Hilbert modular forms over F of parallel weight (k, k, . . . , k) (“of weight k”), where k ≥ 1. There are several competing normalizations and conventions concerning such forms; we shall use the language of representation theory, but for the reader’s convenience we briefly recall the dictionary between the classical and representationtheoretical approaches. The reciprocity isomorphism of class field theory will be normalized by letting uniformizers correspond to geometric Frobenius elements. We shall use it to identify, for any pro-finite abelian group A, a continuous homomorphism α : A∗F /F ∗ −→ A (resp., Fv∗ −→ A) with a continuous homomorphism GF −→ A (resp., Gv −→ A). This applies, in particular, to finite abelian groups A, such as A = µn (C) ⊂ C. (12.3.1) For a non-zero ideal n ⊂ OF let ( ! ) a b Y b U0 (n) = ∈ GL2 (OF ) | c ≡ 0 (mod b n) = U0 (n)v c d v-∞ ( ! ) a b Y bF ) | c, d − 1 ≡ 0 (mod b U1 (n) = ∈ GL2 (O n) = U1 (n)v c d v-∞ bF = OF ⊗ Z b = Q b be the standard open compact subgroups of GL2 (Fb ) (where O n = n ⊗ Z, v-∞ Ov , b Q ∞ b b F = OF ⊗ Q = A ). Write an element of K∞ = SO(2) as k∞ = (r(θv ))v|∞ , where F

v|∞

337

r(θ) = and put ρ(k∞ ) =

Q

cos θ

− sin θ

sin θ

cos θ

! ,

exp(iθv ). Let Z ⊂ GL2 be the subgroup of scalar matrices and define ! y∞ 0 Y Y sgn(z∞ ) = sgn( )= sgn(yv ), (z∞ ∈ Z(F ⊗ R) = Z(Fv )). 0 y∞ v|∞ v|∞ v|∞

A Hilbert modular form of weight k ≥ 1 and level U1 (n) is a continuous function f : GL2 (AF ) −→ C satisfying the functional equation f (γz∞ guk∞ ) = ρ(k∞ )−k sgn(z∞ )k f (g),

γ ∈ GL2 (F ), z∞ ∈ Z(F ⊗ R), u ∈ U1 (n), k∞ ∈ K∞

such that, for fixed g ∞ ∈ GL2 (Fb ), the function −k/2

|y∞ |

f(

y∞

x∞

0

1

! g∞)

(12.3.1.1)

Q is holomorphic in the variables x∞ + iy∞ = (xv + iyv )v|∞ (|y∞ | = v|∞ |yv |) and is bounded as |yv | −→ ∞ (see [Oh 2], Prop. 5.1.2). Such functions form a finite-dimensional complex vector space Mk (n). The subspace of cusp forms Sk (n) ⊂ Mk (n) consists of those functions for which the expression (12.3.1.1) tends to zero as |yv | −→ +∞ for all v|∞. (12.3.2) The operators ! b 0 (S(b)f ) (g) = f ( g) (b ∈ A∗F , f ∈ Mk (n)) 0 b define a representation of the ray class group In∞ = F ∗ \A∗F /(F ⊗ R)∗+ (1 + b n) modulo n∞ (where ∞ is a shorthand for the sum of all archimedean primes of F ) on Mk (n) (resp., Sk (n)). Under this action, the spaces Mk (n) ⊃ Sk (n) decompose into direct sums M M Mk (n) = Mk (n, ϕ), Sk (n) = Sk (n, ϕ), ϕ

ϕ

where ϕ runs through all characters ϕ : In∞ −→ C∗ satisfying the parity condition ϕv (−1) = (−1)k

(∀v|∞)

and Xk (n, ϕ) = {f ∈ Xk (n) | (∀b ∈ A∗F ) S(b)f = ϕ(b)f }

(X = M, S).

In concrete terms, elements of Mk (n, ϕ) satisfy the functional equation f (γgzuk∞) = ϕ(z)ϕ∞ (u)ρ(k∞ )−k f (g),

γ ∈ GL2 (F ), z ∈ Z(AF ), u ∈ U0 (n), k∞ ∈ K∞ ,

where we have denoted ∞

ϕ

∗

: U0 (n)/U1 (n) −→ C , 338

a

b

c

d

! U1 (n) −→

Y v|n

ϕv (dv )

and ∗

ϕ : Z(AF ) −→ C ,

z

0

0

z

! 7→ ϕ(z).

Let h be the narrow class number of F . The recipe from ([Sh], §2; [Oh 2], Prop. 5.1.2) identifies each f ∈ Mk (n, ϕ) with an h-tuple of holomorphic functions (f1 , . . . , fh ) : H[F :Q] −→ Ch on the self-product of the upper half-plane H = {z ∈ C | Im(z) > 0}, which satisfy the classical weight k transformation formulas with respect to suitable congruence subgroups of GL2 (F ). (12.3.3) For each non-archimedean prime v of F we denote by dgv the (bi-invariant) Haar measure on R Q GL2 (Fv ) normalized by GL2 (Ov ) dgv = 1 and by dg = v-∞ dgv the product Haar measure on GL2 (Fb ). The Hecke algebra H(U1 (n)\GL2 (Fb )/U1 (n)) of U1 (n)-biinvariant compactly supported functions α : GL2 (Fb ) −→ C with the convolution product Z (α ∗ β)(h) = α(g)β(g −1 h) dg b) GL2 (F acts on Mk (n) (and on its subspaces Mk (n, ϕ), Sk (n, ϕ)) by the formula Z (α · f )(h) = α(g)f (hg) dg. b) GL2 (F For each square-free ideal a ⊂ OF , the classical Hecke operator Tn (a) : Mk (n) −→ Mk (n) corresponds to the action of ! Z −1 a 0 (N a)k/2−1 · dg · the characteristic function of A = U1 (n) U1 (n), A 0 1 where a ∈ Fb∗ is any finite id`ele satisfying div(a) = a (i.e. ordv (av ) = ordv (a) for all v - ∞). If n0 |n, then Mk (n0 ) ⊂ Mk (n). The operators Tn (a), Tn0 (a) do not always agree on Mk (n0 ); they do if every prime ideal dividing (a, n) also divides n0 . If there is no danger of confusion we shall write T (a) instead of Tn (a). As in the classical case F = Q, the Hecke operators T (a) for (a, n) = (1) acting on Mk (n) satisfy the formal identity X

T (a)(N a)−s =

(a,n)=(1)

Y

1 − T (v)(N v)−s + S(v)(N v)k−1−2s


−1

.

v-n

(12.3.4) Newforms [Miy] Assume that f ∈ Sk (n, ϕ) is a (non-zero) cuspidal eigenform for all operators T (a) with (a, n) = (1): T (a)f = λf (a)f,

((a, n) = (1), λf (a) ∈ C).

If n0 |n and f 0 ∈ Sk (n0 , ϕ) − {0}, we shall write f 0 ∼ f iff T (a)f 0 = λf (a)f 0

((a, n) = (1)).

If v|n is a prime ideal, we say that f is v-new if there is no f 0 ∈ Sk (n/v, ϕ) − {0} satisfying f 0 ∼ f . If this is the case, then f is also an eigenform for T (v) = Tn (v): T (v)f = λf (v)f,

(λf (v) ∈ C).

We say that f is a newform of level n if f is v-new at all prime ideals v|n. If this is the case, then f is an eigenform for all Hecke operators T (a) = Tn (a) 339

T (a)f = λf (a)f

(a ⊂ OF )

and the L-function of f L(f, s) :=

X

λf (a)(N a)−s

a

(absolutely convergent for sufficiently large Re(s) >> 0) is equal to the Euler product Y
−1 Y
−1 L(f, s) = 1 − λf (v)(N v)−s 1 − λf (v)(N v)−s + ϕ(v)(N v)k−1−2s . v|n

v-n

In general, there exists a smallest ideal n0 |n (with respect to divisibility) and f 0 ∈ Sk (n0 , ϕ) − {0} such that f 0 ∼ f . The form f 0 is unique up to a scalar multiple and is a newform of level n(f ) := n0 . We say that f 0 is the newform attached to f . We also put L(f, s) := L(f 0 , s). If f is a newform of level n and χ : A∗F /F ∗ −→ C∗ a character of finite order of conductor cond(χ), then there is a unique newform g (up to a scalar multiple) of level dividing cond(χ)2 n such that λg (v) = χ(v)λf (v)

(∀v - cond(χ)n).

We shall write g = f ⊗ χ. (12.3.5) We shall use the language of representation theory, which we briefly recall. Assume that f ∈ Sk (n, ϕ) (k ≥ 1) is a non-zero cuspidal eigenform for all Hecke operators T (a) with (a, n) = (1). The right regular action of Lie(GL2 (F ⊗ R)) × GL2 (Fb ) on f then generates an irreducible automorphic representation π = π(f ) = ⊗0v πv of GL2 (AF ). This representation has central character ϕ, i.e. ! a 0 π( ) = ϕ(a)I (∀a ∈ A∗F ). 0 a The isomorphism class of π(f ) depends only on the newform f 0 attached to f . The representation attached to g = f ⊗ χ is equal to π(f ⊗ χ) = π(f ) ⊗ χ := π(f ) · (χ ◦ det) (and has central character ϕχ2 ). The dual representation to π = π(f ) is isomorphic to ∼

π e −→ π ⊗ ϕ−1 (and has central character ϕ−1 ). (12.3.6) For each non-archimedean prime v of F , the local representation πv = π(f )v of GL2 (Fv ) is one of the following types: (12.3.6.1) (Irreducible) principal series representation π(µ, µ0 ), in which GL2 (Fv ) acts by right translations on the space B(µ, µ0 ) of locally constant functions f : GL2 (Fv ) −→ C satisfying ! a b f( g) = µ(a)µ0 (a0 )|a/a0 |1/2 f (g). 0 0 a Above, µ, µ0 : Fv∗ −→ C∗ are characters, | · | = | · |v : Fv∗ −→ R∗+ is the normalized valuation (|$v |v = (N v)−1 for any uniformizer $v ∈ Ov ), µ/µ0 6= | · |±1 . The representation π(µ, µ0 ) is isomorphic to π(µ0 , µ); its central character is equal to µµ0 and the dual representation is isomorphic to π(µ−1 , (µ0 )−1 ). If ν : Fv∗ −→ C∗ is ∼ another character, then π(µ, µ0 ) ⊗ ν −→ π(µν, µ0 ν). (12.3.6.2) Twisted Steinberg representation (= special representation) 340

St(µ) = St ⊗ µ ⊂ B(µ| · |1/2 , µ| · |−1/2 ) (where µ : Fv∗ −→ C∗ is a character). The central character of St(µ) is equal to µ2 ; the dual representation ∼ is isomorphic to St(µ−1 ). If ν : Fv∗ −→ C∗ is a character, then St(µ) ⊗ ν −→ St(µν). (12.3.6.3) Supercuspidal representations. (12.3.6.4) For each archimedean prime v|∞, πv is the Harish-Chandra module attached to the discrete series representation of weight k (resp., to the limit discrete series representation) if k ≥ 2 (resp., if k = 1). (12.3.7) The local L-factors as defined by Jacquet and Langlands ([J-L], Prop. 3.5, 3.6) are given by the formulas

Lv (πv , s) =

−s

where ΓC (s) = 2(2π)

 ΓC (s + k−1  2 ),      Lv (µ, s)Lv (µ0 , s),   Lv (µ, s + 12 ),     1,

v|∞ πv = π(µ, µ0 ) πv = St(µ) πv supercuspidal,

Fv∗

Γ(s) and, for each character µ : −→ C∗ (v - ∞), ( (1 − µ(v)(N v)−s )−1 , µ unramified Lv (µ, s) = 1, µ ramified.

(12.3.8) For each non-archimedean prime v - ∞ of F , the local Hecke algebra Hv of locally constant functions α : GL2 (Fv ) −→ C with compact support (with respect to the convolution product defined by the measure dgv ) acts on the representation space of πv by the operator Z α(gv )πv (gv ) dgv . GL2 (Fv )

For each m ≥ 1, the action of the primitive Hecke operator T (v m )prim (T (v)prim = T (v)) corresponds to the action of ! Z −1 $vm 0 m k/2−1 (N v ) · dgv · the characteristic function of A = Uv Uv , A 0 1 where $v is any prime element of Ov and U = U1 (n). If v - n, then πv = π(µ, µ0 ) for unramified characters µ, µ0 : Fv∗ −→ C∗ (“unramified principal series”). The characteristic functions of the double cosets ! ! $v 0 $v 0 Uv Uv , Uv Uv 0 1 0 $v (note that Uv = GL2 (Ov ) in this case) act on the one-dimensional space πvUv of spherical vectors by the scalars (N v)1/2 (µ(v) + µ0 (v)) and (µµ0 )(v), respectively. In other words, the “Hecke polynomial” " ! # " ! # $v 0 $v 0 1 − Uv Uv X + U v Uv (N v)X 2 0 1 0 $v acts on πvUv as Hv (πv , X) = (1 − (N v)1/2 µ(v)X)(1 − (N v)1/2 µ0 (v)X). The classical local L-factor of f 341


−1

Lv (f, s) = 1 − λf (v)(N v)−s + ϕ(v)(N v)k−1−2s is related to the local Hecke L-factor −s −1 LH ) v (πv , s) = Hv (πv , (N v)

and to the local Jacquet-Langlands L-factor  −1 Lv (πv , s) = (1 − µ(v)(N v)−s )(1 − µ0 (v)(N v)−s ) by Lv (f, s) =

LH v

    k k−1 πv , s − ( − 1) = Lv πv , s − ; 2 2

(12.3.8.1)

in other words, λf (v) = (N v)(k−1)/2 (µ(v) + µ0 (v)). If v|n and f is v-new, then it is still true that Lv (f, s) := 1 − λf (v)(N v)−s


−1

  k−1 = Lv πv , s − . 2

(12.3.8.2)

In particular, if πv = St(µ), then ( λf (v) =

µ(v)(N v)k/2−1 ,

µ unramified

0,

µ ramified.

(12.3.8.3)

(12.3.9) For each prime ideal v|n, the exponent of v in n(f ) is determined purely by the local representation πv = π(f )v , namely ordv (n(f )) = o(πv ), where o(πv ) ([Cas]) denotes the minimal integer n ≥ ordv (cond(ϕ)) for which there exists a non-zero vector in the representation space of πv on which U0 (v n )v acts by the character ! a b 7→ ϕv (d) c d (equivalently, for which πv admits a non-zero vector invariant under the action of U1 (v n )v ). If we denote, for any character µ : Fv∗ −→ C∗ , the exponent of the conductor of µ by o(µ) := min{n ≥ 0 | µ((1 + $vn Ov )∗ ) = {1}}, then we have ([Cas], proof of Thm. 1)  o(µ) + o(µ0 ),       1, o(πv ) =   2 o(µ),     ≥ 2,

πv = π(µ, µ0 ) πv = St(µ), µ unramified πv = St(µ), µ ramified

(12.3.9.1)

πv supercuspidal.

In particular, if ϕv is unramified, then o(πv ) = 1 ⇐⇒ πv = St(µ), µ unramified

(12.3.9.2)

2 - o(πv ), o(πv ) > 1 =⇒ πv supercuspidal.

(12.3.9.3)

and

342

(12.3.10) Lemma. Let f ∈ Sk (n, ϕ) (k ≥ 1) be a non-zero eigenform for all T (a) with (a, n) = (1); let π(f ) be the corresponding automorphic representation. (i) For a non-archimedean prime v - ∞ of F , the following conditions are equivalent: π(f )v = St(µ) with µ unramified ⇐⇒ ordv (n(f )) = 1 and ϕv is unramified. If they are satisfied, then λf (v)2 = ϕ(v)(N v)k−2 . (ii) If v|n and f is v-new, then the following conditions are equivalent: ( λf (v) 6= 0 ⇐⇒ π(f )v =

St(µ),

µ unramified

π(µ, µ0 ),

µ unramified.

If λf (v) 6= 0, then the two cases are distinguished as follows: π(f )v = St(µ), µ unramified ⇐⇒ ordv (n) = 1, ordv (cond(ϕ)) = 0

π(f )v = π(µ, µ0 ), µ unramified ⇐⇒ ordv (n) = ordv (cond(ϕ)) (= o(µ0 )). Proof. (i) The two conditions are equivalent, by (12.3.9.2). If they are satisfied, then λf (v)2 = µ(v)2 (N v)k−2 = ϕ(v)(N v)k−2 , by (12.3.8.3). The statement (ii) follows from (i) combined with (12.3.8.2) and (12.3.9.1). (12.3.11) The statement of Lemma 12.3.10(ii) should be compared to the fact that, for v - n(f ), π(f )v = π(µ, µ0 ), µ, µ0 unramified.

ordv (n(f )) = ordv (cond(ϕ)) = 0,

(12.3.12) Assume that f ∈ Sk (n, ϕ) is a (non-zero) newform of level n; let π = π(f ) be the corresponding automorphic representation. The complete L-functions L(π, s) =

Y

Lv (πv , s),

b s) = L(f,

v

Y

ΓC (s)

v|∞

Y

Lv (f, s)

v-∞

are related by   b s) = L π, s − k − 1 L(f, 2 (see (12.3.8.1-2)) and satisfy the functional equation L(π, s) = ε(π, s)L(e π , 1 − s) = ε(π, s)L(π ⊗ ϕ−1 , 1 − s), where ε(π, s) =

Y

εv (πv , s, ψv )

v

is the product of the local ε-factors defined in [JL] (above, ψ = (ψv ) is a non-trivial (unitary) additive character ψ : AF /F −→ C∗ ). Each local ε-factor is of the form a · bs , where a ∈ C∗ and b > 0. Their basic properties are summarized below: 343

(12.3.13) Lemma. (i) If ψv0 (x) = ψv (ax) (a ∈ Fv∗ ), then εv (πv , s, ψv0 ) = ϕv (a)|a|2s−1 εv (πv , s, ψv ). v (ii) (iii) (iv) (v) (vi)

For each character µ : Fv∗ −→ C∗ , εv (πv ⊗ µ, s, ψv ) εv (e πv ⊗ µ−1 , 1 − s, ψv ) = ϕv (−1). 1 If v - n and Ker(ψv ) = Ov , then εv (πv , 2 , ψv ) = 1. If ϕv = 1, then εv (πv , 12 ) := εv (πv , 12 , ψv ) does not depend on ψv and is equal to εv (πv , 12 ) = ±1. If πv = π(µ, µ−1 ), then εv (πv , 12 ) = µ(−1). If πv = St(µ), µ2 = 1, then ( −1, µ=1 εv (πv , 12 ) = µ(−1) × 1, µ 6= 1.

(vii) If v|∞ and 2|k, then εv (πv , 12 ) = (−1)k/2 . Proof. (i),(ii) [JL, remarks after Thm. 2.18] (the formula (ii) follows from the comparison of the local functional equations of Lv (πv ⊗ µ, s) and Lv (e πv ⊗ µ−1 , s)). 0 (iii) As πv = π(µ, µ ) with unramified characters µ, µ0 , we have εv (πv , s, ψv ) = εv (µ, s, ψv ) εv (µ0 , s, ψv ) = 1. (iv) Independence on ψv follows from (i), the equality εv (πv , 12 )2 = 1 from (ii) (with µ = 1). (v) In this case εv (π(µ, µ−1 ), 12 ) = εv (µ, 12 , ψv ) εv (µ−1 , 12 , ψv ) = µ(−1), where the latter equality follows from the comparison of the local functional equations for Lv (µ, s) and Lv (µ−1 , 1 − s). (vi) By definition,   Lv (µ−1 , 12 −s) µ unramified 1 1 1 , εv (St(µ), s, ψv ) = εv (µ, s + 2 , ψv ) εv (µ, s − 2 , ψv ) · L (µ,s− 2 )  v 1, µ ramified. If we let s −→ 12 , then the product of the two ε-factors tends to µ(−1) (cf. the proof of (v)), while the remaining term tends to −1 (resp., 1) if µ = 1 (resp., if µ 6= 1). (vii) This is well-known; one can use, for example, the corresponding formulas for the representations of the Weil group of R. (12.3.14) We shall be particularly interested in the case when f ∈ Sk (n, 1) is a newform of level n with trivial character (=⇒ 2|k). In this case ran (F, f ) := ords=k/2 L(f, s) = ords=1/2 L(π(f ), s) and (−1)ran (F,f ) = ε(π(f ), 12 ) =

Y v

εv (π(f )v , 12 )

(as ΓC (k/2) 6= 0, ∞), with each local ε-factor equal to εv (π(f )v , 12 ) = ±1. (12.4) Galois representations attached to Hilbert modular forms From now on, fix a prime number p and embeddings ι∞ : Q ,→ C, ιp : Q ,→ Qp . (12.4.1) Assume that f ∈ Sk (n, ϕ) (k ≥ 1) is a non-zero cuspidal eigenform for all Hecke operators T (a) with (a, n) = (1). According to ([Shi], Prop. 1.3), there exists a number field L ⊂ Q such that ι∞ (OL ) contains the values of ϕ and all Hecke eigenvalues λf (a) ((a, n) = (1)); fix such a field L. The embedding ιp induces a prime ideal p|p of L. (12.4.2) There exists a continuous two-dimensional representation V (f ) = Vp (f ) of GF = Gal(F /F ) over Lp which is unramified outside np and satisfies 344

det(1 − Fr(v) X | V (f )) = 1 − λf (v)X + ϕ(v)(N v)k−1 X 2 for all prime ideals v - np of OF (here Fr(v) = Fr(v)geom denotes the geometric Frobenius element at v). The representation V (f ) is absolutely irreducible, by [Tay 2, Prop. 3.1] (and the fact that each complex ˇ conjugation acts on V (f ) by a matrix with distinct eigenvalues ±1 ∈ Lp ). This implies, by the Cebotarev density theorem, that V (f ) is unique up to isomorphism, depends only on the newform attached to f , and V (f ⊗ χ) = V (f ) ⊗ χ for every character of finite order χ : A∗F /F ∗ −→ C∗ . The determinant of V (f ) is equal to ∼

det(V (f )) = Λ2 V (f ) −→ Lp (1 − k) ⊗ ϕ, hence the dual of V (f ) is isomorphic to ∼

V (f )∗ −→ V (f )(k − 1) ⊗ ϕ−1 . (12.4.3) For each prime ideal v - p, the restriction V (f )v of V (f ) to the decomposition group Gv = Gal(F v /Fv ) is related to π(f )v by the local Langlands correspondence as follows: for each character χ : Fv∗ −→ C∗ of finite order and each non-trivial additive character ψv : Fv −→ C∗ , Lv (π(f )v ⊗ χ, s) = Lv (V (f )v ⊗ χ, s +

k−1 2 )

εv (π(f )v ⊗ χ, s, ψv ) = εv (V (f )v ⊗ χ, ψv , dxψv , s +

k−1 2 ),

(12.4.3.1)

where Lv (V (f )v ⊗ χ, s) = det(1 − (N v)−s Fr(v) | (V (f )v ⊗ χ)Iv )−1 and the local ε-factor on the R.H.S. is the one defined in [De 2, 8.12] (above, dxψv is the self-dual Haar measure on Fv with respect to ψv ). Moreover, the conductors also agree: o(π(f )v ) = ordv (the Artin conductor of V (f )).

(12.4.3.2)

These results are due, under varying degrees of generality, to Eichler, Shimura, Deligne (F = Q), Langlands, Carayol, Ohta, Rogawski-Tunnell, Wiles, R. Taylor, Blasius-Rogawski. We shall be particularly interested in the ordinary case, treated in [Wi]. (12.4.4) In concrete terms, (12.4.3.1) implies the following (again, for a prime ideal v - p): (12.4.4.1) If π(f )v = π(µ, µ0 ), then Iv acts on V (f )v through a finite quotient and the semi-simplification of V (f )v is isomorphic to ∼ (1−k)/2 V (f )ss ⊕ Lp ⊗ µ0 | · |(1−k)/2 v −→ Lp ⊗ µ| · | (thus Iv acts on V (f )v by µ|Ov∗ ⊕ µ0 |Ov∗ ). (12.4.4.2) If π(f )v = St(µ), then the representation V (f )v is reducible and Iv acts on V (f )v through an infinite quotient. There is an exact sequence of Lp [Gv ]-modules 0 −→ Lp ⊗ µ| · |1−k/2 −→ V (f )v −→ Lp ⊗ µ| · |−k/2 −→ 0, whose extension class in ∼ 1 1 b p −→ Hcont (Gv , Hom(Lp ⊗ µ| · |−k/2 , Lp ⊗ µ| · |1−k/2 )) = Hcont (Gv , Lp (1)) = Fv∗ ⊗L Lp

is non-zero. In particular, if µ is unramified, then Iv acts on V (f )v through its tame quotient Ivt = Iv /Ivw , and any topological generator of Ivt acts on V (f )v by an endomorphism A satisfying (A−1)2 = 0 6= A−1. (12.4.4.3) If π(f )v is supercuspidal, then Iv acts on V (f )v irreducibly, through a finite quotient. More precisely, ∼ v either V (f )v is monomial, i.e. V (f )v −→ IndG Gw (µ), where Ew /Fv is a ramified quadratic extension and ∗ ∗ ∗ µ : Ew −→ Lp does not factor through NEw /Fv : Ew −→ Fv∗ (hence µ is ramified), 345

or V (f )v is exceptional (=⇒ v | 2), in which case there exists a Galois extension of Fv with Galois group isomorphic to A3 or S3 , over which V (f )v becomes monomial ([W]). (12.4.4.4) Let k be even. It follows from ([Wa 1], Lemma I.2.3) that, for each prime v - pn∞, the local representation π(f )v is defined over L, in the sense of ([Wa 1], I.1). According to ([Wa 1], Cor. I.8.3), the global representation π(f ) is also defined over L, and so are the local representation π(f )v for all v - ∞. As k is even, the correspondence between V (f )v and π(f )v normalized as in (12.4.3.1) commutes with the action of automorphisms of Q, which implies that (∀v - p∞) (∀g ∈ W (F v /Fv ))

det(1 − gX | V (f )v ) ∈ L[X].

(12.4.5) Lemma. Assume that f ∈ Sk (n, ϕ) (k ≥ 1) is a newform of level n. Then, for each prime ideal v - p of OF , (i) Iv acts trivially on V (f ) ⇐⇒ π(f )v is in the unramified principal series ⇐⇒ v - n. (ii) If ϕv is unramified, then the following conditions are equivalent: dimLp (V (f )Iv ) = 1 ⇐⇒ π(f )v = St(µ), µ unramified ⇐⇒ ordv (n) = 1. Proof. (i) This is a consequence of (12.4.3.1). The second equivalence in (ii) is just (12.3.9.2), while the first implication ‘⇐=’ follows from 12.4.4.2. If πv (f ) = St(µ) with µ ramified or if πv (f ) is supercuspidal, then V (f )Iv = 0. If πv (f ) = π(µ, µ0 ), then µµ0 = ϕv is unramified; thus ( V (f ), µ, µ0 unramified Iv V (f ) = 0, µ, µ0 ramified. This proves the first implication ‘=⇒’. (12.4.6) Corollary. Assume that f ∈ Sk (n, 1) is a newform of level n and trivial central character (=⇒ 2|k). Then, for each prime ideal v|n, v - p, ( det(−Fr(v) | V (f )(k/2)Iv ) =

−(N v)−k/2 λf (v) = −(N v)−1 µ(v),

π(f )v = St(µ) (µ2 = 1), µ unramified

1,

otherwise.

(12.4.7) The Jacquet-Langlands correspondence. Fix an archimedean prime τ1 : F ,→ R of F . If R is a finite set of non-archimedean primes of F satisfying |R| ≡ [F : Q] − 1 (mod 2),

(12.4.7.1)

then there exists a unique quaternion algebra B over F ramified at the set Ram(B) = {v|∞, v 6= τ1 } ∪ R. Assume that the form f satisfies (∀v ∈ R)

π(f )v is not in the principal series

(12.4.7.2)

(if 2 - [F : Q], then the conditions (12.4.7.1-2) are automatically satisfied for R = ∅). According to [JL, ∗ Thm. 16.1] (see also [G-J], §8; [Ro], §3), there exists an irreducible automorphic representation π 0 of BA such that ∼

(∀v 6∈ Ram(B)) πv0 −→ π(f )v . If k ≥ 2, then the Galois representation V (f ) can be constructed as a quotient of the ´etale cohomology group 1 Het (X ⊗F F , F ) of a suitable local system F on a Shimura curve X attached to B and π 0 ([Oh 1,2]; [Car], §2). If v - n(f ), then there exists X as above with good reduction at v. (12.4.8) Purity (Ramanujan’s Conjecture) 346

(12.4.8.1) Definition. Let v be a non-archimedean prime of a number field K and n ∈ Z. A v-Weil number of weight n is an algebraic number α ∈ Q such that (∃i ∈ Z)

(N v)i α is an algebraic integer;

(∀σ : Q ,→ C) |σ(α)| = (N v)n/2 . (12.4.8.2) Theorem. Let f ∈ Sk (n, ϕ) (k ≥ 2) be a non-zero cuspidal eigenform for all Hecke operators T (a) with (a, n) = (1). Let v - p∞ be a prime of F and g ∈ Gv a lift of the geometric Frobenius Fr(v) ∈ Gv /Iv . Then the two eigenvalues of g acting on V (f ) are v-Weil numbers of weights ( k − 1, k − 1 if π(f )v 6= St(µ) k, k − 2

if π(f )v = St(µ).

Proof. The case π(f )v = St(µ) is easy, thanks to 12.4.4.2. Assume π(f )v 6= St(µ). In the case F = Q and v - (f ) (resp., 2 - [F : Q] and v - (f )) the statement is deduced from Weil’s conjectures in [De 1] (resp., in [Oh 2], Thm. 1.4.1). If v does not lie in a certain exceptional finite set, the result is proved in [B-L]. The general case is treated in [Bl]. (12.4.8.3) Letting p (and p) vary, one can deduce from Theorem 12.4.8.2 and 12.4.3-4 an ‘automorphic’ version of Ramanujan’s Conjecture. For example, if v is a non-archimedean prime of F such that π(f )v = π(µ, µ0 ) is in the principal series, then, for each uniformizer $v ∈ Ov , both µ($v ) and µ0 ($v ) are v-Weil numbers of weight 0. (12.4.8.4) Proposition. Let f ∈ Sk (n, ϕ) be as in 12.4.8.2; assume that the weight k ∈ 2Z is even. Let v - p∞ be a prime of F and Ew /Fv a finite extension. Set ( ∅, if π(f )v 6= St(µ) J= {k/2 − 1, k/2 + 1}, if π(f )v = St(µ). Then, for each character of finite order β : Gw = Gal(F v /Ew ) −→ L∗p , we have (∀j ∈ Z − J)

∼

RΓcont (Gw , V (f )(j) ⊗ β) −→ 0

in Dfb t (Lp Mod). Proof. Let g ∈ Gw be a lift of the geometric Frobenius Fr(w) ∈ Gw /Iw . If π(f )v 6= St(µ), then all eigenvalues of g acting on V = V (f )(j) ⊗ β are w-Weil numbers of weight k − 1 − 2j 6= 0 (as k is even); ∼ 0 thus Hcont (Gw , V ) = 0. As V (f )(j)∗ (1) = V (f )∗ (1 − j) −→ V (f ⊗ ϕ−1 )(k − j), the same argument applied 2 0 1 to f ⊗ ϕ−1 shows that Hcont (Gw , V ) = Hcont (Gw , V ∗ (1))∗ = 0. The vanishing of Hcont (Gw , V ) then follows from the Euler characteristic formula (Theorem 4.6.9 and 5.2.11) 2 X

q (−1)q dimLp Hcont (Gw , V ) = 0.

q=0

If π(f )v = St(µ), then V Iw ⊂ Lp (1 − k/2 + j) ⊗ β · (µ ◦ Nw ), where Nw denotes the norm from the extension Ew /Fv . This means that either V Iw = 0, or g acts on the one-dimensional Lp -vector space V Iw by a w-Weil 0 number of weight k − 2 − 2j; thus Hcont (Gw , V ) = 0 for j 6= k/2 − 1. The same argument as above shows 2 1 that Hcont (Gw , V ) = 0 for j 6= k/2 + 1, hence Hcont (Gw , V ) = 0 for j 6= k/2 ± 1. (12.4.9) p-adic results. Let f ∈ Sk (n, ϕ) (k ≥ 2) be as in 12.4.8.2. (12.4.9.1) One can formulate a conjectural variant of (12.4.3.1) for a prime v|p of F dividing p. If the conditions (12.4.7.1-2) hold for suitable R, then the corresponding analogue of the first equality in (12.4.3.1) was proved by T. Saito ([Sa], Thm. 1) (1) . If the form f is p-ordinary, then this compatibility can be established directly (even in the case when R does not exist); see Lemma 12.5.4(iii) below. (1)

The general case was recently proved by M. Kisin [Kis]; this result implies that the assumptions (1)–(3) in Proposition 12.4.9.2 can be removed. 347

(12.4.9.2) Proposition. Let v|p be a prime of F which does not divide n(f ). Assume that at least one of the following conditions holds: (1) (12.4.7.1-2) hold for suitable R. (2) k = 2 and the residual representation of V (f ) is absolutely irreducible. (3) v is unramified in F/Q. Then V (f )v is a crystalline representation of Gv whose Hodge-Tate weights are contained in the interval [1 − k, 0] (if (1) holds, then the Hodge-Tate weights are known to be equal to 1 − k and 0). Proof. In the case (1), the result follows from the geometric construction of V (f )v ([Oh 1,2]) and the p-adic comparison results of Fontaine-Messing, Faltings and Tsuji. In general, V (f )v is constructed as a “limit” of finite subquotients of crystalline representation of Gv whose Hodge-Tate weights are contained in the interval [1 − k, 0]; a conjecture of Fontaine asserts that V (f )v itself is crystalline (and its Hodge-Tate weights are contained in the same interval). This was proved by R. Taylor [Tay 2] in the case (2). The case (3) is due to Breuil [Bre] if k < p, and to Berger [Be] without restriction on k (a forthcoming work of Berger-Colmez is expected to treat the case of ramified v). (12.4.10) Tamagawa factors. (12.4.10.1) Definition. If L ⊂ Q is a number field and j ≥ 1, denote by wj (L) the order of the (finite cyclic) group H 0 (GL , Q/Z(j)). (12.4.10.2) In particular, if ζ ∈ Q is a primitive root of unity of order n ≥ 1, then ζ ∈ L ⇐⇒ n | w1 (L) ζ +ζ

−1

∈ L =⇒ n | w2 (L).

(12.4.10.3) Proposition. Let f ∈ Sk (n, ϕ) (k ≥ 1) be a newform and L the number field generated by the Hecke eigenvalues of f and the values of ϕ. Let p | p be a prime of L and v - p∞ a prime of F such that ϕv is unramified. Let T (f ) ⊂ V (f ) be a GF -stable OL,p -lattice. Assume that p - w2 (L) and that either π(f )v is in the ramified principal series, or π(f )v is supercuspidal. Then (V (f )/T (f ))Iv = 0,

Tamv (T (f ), p) = 0.

Proof. The assumptions imply that there exists an element g ∈ Iv ⊂ Gv which acts on V (f )v non-trivially (necessarily by an automorphism of finite order). As det(g | V (f )v ) = ϕv (g) = 1, the eigenvalues of g are equal to ζ, ζ −1 , where ζ is a primitive root of unity of order n ≥ 2. As ζ + ζ −1 = Tr(g | V (f )v ) ∈ L (by (12.4.4.4)), we have n | w2 (L). The assumption p - w2 (L) implies that p - n, hence (1 − ζ)(1 − ζ −1 ) ∈ L∗ is a p-adic unit. In particular, g − 1 acts invertibly on T (f ) and V (f )/T (f ), hence V (f )Iv = 0,

1 Hcont (Iv , T (f )) = (V (f )/T (f ))Iv = 0,

Tamv (T (f ), p) = 0.

(12.5) Ordinary Hilbert modular forms Recall that we have fixed a prime number p and embeddings ι∞ : Q ,→ C, ιp : Q ,→ Qp . For non-zero ideals a, b ⊂ OF , denote by a(b) := a/(a, b∞) the prime-to-b part of a. (12.5.1) Definition. A non-zero cusp form f ∈ Sk (n, ϕ) (k ≥ 2) is p-ordinary if it is an eigenform for all Hecke operators T (a) = Tn (a) with (a, n(p) ) = (1): (a, n(p) ) = (1)

T (a)f = λf (a)f, and if (∀v|p)

|ιp (λf (v))|p = 1. 348

(12.5.2) If v|p but v - n, then the restriction of the Hecke operator Tnv (v) to Sk (n, ϕ) does not coincide with Tn (v); this implies that the inclusion Sk (n, ϕ) ⊂ Sk (nv, ϕ) does not preserve p-ordinary forms. However, there is a canonical procedure of “v-stabilization” ([Wi], p. 538) which attaches to f from 12.5.1 an ordinary form f 0v ∈ Sk (nv, ϕ) satisfying Tnv (a)f 0v = λf (a)f 0v

(a, n(v) ) = (1)

(hence f 0v ∼ f ). In the classical case F = Q (v = p), f (resp., f 0v ) corresponds to a classical cusp form Φ(z) (resp., Φ0 (z)) defined on the upper half plane and Φ0 (z) = Φ(z) − λf (p)−1 ϕ(p) pk−1 Φ(pz). In general, one can apply the v-stabilization with respect to all primes v|p which do not divide n and obtain the “p-stabilization” of f , which is an ordinary form f 0 ∈ Sk (lcm(n, (p)), ϕ) satisfying T(p)n (a)f 0 = λf (a)f 0

(a, n(p) ) = (1)

(hence f 0 ∼ f ). There is a theory of ‘newforms’ for v- and p-stabilized cusp forms (see [Hi 3], p. 318; [Wi], (1.2.2)). (12.5.3) For p-ordinary cusp forms f the representation V (f ) was constructed in full generality by Wiles [Wi], who also described its restriction V (f )v to Gv for all v|p (note that our representation is dual to the one considered in [Wi], as we prefer geometric Frobenius elements). Namely, for each v|p there is an exact sequence of Lp [Gv ]-modules − 0 −→ V (f )+ v −→ V (f )v −→ V (f )v −→ 0,

dimLp V (f )± v = 1,

in which ∼

Iv ∗ V (f )+ −→ (V (f )− v = V (f ) v ) (1 − k) ⊗ ϕv

(12.5.3.1)

is an unramified Gv -module on which Fr(v) (= Fr(v)geom ) acts by the scalar αv (f ) ∈ (

∗ OL,p ,

where

αv (f ) = λf (v),

if v|n

(1 − αv (f )X)(1 − ϕ(v)αv (f )−1 (N v)k−1 X) = 1 − λf (v)X + ϕ(v)(N v)k−1 X 2 ,

if v - n.

The subspace V (f )+ v ⊂ V (f ) is uniquely determined by these properties. The ordinarity assumption implies, by 12.3.10-11, that either ordv (n(f )) = ordv (cond(ϕ)) ≥ 0

or

ordv (n(f )) = 1, ordv (cond(ϕ)) = 0.

More precisely, we have the following dichotomy: (12.5.4) Lemma. If f ∈ Sk (n, ϕ) (k ≥ 2) is p-ordinary, then for each v|p there are two mutually exclusive possibilities: (i) If ordv (n(f )) = ordv (cond(ϕ)) ≥ 0, then V

(f )+ v

π(f )v = π(µ+ , µ− ), = Lp ⊗ µ+ | · |(1−k)/2 ,

µ+ µ− = ϕv , µ+ is unramified, − V (f )v = Lp (1 − k) ⊗ µ− | · |(k−1)/2 ,

o(µ− ) = ordv (n(f )), αv (f ) = (N v)(k−1)/2 µ+ (v)

and αv (f ) is a v-Weil number of weight (k − 1)/2. The representation V (f )v is de Rham; it is crystalline ∗ over a finite extension Ew of Fv ⇐⇒ the character µ− ◦ NEw /Fv = µ− ◦ Nw : Ew −→ C∗ is unramified. (ii) If ordv (n(f )) = 1, ordv (cond(ϕ)) = 0, then µ2 = ϕv ,

π(f )v = St(µ), V

(f )+ v

= Lp ⊗ µ,

V

(f )− v

µ is unramified,

= Lp (−1) ⊗ µ, 349

k = 2,

αv (f ) is a root of unity.

The representation V (f )v is semistable, but not crystalline (over any finite extension Ew of Fv ): its extension class in ∼ 1 + 1 ∗b Hcont (Gv , HomLp (V (f )− v , V (f )v )) = Hcont (Gv , Lp (1)) −→ Fv ⊗Lp ∼ b p. is not contained in Hf1 (Gv , Lp (1)) −→ Ov∗ ⊗L (iii) In the case (i) (resp., (ii)), the representation of the Weil-Deligne group of Fv attached to Dpst (V (f )v ) as in ([Fo-PR], I.2.2.5, I.1.3.2) is given by the direct sum of characters µ+ | · |(1−k)/2 and µ− | · |(1−k)/2 (resp., is equal to µ| · |−1 ⊗ sp(2)), hence it corresponds to π(f )v = π(µ+ , µ− ) (resp., to π(f )v = St(µ)) by the local Langlands correspondence (normalized as in 12.4.3).

Proof. Fix v|p. In the case (i), we know from 12.3.10-11 that π(f )v = π(µ, µ0 ) with µ unramified and (1−k)/2 µµ0 = ϕv . Denote by µ± : Fv∗ −→ L∗p the characters such that Gv acts on V (f )+ , and on v by µ+ | · | − (k−1)/2 V (f )v by Lp (1 − k) ⊗ µ− | · | . If v|n(f ), then Lv (µ, s) = Lv (π(f )v , s) = (1 − λf (v)(N v)−s−(k−1)/2 )−1 , which implies that (µ| · |(1−k)/2 )(v) = λf (v) = αv (f ) = (µ+ | · |(1−k)/2 )(v) =⇒ µ = µ+ . It follows from (12.5.3.1) that µ− = ϕv /µ+ = µ0 . Finally, 12.4.8.3 implies that µ+ (v) is a v-Weil number of weight 0 (hence αv (f ) is a v-Weil number of weight (k − 1)/2). If v - n(f ), but v|n, then the theory of ‘newforms’ for v-stabilized cusp forms implies that there exists a p-ordinary form g ∈ Sk (n(v) , ϕ) such that, for each ideal a with (a, n(v) ) = (1), Tn (a)f = λf (a)f,

Tn(v) (a)g = λf (a)g.

This implies that it is sufficient to prove (i) for g, i.e. we can assume that v - n. In this case µ0 is also unramified and Lv (µ, s)Lv (µ0 , s) = Lv (π(f )v , s) = (1 − λf (v)(N v)−s−(k−1)/2 + ϕ(v)(N v)−2s )−1 , hence {µ(v), µ0 (v)} = {αv (f )(N v)−(k−1)/2 , ϕ(v)αv (f )−1 (N v)(k−1)/2 } = {µ+ (v), µ− (v)}. Exchanging µ and µ0 if necessary (and noting that µ+ (v) 6= µ− (v), as their p-adic valuations are distinct), we obtain µ = µ+ , µ0 = µ− . Again, 12.4.8.3 implies that both µ± (v) are v-Weil numbers of weight 0. The extension class of − 0 −→ V (f )+ v −→ V (f )v −→ V (f )v −→ 0

(12.5.4.1)

is contained in 1 + 1 + ⊗2 1 Hcont (Gv , HomLp (V (f )− ⊗ ϕ−1 v , V (f )v )) = Hcont (Gv , (V (f )v ) v (k − 1)) = Hcont (Gv , Lp (k − 1) ⊗ β),

where β : Gv /Iv −→ L∗p is an unramified character whose value at the geometric Frobenius β(v) = β(Fr(v)) = αv (f )2 ϕ(v)−1 = (N v)k−1 µ+ (v)/µ− (v) is a v-Weil number of weight k − 1 ≥ 1 (thus β 6= 1). It follows from [B-K, 3.9] that, for any unramified character ν : Gv /Iv −→ L∗p , we have 1 (∀n > 1) Hcont (Gv , Lp (n) ⊗ ν) = Hf1 (Fv , Lp (n) ⊗ ν) 1 ν 6= 1 =⇒ Hcont (Gv , Lp (1) ⊗ ν) = Hf1 (Fv , Lp (1) ⊗ ν).

350

(12.5.4.2)

Applying (12.5.4.2) to ν = β, we deduce that V (f )v is a crystalline representation of Gv . 1 Back to the case v|n(f ); we see that the extension class of (12.5.4.1) is contained in Hcont (Gv , Lp (k − 1) ⊗ β), where 1−k β = (µ+ /µ− )| · |1−k = µ2+ ϕ−1 : Gv −→ L∗p v |·|

is a product of an unramified character by a character of finite order; [B-K, 3.9] then implies that (∀n ≥ 1)

1 Hcont (Gv , Lp (n) ⊗ β) = Hg1 (Fv , Lp (n) ⊗ β),

hence V (f )v is a de Rham representation of Gv . If V (f )v is crystalline over Ew , so is V (f )− v = Lp (1 − k) ⊗ µ− | · |(k−1)/2 , hence µ− ◦ Nw is unramified. Conversely, if µ− ◦ Nw is unramified, applying (12.5.4.2) to Ew and to the restriction β 0 of β to Gw , we deduce that V (f )v is a crystalline representation of Gw (note that β 0 6= 1, as β 0 (Fr(w)) is a w-Weil number of weight k − 1 ≥ 1). (ii) We have π(f )v = St(µ), by Lemma 12.3.10. The formula Lv (µ, s + 12 ) = Lv (π(f )v , s) = (1 − λf (v)(N v)−s−(k−1)/2 )−1 implies that (µ| · |1−k/2 )(v) = λf (v) = αv (f ), ∼

1−k/2 + ∗ hence Gv acts on V (f )+ (=⇒ µ is unramified) and on V (f )− v by µ| · | v −→ (V (f )v ) (1 − k) ⊗ ϕv by 1−k/2 −1 2 k/2−1 Lp (1 − k) ⊗ (µ| · | ) µ = Lp (1 − k) ⊗ µ| · | . As

|ιp (ϕ(v)(N v)k−2 )|p = |ιp (λf (v)2 )|p = 1, it follows that k = 2, hence αv (f ) = µ(v), which is a root of unity (as µ2 = ϕv has finite order). The fact that the representation V (f )v is semistable, but not crystalline, follows from the comparison result of T. Saito ([Sa], Thm. 1). One can also argue geometrically: V (f )v is a direct factor of the first ´etale cohomology group of the Jacobian of a Shimura curve attached to a quaternion algebra over F ramified at ˇ ˇ (see also [Dr] and [Bo-Car] in the v, but whose level subgroup is maximal at v. The results of Cerednik [Ce] case F = Q) imply that the Jacobian in question has purely toric reduction at v, which proves the desired result. (iii) This follows from the following two facts: firstly, if χ : Gv −→ L∗p is a potentially unramified onedimensional representation and if n ∈ Z, then the representation of the Weil-Deligne group of Fv attached to Lp (n) ⊗ χ is the one-dimensional representation given by χ| · |n ; secondly, that the monodromy operator N on Dpst (V (f )v ) satisfies N = 0 (resp., N 6= 0) in the case (i) (resp., (ii)), as V (f )v is (resp., is not) potentially crystalline. (12.5.5) Assume that f ∈ Sk (n, ϕ) is p-ordinary and there exists a character χ : A∗F /F ∗ −→ C∗ (with values contained in ι∞ (L)) such that ϕ = χ−2 (=⇒ k is even). Fix such a character χ; it is determined up to multiplication by a quadratic character A∗F /F ∗ −→ {±1}. The form g := f ⊗ χ ∈ Sk (n(g), 1) is a newform of level n(g) dividing cond(χ)2 n and trivial character. The twisted Galois representation V = V (g)(k/2) = V (f )(k/2) ⊗ χ of GF is self-dual in the sense that ∼

Λ2 V −→ Lp (1),

∼

V −→ V ∗ (1)

and, for each prime v|p of F , the twisted representations of Gv Vv± = V (f )± v (k/2) ⊗ χv 351

(12.5.5.1)

sit in an exact sequence of Lp [Gv ]-modules 0 −→ Vv+ −→ Vv −→ Vv− −→ 0,

(12.5.5.2) ∼

which is self-dual with respect to the isomorphism (12.5.5.1), i.e. Vv± −→ (Vv∓ )∗ (1). If Ew /Fv (again for v|p) is a finite extension, we put αw (f ) = αv (f )[κ(w):κ(v)] , where κ(w)/κ(v) is the corresponding extension of residue fields. If, in addition, the character ∗ χv ◦ Nw : Ew −→ Fv∗ −→ C∗

(Nw = NEw /Fv )

is unramified, then we put αw (g) = αw (f )(χv ◦ Nw )(w). For example, if π(f )v = St(λ) with λ unramified, then αv (f ) = λ(v) and αw (f ) = (λ ◦ Nw )(w); if, in addition, χv ◦ Nw is unramified, then π(g)v = St(µ),

µ2 = 1,

µ = λχv ,

µ ◦ Nw is unramified,

αw (g) = ((λχv ) ◦ Nw )(w) = (µ ◦ Nw )(w).

In general, Lemma 12.5.4 implies that we have, for each prime v | p of F , either π(g)v = π(µ+ χv , (µ+ χv )−1 ),

µ+ unramified,

ordv (n(g)) = 2 ordv (cond(χ))

or π(g)v = St(µ),

µ2 = 1,

k = 2,

µχ−1 v unramified,

ordv (n(g)) = max (1, 2 ordv (cond(χ))) .

(12.5.6) We wish to relate two kinds of Selmer groups attached to V ⊗ β (where β is a character of finite e 1 (V ⊗ β) (with respect to Greenberg’s local conditions defined by order) over a finite extension of F : our H f 1 (12.5.5.2)) and Bloch-Kato’s Hf (V ⊗ β) (cf. [N-P, 3.1.7] for F = Q). Recall first the local situation: let v|p be a prime of F and W a continuous Qp -representation of Gv ; put W ∗ (1) = HomQp (W, Qp (1)). Bloch-Kato [B-K, 3.7.2] defined subspaces 1 He1 (Fv , W ) ⊂ Hf1 (Fv , W ) ⊂ Hg1 (Fv , W ) ⊂ H 1 (Fv , W ) = Hcont (Gv , W )

satisfying the following properties: if W is a de Rham representation of Gv , then (12.5.6.1) The extension class of a short exact sequence of continuous Qp -representation of Gv 0 −→ W −→ W 0 −→ Qp −→ 0 is contained in Hg1 (Fv , W ) (resp., in Hf1 (Fv , W )) iff W 0 is a de Rham representation of Gv (resp., a crystalline representation of Gv , provided that W is crystalline). (12.5.6.2) He1 (Fv , W ) (resp., Hf1 (Fv , W )) is the exact annihilator of Hg1 (Fv , W ∗ (1)) (resp., of Hf1 (Fv , W ∗ (1))) under the local Tate duality ∪

∼

H 1 (Fv , W ) × H 1 (Fv , W ∗ (1))−→H 2 (Fv , Qp (1)) −→ Qp . (12.5.6.3) He1 (Fv , W ) is a quotient of DdR (W )/F 0 . (12.5.6.4) Hf1 (Fv , W )/He1 (Fv , W ) is a quotient of Dcris (W )/(f − 1). 352

(12.5.6.5) For each exact sequence of Qp -representations of Gv 0 −→ W + −→ W −→ W − −→ 0 (=⇒ both W + and W − are de Rham representations), the induced sequence 0 −→ H 0 (Fv , W + ) −→ H 0 (Fv , W ) −→ H 0 (Fv , W − ) −→ Hg1 (Fv , W + ) −→ Hg1 (Fv , W ) −→ Hg1 (Fv , W − ) is also exact. We shall also study the subspaces H∗1 (Ew , W ) over finite extensions Ew of Fv ; in this case we use the notation Dcris,w (W ) = H 0 (Ew , W ⊗Qp Bcris ),

DdR,w (W ) = H 0 (Ew , W ⊗Qp BdR )

for Fontaine’s functors over Ew . (12.5.7) Lemma. Under the assumptions of 12.5.5, let Ew /Fv (where v|p) be a finite extension and β : Gw = Gal(F v /Ew ) −→ L∗p a character of finite order. Then: (1) (i) He1 (Ew , Vv− ⊗ β) = 0, Hg1 (Ew , Vv+ ⊗ β) = H 1 (Ew , Vv+ ⊗ β). (ii) Dcris,w (Vv+ ⊗ β)f =1 = H 0 (Ew , Vv+ ⊗ β) = 0. (iii) He1 (Ew , Vv+ ⊗ β) = Hf1 (Ew , Vv+ ⊗ β), Hf1 (Ew , Vv− ⊗ β) = Hg1 (Ew , Vv− ⊗ β). (2) If Dcris,w (Vv− ⊗ β ±1 )f =1 = 0, then: (i) Dcris,w (V ⊗ β ±1 )f =1 = 0. (ii) (∀W = V, Vv± ) H 0 (Ew , W ⊗β ±1 ) = 0, He1 (Ew , W ⊗β ±1 ) = Hf1 (Ew , W ⊗β ±1 ) = Hg1 (Ew , W ⊗β ±1 ). 1 − ±1 (iii) (∀∗ = e, f, g) H∗ (Ew , Vv ⊗ β ) = 0, H∗1 (Ew , Vv+ ⊗ β ±1 ) = H 1 (Ew , Vv+ ⊗ β ±1 ). 1 + ±1 1 (iv) The map H (Ew , Vv ⊗β ) −→ H (Ew , V ⊗β ±1 ) is injective and its image is equal to Hf1 (Ew , V ⊗β ±1 ). Proof. (1) As Vv− (k/2 − 1) ⊗ χv is an unramified representation of Gv and k/2 − 1 ≥ 0, it follows that DdR,w (Vv− ⊗ β)/F 0 = 0, hence He1 (Ew , Vv− ⊗ β) = 0, by 12.5.6.3. Applying the duality 12.5.6.2 and replacing β by its inverse we obtain the second statement of (i). As Vv+ (−k/2) ⊗ χ−1 v is an unramified representation and k 6= 0, we have H 0 (Ew , Vv+ ⊗ β) ⊆ Dcris,w (Vv+ ⊗ β)f =1 = 0, proving (ii). Combining (ii) with 12.5.6.4 yields the first statement of (iii); the second follows from the duality 12.5.6.2 (replacing β by its inverse). (2) As both Dcris,w (Vv± ⊗β ±1 )f =1 = 0 vanish, so do Dcris,w (V ⊗β ±1 )f =1 , H 0 (Ew , W ⊗β ±1 ) and Hf1 (Ew , W ⊗ β ±1 )/He1 (Ew , W ⊗ β ±1 ) (W = V, Vv± ), thanks to 12.5.6.4. Applying the duality 12.5.6.2, we obtain the equality Hf1 = Hg1 in (ii). The statements in (iii) follow from (ii) and (1)(i). Finally, (iv) is a consequence of (iii) and the exact sequence 12.5.6.5 attached to 0 −→ Vv+ ⊗ β ±1 −→ V ⊗ β ±1 −→ Vv− ⊗ β ±1 −→ 0. (12.5.8) Proposition. Under the assumptions of 12.5.5, let Ew /Fv (where v|p) be a finite extension and β : Gw = Gal(F v /Ew ) −→ L∗p a character of finite order. Then the kernel (resp., the image) of the map H 1 (Ew , Vv+ ⊗ β) −→ H 1 (Ew , V ⊗ β) is equal to H 0 (Ew , Vv− ⊗ β) (resp., to Hf1 (Ew , V ⊗ β)), i.e. there is an exact sequence 0 −→ H 0 (Ew , Vv− ⊗ β) −→ H 1 (Ew , Vv+ ⊗ β) −→ Hf1 (Ew , V ⊗ β) −→ 0. Moreover,

( dimLp H

0

(Ew , Vv−

In particular, if β = 1, then (

1,

π(g)v = St(µ) (µ2 = 1), β · (µ ◦ Nw ) = 1

0,

otherwise.

1,

π(f )v = St(λ) (λ unramified), χv ◦ Nw unramified, αw (g) = 1

(

0, 1,

otherwise π(g)v = St(µ) (µ = λχv , µ2 = 1), µ ◦ Nw unramified, αw (g) = 1

(

0, 1,

otherwise π(g)v = St(µ) (µ2 = 1), µ ◦ Nw = 1

0,

otherwise

dimLp H 0 (Ew , Vv− ) = = =

⊗ β) =

353

Proof. We analyze separately the two cases distinguished in Lemma 12.5.4. If π(f )v = π(µ+ , µ− ), then π(g)v = π(µ+ χv , µ− χv ); we claim that Dcris,w (Vv− ⊗ β ±1 )f =1 = 0

(12.5.8.1)

in this case (which implies the statement of the Proposition, thanks to Lemma 12.5.7(2)(ii),(iv)). Fix ν ∈ {β, β −1 }; if the character ν · ((µ− χv ) ◦ Nw ) is ramified, then Dcris,w (Vv− ⊗ ν) = 0. If it is unramified, then Dcris,w (Vv− ⊗ ν) is a free Lp ⊗Qp (Ew ∩ Qur p )-module of rank one, on which f acts by a ⊗ σ, where σ is the absolute (arithmetic) Frobenius and a = (νµ− χv )(Fr(w))(N w)−1/2 is w-Weil number of weight −1, by 12.4.8.3; thus (12.5.8.1) holds in this case, too. If π(f )v = St(λ) (λ unramified), then π(g)v = St(µ), where µ = λχv , µ2 = 1. As Vv+ = Lp (1) ⊗ µ and Vv− = Lp ⊗ µ, we have, for each ν ∈ {β, β −1 }, ( 1, ν · (µ ◦ Nw ) = 1 − f =1 dimLp Dcris,w (Vv ⊗ ν) = (12.5.8.2) 0, otherwise. If β · (µ ◦ Nw ) 6= 1, then β −1 · (µ ◦ Nw ) 6= 1 either, since µ2 = 1; thanks to (12.5.8.2), we conclude again by appealing to Lemma 12.5.7(2)(ii),(iv). In the remaining case β · (µ ◦ Nw ) = 1, the exact sequence of Lp [Gw ]-modules 0 −→ Vv+ ⊗ β −→ V ⊗ β −→ Vv− ⊗ β −→ 0 is isomorphic to a non-crystalline Kummer extension 0 −→ Lp (1) −→ V ⊗ β −→ Lp −→ 0. The exactness of the sequence 0 −→ Lp = H 0 (Ew , Vv− ⊗ β) −→ H 1 (Ew , Vv+ ⊗ β) −→ Hf1 (Ew , V ⊗ β) −→ 0 then follows from the calculation in ([N-P], 3.1.6). In the special case β = 1, we have µ ◦ Nw = 1 ⇐⇒ µ ◦ Nw is unramified and (µ ◦ Nw )(w) = 1 ⇐⇒ χv ◦ Nw is unramified and αv (g) = 1, which proves the remaining statements. (12.5.9) Comparison of Selmer groups Let f, χ, g = f ⊗ χ and V = V (g)(k/2) be as in 12.5.5. (12.5.9.1) Fix a finite set S of primes of F such that S ⊃ {v|pn(g)∞}, a finite subextension E/F of FS /F and a (continuous) character of finite order β : Gal(FS /E) = GE,S −→ L∗p . The Bloch-Kato Selmer group  1 Hf1 (E, V ⊗ β) = Ker Hcont (GE,S , V ⊗ β) −→

MM

 1 Hcont (Gw , V ⊗ βw )/Hf1 (Ew , V ⊗ βw )

v∈Sf w|v 1 (where w runs through all primes of E above v, and Hf1 (Ew , −) = Hur (Ew , −) if v - p) does not depend on the choice of S, by ([Fo-PR], II.1.3.2). Fix a subset Σ ⊂ Sf containing all primes dividing p, set Σ0 = Sf − Σ and define, for each v ∈ Sf and w|v, Greenberg’s local conditions ∆Σ,w (V ⊗ β) for V ⊗ β as follows:  • Ccont (Gw , Vv+ ⊗ βw ), v|p    Uw+ (V ⊗ β) = 0, v ∈ Σ, v - p    • Ccont (Gw /Iw , (V ⊗ βw )Iw ), v ∈ Σ0 .

354

(12.5.9.2) Proposition-Definition. Under the assumptions of 12.5.9.1, • • (i) Let w - p∞ be a prime of E; then the complexes Ccont (Gw , V ⊗ βw ) and Ccont (Gw /Iw , (V ⊗ βw )Iw ) (where Gw = Gal(E w /Ew )) are acyclic. gf (GE,S , V ⊗ β; ∆Σ (V ⊗ β)) does not depend on (ii) Up to a canonical isomorphism, the Selmer complex RΓ gf (E, V ⊗ β) and its cohomology groups by H e i (E, V ⊗ β). the choice of S and Σ; we denote it by RΓ f (iii) There is an exact sequence MM e f1 (E, V ⊗ β) −→ Hf1 (E, V ⊗ β) −→ 0, 0 −→ H 0 (Ew , Vv− ⊗ βw ) −→ H v|p w|v

in which v|p is a prime of F and ( dimLp H

0

(Ew , Vv−

⊗ βw ) =

1,

π(g)v = St(µ) (µ2 = 1), βw · (µ ◦ Nw ) = 1

0,

otherwise.

(iv) Let E 0 /E be a finite subextension of FS /E such that β factors through β : Gal(E 0 /E) −→ L∗p . Then, in the notation of Proposition 8.8.7, there are canonical isomorphisms (β ∼ eq e q (E, V ⊗ β) −→ H Hf (E 0 , V ) f

−1

)

∼

Hf1 (E, V ⊗ β) −→ Hf1 (E 0 , V )(β

,

−1

)

.

Proof. (i) This follows from the fact that the inflation map • • inf : Ccont (Gw /Iw , (V ⊗ βw )Iw ) −→ Ccont (Gw , V ⊗ βw ) • induces monomorphisms on cohomology and that the complex Ccont (Gw , V ⊗ βw ) is acyclic (by Proposition 12.4.8.4). (ii) Independence on S (resp., on Σ) follows from Proposition 7.8.8 (resp., from (i)). (iii) It follows from (i) that the bottom row of (6.1.3.2) boils down to the following exact triangle: MM gf (E, V ⊗ β) −→ RΓcont (GE,S , V ⊗ β) −→ RΓ RΓcont (Gw , Vv− ⊗ βw ).

v|p w|v

In the corresponding cohomology sequence H 0 (GE,S , V ⊗ β) −→

MM

r e 1 (E, V ⊗ β) −→ H 1 (GE,S , V ⊗ β)−→ H 0 (Ew , Vv− ⊗ βw ) −→ H f cont

v|p w|v r

−→

MM

1 Hcont (Gw , Vv− ⊗ β)

v|p w|v

the group H 0 (GE,S , V ⊗ β) = 0 vanishes by Proposition 12.4.8.4, H 0 (Ew , Vv− ⊗ βw ) was computed in Proposition 12.5.8 and Ker(r) = Hf1 (E, V ⊗ β) (by combining (i) with Proposition 12.5.8). (iv) This follows from Proposition 8.8.7 (the same proof also applies to the Bloch-Kato Selmer groups). (12.5.9.3) Corollary-Definition. Denote e e fi (E, V ⊗ β), hif (E, V ⊗ β) = dimLp H then

h1f (E, V ⊗ β) = dimLp Hf1 (E, V ⊗ β);

1 1 2 e hf (E, V ⊗ β) − hf (E, V ⊗ β) = {w : w|v|p, π(g)v = St(µ) (µ = 1), βw · (µ ◦ Nw ) = 1} .

(12.5.9.4) Proposition-Definition. Under the assumptions of 12.5.9.1, define ( −1, if v | p, π(g)v = St Y 1 1 1 1 εe(π(g), 2 ) = εev (π(g)v , 2 ), εev (π(g)v , 2 ) = εv (π(g)v , 2 ) · 1, otherwise. v 355

Then we have and

1 1 εe(π(g), 12 )/ε(π(g), 12 ) = (−1)ehf (F,V )−hf (F,V )

Y Y 1 1 (−1)ehf (F,V )−hf (F,V )+ran (F,g) = εe(π(g), 12 ) = χv (−1) εv (π(g)v , 12 ) χv (−1)(−1)k/2 . v-p∞

v|∞

Proof. Applying Proposition 12.5.9.2(iii) to E = F and β = 1, we obtain Y

1 1 (−1)ehf (F,V )−hf (F,V ) =

(−1) = εe(π(g), 12 )/ε(π(g), 12 ).

v|p π(g)v =St

Fix a prime v|p of F . According to Lemma 12.5.4 there are two possible cases: either π(f )v = π(µ+ , µ− ) with µ+ unramified, which implies that π(g)v = π(µ+ χv , (µ+ χv )−1 ) and εv (π(g)v , 12 ) = (µ+ χv )(−1) = χv (−1) (by Lemma 12.3.13(v)), or π(f )v = St(λ) with λ unramified, which implies that π(g)v = St(µ) with µ = λχv and ( εv (π(g)v , 12 ) = χv (−1) ·

−1,

if µ = 1

1,

if µ 6= 1

(by Lemma 12.3.13(vi), as µ(−1) = χv (−1)). Combining the last two formulas, we obtain Y

(−1) =

Y

χv (−1) εv (π(g)v , 12 ).

v|p

v|p π(g)v =St

The second statement follows from the product formulas Y

Y

χv (−1) = 1,

v

v

εv (π(g)v , 12 ) = (−1)ran (F,g)

and Lemma 12.3.13(vii). (12.5.9.5) Proposition. Let K (resp., Lp ) be a finite extension of Q (resp., of Qp ), S ⊃ {v | p∞} a finite set of primes of K and W a finite-dimensional vector space over Lp equipped with a continuous Lp -linear action of GK,S . Assume that, for each prime v | p of K, the representation Wv (= W considered as a representation of Gv ) is de Rham. Denote by (−)∗ = HomLp (−, Lp ) the Lp -dual. (i) For each prime v of K, denote i hiv (W ) := dimLp Hcont (Gv , W ),

Then

( h0v (W )

−

h1v,f (W )

=

h1v,f (W ) := dimLp Hf1 (Kv , W ) 0, − dimLp DdR,v (Wv )/F


0

(v - ∞). v - p∞

,

v|p

(ii) There is an exact sequence 0 −→ H 0 (GK,S , W ) −→ −→

M

M

2 H 0 (Gv , W ) −→ Hcont (GK,S , W ∗ (1))∗ −→ Hf1 (K, W ) −→

v∈Sf

Hf1 (Kv , W )

1 −→ Hcont (GK,S , W ∗ (1))∗ −→ Hf1 (K, W ∗ (1))∗ −→ 0.

v∈Sf

356

i (iii) Denoting hi (W ) := dimLp Hcont (GK,S , W ) and h1f (W ) := dimLp Hf1 (K, W ), then the Euler-Poincar´e 0 1 characteristic χf (W ) := h (W ) − hf (W ) + h1f (W ∗ (1)) − h0 (W ∗ (1)) is equal to

χf (W ) =

X

h0v (W ) −

X

v|∞


dimLp DdR,v (Wv )/F 0 .

v|p ∼

(iv) If W = V ⊗ α, where α : GK,S −→ L∗p is a character of finite order and V −→ V ∗ (1) is self-dual, then χf (W ) = 0. In particular, if H 0 (GK,S , V ⊗ α) = H 0 (GK,S , V ⊗ α−1 ) = 0, then h1f (V ⊗ α) = h1f (V ⊗ α−1 ). Proof. (i) For v - p∞ (resp., for v | p) the result follows from the exact sequence Fr(v)−1

0 −→ H 0 (Gv , W ) −→ W Iv −−−−→W Iv −→ Hf1 (Kv , W ) −→ 0 (resp., from the bottom exact sequence in [B-K, 3.8.4]). (ii) See [Fo-PR], II.2.2.1. 3−i (iii) The duality Theorem 5.4.5 implies that hi (W ∗ (1)) = h3−i c (W ) := dimLp Hc,cont (GK,S , W ). Applying the Euler characteristic formula 5.3.6, we deduce that χ(W ∗ (1)) := h0 (W ∗ (1)) − h1 (W ∗ (1)) + h2 (W ∗ (1)) = h1c (W ) − h2c (W ) + h3c (W ) = −

X

h0v (W ).

v|∞

On the other hand, it follows from the exact sequence (ii) that χf (W ) + χ(W ∗ (1)) =

X


(i) X
h0v (W ) − h1v,f (W ) = − dimLp DdR,v (Wv )/F 0 .

v∈Sf

v|p

Subtracting the two formulas yields the result. (iv) In this case W ∗ (1) = V ⊗ α−1 and (∀v | ∞)

h0v (V ⊗ α) = [Kv : R] dim(V )/2.

If v | p, then 2 dimLp F 0 DdR,v (Vv ) = dimLp F 0 DdR,v (Vv ) + dimLp F 0 DdR,v (V ∗ (1)v ) = = dimLp DdR,v (Vv ) = [Kv : Qp ] dim(V )

dimLp DdR,v (Wv )/F 0 = dimLp DdR,v (Vv )/F 0 = [Kv : Qp ] dim(V )/2, hence  (iii)

χf (W ) = 

X

[Kv : R] −

v|∞

X

 [Kv : Qp ] dim(V )/2 = 0.

v|p

(12.5.10) Proposition. Let g ∈ Sk (n, ϕ) (k ≥ 2) be a newform. The following conditions are equivalent: (i) There exists a character of finite order χ : A∗F /F ∗ −→ C∗ such that g ⊗ χ−1 is p-ordinary. (ii) There exists a finite totally real solvable extension F 0 /F such that the form g 0 := BCF 0 /F (g) is pordinary. Proof. (i) =⇒ (ii): According to ([A-T], ch. 10, Thm. 5), there exists a character of finite order χ0 : A∗F /F ∗ −→ C∗ such that (∀v | p) χ0v = χv ,

(∀v | ∞) χ0v = 1. 357

Ker(χ0 )

For such χ0 , the form g ⊗ χ0−1 is p-ordinary, F 0 := F is a totally real cyclic extension of F and the base change form BCF 0 /F (g) = BCF 0 /F (g ⊗ χ0−1 ) is p-ordinary. (ii) =⇒ (i) Fix a number field L as in 12.4.1; let u - p∞ be a prime of L and V (g) = Vu (g) the Lu representation of GF attached to g. Let v | p be a prime of F and v 0 | v a prime of F 0 . We claim that 0 0 π(g)v is not supercuspidal: if it were, then there would exist intermediate fields Fv ⊂ Kw ⊂ Kw 0 ⊂ Fv 0 ∼ ∼ Gw 0 with Gal(Kw /Fv ) −→ {1}, A3 or S3 , Gal(Kw0 /Kw ) = {1, σ} and V (g)w −→ IndGw0 (µ) monomial, where ∗

0 ∗ µ : (Kw 0 ) −→ Q ,

π(g 0 )v0 = π(µ1 , µ2 ),

µ2 = σµ ◦ NF 0 0 /K 0 0

µ1 = µ ◦ NF 0 0 /K 0 0 , v

w

v

w

(σµ(h) = µ(σ −1 hσ)).

0 There exists an integer n ≥ 1 such that µn is unramified. If π is a uniformizer of Kw 0 , then

(σµ)n (π) = µn (σ −1 (π)) = µn (σ) µn (σ −1 (π)/π) = µn (π), which implies that (σµ/µ)n = 1, hence (µ2 /µ1 )n = 1. In particular, the numbers ιp (µn1 (v 0 )) and ιp (µn2 (v 0 )) have the same p-adic valuations, which contradicts the p-ordinarity of g 0 . This contradiction shows that π(g)v is not supercuspidal. As g 0 is p-ordinary, we must have     π(µ(v) , µ(v) ), ιp (µ(v) ◦ NF 0 /F )(v 0 )(N v 0 )(k−1)/2 = 1 1 2 1 v v0 p π(g)v =  (v) (v) St(µ1 ), µ1 of finite order. Applying again ([A-T], ch. 10, Thm. 5), there exists a character of finite order χ : A∗F /F ∗ −→ C∗ such that (v) −1 µ1 χ−1 is then p-ordinary. v is unramified, for all primes v | p; the form f ⊗ χ (12.6) L-functions over ring class fields Throughout this section F is a totally real number field, K/F a totally imaginary quadratic extension, η = ηK/F : A∗F /F ∗ −→ {±1} the quadratic character attached to K/F and τ the non-trivial element of Gal(K/F ). We denote N = NK/F and Nw = NKw /Fv , Trw = TrKw /Fv (if w is a prime of K above a prime v of F ). (12.6.1) Ring class fields and characters (12.6.1.1) If β : A∗K /K ∗ −→ C∗ is a (continuous) character of finite order, then there is a unique automorphic representation θ(β) of GL2 (AF ) satisfying Y Lv (θ(β) ⊗ µ, s) = Lw (βw · (µ ◦ Nw ), s) w|v

Fv∗

for all primes v of F and all characters µ : −→ C∗ ; it corresponds to the two-dimensional representation GK IndGF (β) of GK . The representation θ(β) has central character η · (β|A∗F ); it is cuspidal iff β does not factor as χ

N

A∗K /K ∗ −→A∗F /F ∗ −→C∗ . If µ : A∗F /F ∗ −→ C∗ is a character of finite order, then θ(β) ⊗ µ = θ(β · (µ ◦ N )). In the classical case F = Q, θ(β) corresponds to the theta series of weight 1 X β(a) q N a (−L(0, χ)/2 if β = χ ◦ N ). More generally, one can define θ(β) for algebraic Hecke characters of K; see 12.6.5 below. (12.6.1.2) The local component of θ(β) at a prime v of F has the following properties ([J-L], Thm. 4.6): (12.6.1.2.1) If v|∞, then θ(β)v is the limit discrete series representation. (12.6.1.2.2) If v - ∞ splits in K/F into vOK = ww0 , then θ(β)v = π(βw , βw0 ), 358

∗ ∗ where βw (resp., βw0 ) is considered as a character of Fv∗ = Kw (resp., of Fv∗ = Kw 0 ). 2 (12.6.1.2.3) If v - ∞ is inert or ramified in K/F (vOK = w or w ), then

( θ(β)v =

π(µ, µηv ),

if βw = µ ◦ Nw

supercuspidal,

if βw 6= µ ◦ Nw

and εv (θ(β)v , s, ψv ) = εv (ηv · βw |Fv∗ , s, ψv ) εw (βw , s, ψv ◦ Trw ). (12.6.1.3) In the situation of (12.6.1.2.3) there is a purely local version of θ(β): one can attach to every ∗ continuous character β : Kw −→ C∗ an irreducible admissible representation θv (β) of GL2 (Fv ) satisfying Lv (θv (β) ⊗ µ, s) = Lw (β · (µ ◦ Nw ), s)

(∀µ : Fv∗ −→ C∗ )

and (12.6.1.2.3). bc = Oc ⊗ Z b=O bF + cO bK ⊂ O bK = (12.6.1.4) For a non-zero ideal c ⊂ OF put Oc = OF + cOK ⊂ OK and O ∗ ∗ ∗ b⊂K b = K ⊗ Z. b The abelian extension K[c]/K corresponding to the subgroup K ·A ·(K ⊗R) · O b∗ ⊂ OK ⊗ Z c F ∗ AK is called the ring class field of K of conductor c. The extension K[c]/F is Galois; more precisely, ∼

Gal(K[c]/F ) −→ Gal(K[c]/K) o {1, τ } is a generalized dihedral group, since each lift of τ to Gal(K[c]/F ) acts by conjugation on ∼ ∼ b∗ bc∗ −→ bc∗ Gal(K[c]/K) −→ A∗K /K ∗ A∗F (K ⊗ R)∗ O K /K ∗ Fb ∗ O

as g 7→ g −1 . (12.6.1.5) We say that β from 12.6.1.1 is a ring class character if β|A∗F = 1 (=⇒ β ◦ τ = β −1 ). We shall denote by c(β) the smallest non-zero ideal c ⊂ OF (with respect to divisibility) such that β|O b∗ = 1. In this c

case the level of θ(β) divides dK/F c(β)2 and its central character is equal to η, which implies that g = θ(β) ⊗ η −1 = θ(β · (η ◦ N )−1 ) = θ(β). θ(β)

(12.6.1.6) Proposition. Let β : A∗K /K ∗ A∗F −→ C∗ be a ring class character of finite order, v - ∞ a prime of F which does not split in K/F , w the unique prime of K above v and µ : Fv∗ −→ {±1} a character. (i) If v is inert in K/F and v - c(β), then βw = 1. In particular, βw = µ ◦ Nw ⇐⇒ µ = 1 or µ = ηv ⇐⇒ µ is unramified. (ii) If v | c(β), then βw is ramified. In particular, βw = µ ◦ Nw =⇒ µ is ramified. 2 (iii) If v is ramified in K/F and v - c(β), then βw = 1 and there is a unique unramified character λ : Fv∗ −→ {±1} such that βw = λ ◦ Nw .

Proof. Easy exercise. (12.6.2) Rankin-Selberg L-functions (12.6.2.1) Let f ∈ Sk (n, ϕ) − {0} (k ≥ 1) be a newform of level n and π = π(f ) the corresponding automorphic representation of GL2 (AF ). One can attach to π its base change to K, which is an automorphic representation πK of GL2 (AK ) ([Jac], Thm. 20.6). If β : A∗K /K ∗ −→ C∗ is a character of finite order, then the local L-factors and ε-factors of πK ⊗ β coincide with the corresponding factors of the Rankin-Selberg product π × θ(β): 359

Lv (π × θ(β), s) =

Y

Lw (πK ⊗ β, s)

w|v

εv (π × θ(β), s, ψv ) =

Y

εw (πK ⊗ β, s, ψv ◦ Trw ).

w|v

The functional equation g 1 − s) L(π × θ(β), s) = ε(π × θ(β), s) L(e π × θ(β), Y ε(π × θ(β), s) = εv (π × θ(β), s, ψv ) v

yields a functional equation for the L-function   Y   Y k−1 k−1 L(fK , β, s) := Lv πK ⊗ β, s − = Lv π × θ(β), s − , 2 2 v-∞

v-∞

in particular for L(fK , s) =

Y v-∞

  Y     k−1 k−1 Y k−1 Lv πK , s − = Lv π, s − Lv π ⊗ η, s − . 2 2 2 v-∞

v-∞

If V (f ) is the p-adic representation of GF attached to f , then we have, for each prime w - p∞ of K,   k−1 GK Lw (πK ⊗ β, s) = Lw ResGF (V (f )) ⊗ β, s + . 2 (12.6.2.2) We shall use the following explicit formulas ([Jac], Thm. 15.1): (12.6.2.2.1) If v|∞, then

Lv (π × θ(β), s) = ΓC s +

k−1 2




,

εv (π × θ(β), 12 , ψv ) = 1.

(12.6.2.2.2) If v - ∞ and πv = π(µ, µ0 ), then Lv (π × π 0 , s) = Lv (π 0 ⊗ µ, s) Lv (π 0 ⊗ µ0 , s) εv (π × π 0 , s, ψv ) = εv (π 0 ⊗ µ, s, ψv ) εv (π 0 ⊗ µ0 , s, ψv ). (12.6.2.2.3) If v - ∞ and πv = St(µ), then Lv (π × π 0 , s) = Lv (π 0 ⊗ µ, s + 12 ) εv (π × π 0 , s, ψv ) = εv (π 0 ⊗ µ, s + 12 , ψv ) εv (π 0 ⊗ µ, s − 12 , ψv )

Lv ( 12 − s, π e0 ⊗ µ−1 ) 1 Lv (s − 2 , π 0 ⊗ µ)

(12.6.2.3) Let g ∈ Sk (n, 1) be a newform of level n with trivial central character (=⇒ k ∈ 2Z), π = π(g) the corresponding automorphic representation and β : A∗K /K ∗ A∗F −→ C∗ a ring class character of finite order. We denote ran (K, g, β) := ords=k/2 L(gK , β, s) = ords=1/2 L(π × θ(β), s) ran (K, g) := ran (K, g, 1) = ords=k/2 L(g, s)L(g ⊗ η, s) = ords=1/2 L(π, s)L(π ⊗ η, s). The functional equation L(π × θ(β), s) = ε(π × θ(β), s) L(π × θ(β), 1 − s) Q implies (as in 12.3.4, we use the fact that the archimedean L-factors v|∞ Lv (π × θ(β), s) have neither zero nor pole at s = 1/2) that 360

(−1)ran (K,g,β) = ε(β) =

Y

ε(β)v ,

(−1)ran (K,g) = ε =

v

Y

εv ,

(12.6.2.3.1)

v

where we have denoted, for each prime v of F , ε(β)v := εv (π × θ(β), 12 , ψv ),

εv := ε(1)v

∗

(for a fixed additive character ψ : AF /F −→ C ). (12.6.2.4) Proposition. In the notation of 12.6.2.3, let v - ∞ be a prime of F . (i) If πv = π(µ, µ−1 ), then ε(β)v = ηv (−1). (ii) If πv = St(µ) (µ2 = 1), then ( ε(β)v = ηv (−1) ·

−1,

if v is not split in K/F and βw = µ ◦ Nw

1,

otherwise.

(iii) If πv is supercuspidal, assume that v - dK/F and, if v is inert in K/F , then v - c(β). Then ε(β)v = ηv (v)o(πv ) = ηv (v)ordv (n) . Proof. (i) It follows from (12.6.2.2.2) that ε(β)v = εv (θ(β)v ⊗ µ, 12 , ψv ) εv (θ(β)v ⊗ µ−1 , 12 , ψv ). −1 If vOK = ww0 (w 6= w0 ), then θ(β)v = π(βw , βw0 ) = π(βw , βw ), hence

−1 1 −1 1 −1 ε(β)v = εv (µβw , 12 ) εv (µ−1 βw , 2 ) εv (µβw , 2 ) εv (µ−1 βw , 12 ) = (µβw )(−1) (µβw )(−1) = 1 = ηv (−1).

If v does not split in K/F and w is the unique prime of K above v, then we obtain from (12.6.1.2.3) εv (θ(β)v ⊗ µ, 12 , ψv ) = εv (θv (βw · (µ ◦ Nw )), 12 , ψv ) = εv (ηv µ2 · βw |Fv∗ , 12 , ψv ) εw (βw · (µ ◦ Nw ), 12 , ψv ◦ Trw ) εv (θ(β)v ⊗ µ−1 , 12 , ψv ) = εv (ηv µ−2 · βw |Fv∗ , 12 , ψv ) εw (βw · (µ ◦ Nw )−1 , 12 , ψv ◦ Trw ). As βw |Fv∗ = 1 and −1 εw (βw · (µ ◦ Nw )−1 , 12 , ψv ◦ Trw ) = εw ((βw · (µ ◦ Nw )−1 ) ◦ τ, 12 , ψv ◦ Trw ) = εw (βw · (µ ◦ Nw )−1 , 12 , ψv ◦ Trw ),

it follows that −1 ε(β)v = εv (ηv µ2 , 12 )εv (ηv µ−2 , 12 )εw (βw ·(µ◦Nw ), 12 )εw (βw ·(µ◦Nw )−1 , 12 ) = (ηv µ2 βw ·(µ◦Nw ))(−1) = ηv (−1).

(ii) It follows from (12.6.2.2.3) that ε(β)v = A · B, where A = εv (θ(β)v ⊗ µ| · |1/2 , 12 , ψv ) εv (θ(β)v ⊗ µ| · |−1/2 , 12 , ψv ) B = lim

s→1/2

Lv (θ(β)v ⊗ µ, 12 − s) Lv (θ(β)v ⊗ µ, s − 12 )

The same calculation as in (i) shows that A = ηv (−1). In order to compute B, we use the fact that, for any character χ : Fv∗ −→ C∗ , 361

Lv (χ, s) B(χ) := lim = s→0 Lv (χ, −s)

(

−1,

χ=1

1,

χ 6= 1.

If v splits in K/F , then (12.6.1.2.2) implies that −1 B = B(βw ) B(βw0 ) = B(βw ) B(βw ) = 1.

If v does not split in K/F , let w be the unique prime of K above v. If βw 6= ν ◦ Nw , then θ(β)v is supercuspidal, hence B = 1. If βw = ν ◦ Nw (ν : Fv∗ −→ C∗ ), then (12.6.1.2.3) implies that ( −1, ν = µ, µηv B = B(µν) B(µνηv ) = 1, ν 6= µ, µηv . ∼

∗ As ηv induces an isomorphism ηv : Fv∗ /Nw (Kw ) −→ {±1}, we have

ν = µ, µηv ⇐⇒ η ◦ Nw = µ ◦ Nw . Putting everything together, we obtain the formula (ii). −1 (iii) By assumption, v is unramified in K/F . If vOK = ww0 , then θ(β)v = π(βw , βw ), hence −1 1 ε(β)v = εv (πv ⊗ βw , 12 , ψv ) εv (πv ⊗ βw , 2 , ψv ) = 1 = ηv (v),

by Lemma 12.3.13(ii). ∗ If v is inert in K/F and w is the unique prime of K above v, then βw : Kw −→ C∗ is unramified and trivial ∗ on Fv ; it follows that βw = 1, hence θ(β)v = π(1, ηv ) and ε(β)v = εv (πv , 12 , ψv ) εv (πv ⊗ ηv , 12 , ψv ). Let n(ψv ) be the minimal integer n ∈ Z such that ψv (v −n OF,v ) = 1. As ηv is unramified, we have εv (πv ⊗ ηv , 12 , ψv ) = ηv (v)o(πv )+2n(ψv ) = ηv (v)o(πv ) = (−1)o(πv ) , εv (πv , 12 , ψv ) which implies that ε(β)v = εv (πv , 12 , ψv )2 ηv (v)o(πv ) = ηv (v)o(πv ) . (12.6.3) Global ε-factors for ring class characters Let g ∈ Sk (n, 1), π = π(g) and β : A∗K /K ∗ A∗F −→ C∗ be as in 12.6.2.3. (12.6.3.1) Define the “Steinberg part” of the level of g by Y nSt = n(a) , where a = v (ordv (nSt ) > 0 ⇐⇒ πv = St(µ)). πv 6=St(µ)

(12.6.3.2) Definition. We define R(β)0 := {v | nSt : πv = St(µ), βw = µ ◦ Nw , v ramified in K/F }

R(β)− := {v | nSt : πv = St(µ), βw = µ ◦ Nw , v inert in K/F } (above, w is the unique prime of K above v). Note that R(1)0 = {v | nSt : πv = St or St(ηv ), v ramified in K/F } R(1)− = {v | nSt : πv = St or St(ηv ), v inert in K/F } (12.6.3.3) Proposition. (i) R(β)− ∩ {v - c(β)} = R(1)− ∩ {v - c(β)}. (ii) R(β)0 ∩ {v | c(β)} ∩ R(1)0 = R(β)− ∩ {v | c(β)} ∩ R(1)− = ∅. Proof. This follows from Proposition 12.6.1.6(i-ii). 362

(12.6.3.4) Proposition. If the order of β is odd, then (i) R(β)0 ∩ {v - c(β)} = R(1)0 ∩ {v - c(β)}. (ii) (R(β)0 ∪ R(β)− ) ∩ {v | c(β)} = ∅. (iii) R(β)0 = R(1)0 ∩ {v - c(β)}, R(β)− = R(1)− ∩ {v - c(β)}. Proof. The statement (i) (resp., (ii)) follows from Proposition 12.6.1.6(iii) (resp., (ii)), while (iii) is a consequence of (i) and (ii). (12.6.3.5) Consider the following conditions: H(K): H(K, β):

(∀v | dK/F ) πv is not supercuspidal. H(K) ∧ ((∀v | c(β) inert in K/F ) πv is not supercuspidal).

(12.6.3.6) Proposition. Assume that H(K) holds. Then   0 ε = (−1)[F :Q] η n(dK/F ) (−1)|R(1) | . Proof. Proposition 12.6.2.4 (for β = 1) implies that Y (−1)[F :Q] ε = ε ηv (−1) = v|∞

Y

ηv (v)o(πv )

Y

(−1).

v∈R(1)0

v-dK/F ∞


(12.6.3.7) Corollary. If (∀v | dK/F ) πv is in the principal series, then ε = (−1)[F :Q] η n(dK/F ) . In particular, if (dK/F , n) = (1), then ε = (−1)[F :Q] η(n). (12.6.3.8) Proposition. Assume that H(K, β) holds. Then   0 − − ε(β) = (−1)[F :Q] η n(dK/F ) (−1)|R(β) | (−1)|R(β) ∩{v|c(β)}| (−1)|R(1) ∩{v|c(β)}| . Proof. Proposition 12.6.2.4 implies that (−1)|R(β)

0

|+|R(β)− |

ε(β) = (−1)|R(1)

0

|+|R(1)− |

ε.

(12.6.3.8.1)

Combining this formula with Proposition 12.6.3.3(i) and 12.6.3.6 yields the result. (12.6.3.9) Corollary. (i) If H(K, β) holds and β has odd order, then   0 − ε(β) = (−1)[F :Q] η n(dK/F ) (−1)|R(1) ∩{v-c(β)}| (−1)|R(1) ∩{v|c(β)}| .
(ii) If H(K, β) holds and (dK/F c(β), nSt ) = (1), then ε(β) = ε = (−1)[F :Q] η n(dK/F ) . (12.6.3.10) Define, for each prime v of F , ( −1, if v | p, v is not split in K/F , πv = St(µ), βw = µ ◦ Nw g := ε(β)v · ε(β) v 1, otherwise (where w is the unique prime of K above v) and set Y Y g , g = g , εev = ε(1) ε(β) ε(β) εe = εev . v v v

v

It follows from the definitions that 0 − g ε(β)/ε(β) = (−1)|R(β) ∩{v|p}| (−1)|R(β) ∩{v|p}| .

(12.6.3.10.1)

Combining this formula with (12.6.3.8.1), we obtain, assuming H(K, β), that (−1)|R(β)

0

∩{v-p}|

(−1)|R(β)

−

∩{v-p}|

g = (−1)|R(1)0 ∩{v-p}| (−1)|R(1)− ∩{v-p}| εe. ε(β)

(12.6.3.11) Consider the following conditions: H(K, p): H(K, β, p):

(p)

H(K) ∧ (∀v | dK/F ) πv is in the principal series. H(K, β) ∧ H(K, p). 363

(12.6.3.10.2)

(12.6.3.12) Proposition. Assume that H(K) holds. Then   0 − εe = (−1)[F :Q] η n(dK/F ) (−1)|R(1) ∩{v-p}| (−1)|R(1) ∩{v|p}| . Proof. Combine (12.6.3.10.1) (for β = 1) with Proposition 12.6.3.6. (12.6.3.13) Corollary. Assume that H(K, p) holds. Then   − εe = (−1)[F :Q] η n(dK/F ) (−1)|R(1) ∩{v|p}| .
If, in addition, (∀v | p) πv is not supercuspidal, then εe = (−1)[F :Q] η n(pdK/F ) . Proof. The first formula follows from Proposition 12.6.3.12. If v | p is unramified in K/F , then

ηv (v)o(πv ) =

 −1,       1,   1,     1,

if πv = St ⊗ µ, µ unramified, v inert in K/F if πv = St ⊗ µ, µ ramified if v is split in K/F if πv = π(µ, µ−1 ),

which proves the second formula. (12.6.3.14) Proposition. Assume that H(K, β) holds. Then   g = (−1)[F :Q] η n(dK/F ) (−1)|R(1)− ∩{v|pc(β)}| (−1)|R(β)0 ∩{v-p}| (−1)|R(β)− ∩{v|c(β), v-p}| . ε(β) Proof. Combining Proposition 12.6.3.8 with (12.6.3.10.1), we obtain   g (−1)[F :Q] η n(dK/F ) (−1)|R(β)0 ∩{v-p}| = (−1)|R(β)− ∩{v|p}| (−1)|R(β)− ∩{v|c(β)}| (−1)|R(1)− ∩{v|c(β)}| . ε(β) The R.H.S. can be simplified as follows: we have |R(β)− ∩ {v | p}| + |R(β)− ∩ {v | c(β)}| = |R(β)− ∩ {v | p, v - c(β)}| + |R(β)− ∩ {v | c(β), v - p}| and |R(β)− ∩ {v | p, v - c(β)}| = |R(1)− ∩ {v | p, v - c(β)}| (using Proposition 12.6.3.4(i)), hence |R(β)− ∩ {v | p, v - c(β)}| + |R(1)− ∩ {v | c(β)}| = |R(1)− ∩ {v | pc(β)}|, which proves the desired formula. (12.6.3.15) Corollary. (i) If H(K, β) holds and β has odd order, then   g = (−1)[F :Q] η n(dK/F ) (−1)|R(1)0 ∩{v-pc(β)}| (−1)|R(1)− ∩{v|pc(β)}| . ε(β) (ii) If H(K, β, p) holds and β has odd order, then   g = (−1)[F :Q] η n(dK/F ) (−1)|R(1)− ∩{v|pc(β)}| . ε(β) 364

(p)

(iii) Assume that H(K, β) holds, (c(β), nSt ) = (1) and either H(K, p) holds or β has odd order. Then g = εe. ε(β) (12.6.4) Ring class characters and twists of p-ordinary forms (12.6.4.1) Let f ∈ Sk (n, ϕ), g = f ⊗ χ ∈ Sk (n(g), 1) and V = V (g)(k/2) = V (f )(k/2) ⊗ χ be as in 12.5.5 (i.e. f is p-ordinary and ϕ = χ−2 ). Fix a ring class character of finite order β : A∗K /K ∗ A∗F −→ C∗ . (12.6.4.2) Using the embeddings ι∞ and ιp , we consider β as a character with values in L∗p (after enlarging L, if necessary). Denote the fixed field of β by Hβ . According to 12.5.9.2(iv) there are canonical isomorphisms (β ∼ e1 e 1 (K, V ⊗ β) −→ H Hf (Hβ , V ) f

−1

)

(β) ∼ e1 e 1 (K, V ⊗ β −1 ) −→ H Hf (Hβ , V ) . f

,

The action of Adf (τ ) (for any lift τ to Gal(Hβ /F )) induces an isomorphism e 1 (Hβ , V )(β H f

−1

)

(β) ∼ e1 −→ H , f (Hβ , V )

hence e h1f (K, V ⊗ β) = e h1f (K, V ⊗ β −1 ). The same results hold for Hf1 (and h1f ). (12.6.4.3) Proposition. Under the assumptions of 12.6.4.1 we have 1 1 g (−1)ehf (K,V ⊗β)−hf (K,V ⊗β) = ε(β)/ε(β), 1 1 g (−1)ehf (K,V ⊗β)−hf (K,V ⊗β)+ran (K,g,β) = ε(β).

Proof. It follows from (12.6.2.3.1) that it is enough to prove the first formula. According to 12.5.9.3,

e h1f (K, V ⊗ β) − h1f (K, V ⊗ β) =

XX

dimLp H

0

(Kw , Vv−

v|p w|v

X

X

v|p πv =St(µ)

w|v

⊗ βw ) =

(

1,

βw = µ ◦ N w

0,

βw 6= µ ◦ Nw .

−1 The contribution from each prime v | p that splits in K/F (vOK = ww0 ) is even, as βw0 = βw , hence

βw0 = µ ◦ Nw0 ⇐⇒ βw = µ ◦ Nw . g The parity of the contributions from the remaining primes yields exactly the sign ε(β)/ε(β). (12.6.4.4) We shall be interested in the validity of the following congruence: ? e h1f (K, V ⊗ β) − h1f (K, V ⊗ β) + ran (K, g, β) ≡ e h1f (K, V ) − h1f (K, V ) + ran (K, g) (mod 2).

(?)

g = εe. It follows from 12.6.4.3 that (?) holds ⇐⇒ ε(β) (12.6.4.5) Proposition. The congruence (?) holds in any of the following cases: (p) (i) H(K, β) holds, (c(β), n(g)St ) = (1) and either H(K, p) holds or β has odd order. (ii) H(K) holds, c(β) divides a power of p and either H(K, p) holds or β has odd order. (iii) (n(g), dK/F ) = (1) and c(β) divides a power of p. Proof. (i) Combine Proposition 12.6.4.3 and Corollary 12.6.3.15(iii). The statement (ii) is a special case of (i), as πv is not supercuspidal for any v | p (hence H(K, β) follows from H(K) in this case). Similarly, (iii) is a special case of (ii). 365

(12.6.4.6) Fix a non-zero ideal c ⊂ OF prime to p and a non-empty set P1 , . . . , Ps | p of prime ideals above p in F . Set [ K[cP1∞ · · · Ps∞ ] = K[cP1n · · · Psn ], G(cP1∞ · · · Ps∞ ) = Gal(K[cP1∞ · · · Ps∞ ]/K). n≥1

The torsion subgroup G(cP1∞ · · · Ps∞ )tors of G = G(cP1∞ · · · Ps∞ ) is finite and the quotient group G/Gtors is isomorphic to Zap , where a=

s X

[FPi : Qp ].

i=1 ∗

(12.6.4.7) Proposition. Assume that β, β 0 : G = G(cP1∞ · · · Ps∞ ) −→ Lp are characters of finite order satisfying β|Gtors = β 0 |Gtors . Then: 0 ∗ ∗ (i) If w - p∞ is a prime of K, then βw |OK,w = βw |OK,w . (ii) The ideals c(β)(p) = c(β 0 )(p) are equal and they divide c. (iii) R(β)0 ∩ {v - p} = R(β 0 )0 ∩ {v - p} and R(β)− ∩ {v - p} = R(β 0 )− ∩ {v - p}. (iv) If H(K, β) holds, so does H(K, β 0 ) and we have e h1f (K, V ⊗ β) − h1f (K, V ⊗ β) + ran (K, g, β) ≡ e h1f (K, V ⊗ β 0 ) − h1f (K, V ⊗ β 0 ) + ran (K, g, β 0 ) (mod 2). (v) If H(K, β) holds, then h1f (K, V ⊗ β) − ran (K, g, β) ≡ h1f (K, V ⊗ β 0 ) − ran (K, g, β 0 ) (mod 2). ∗ ∗ ∗ Proof. (i) As the pro-p-part of OK,w is finite, the image of OK,w /OF,v (where v is the prime of F induced by 0 w) in G is contained in Gtors , which implies that βw = βw . The statements (ii) and (iii) are straightforward consequences of (i). (iv) As c(β)(p) = c(β 0 )(p) and πv is not supercuspidal at any prime v | p, H(K, β 0 ) follows from H(K, β). g = ε(β 0 ), which is equivalent to the statement ] Applying Proposition 12.6.3.14 to β and β 0 , we obtain ε(β) (iv), by Proposition 12.6.4.3. ∼ (v) We use the fundamental parity result proved in Theorem 10.7.17. Fix an isomorphism G −→ Gtors × H, [F :Q] H = Zp , and put K0 = K[cP1∞ · · · Ps∞ ]H . The character β (resp., β 0 ) decomposes as a product β = α0 ×α 0 (resp., β = α0 × α0 ), where ∗

α0 = β|Gtors = β 0 |Gtors : Gtors −→ Lp ,

∗

α : H −→ Lp ,

∗

α0 : H −→ Lp .

The parity result 10.7.17(iii) then yields e h1f (K, V ⊗ β) ≡ e h1f (K, V ⊗ α0 ) (mod 2) e h1f (K, V ⊗ β 0 ) ≡ e h1f (K, V ⊗ α0 ) (mod 2), hence e h1f (K, V ⊗ β) ≡ e h1f (K, V ⊗ β 0 ) (mod 2). Combined with (iv), this proves (v). (P ···P )

∗

1 s (12.6.4.8) Corollary. Assume that (n(g), c dK/F ) = (1) and that β, β 0 : G = G(cP1∞ · · · Ps∞ ) −→ Lp are characters of finite order satisfying β|Gtors = β 0 |Gtors . Then

h1f (K, V ⊗ β) − ran (K, g, β) ≡ h1f (K, V ⊗ β 0 ) − ran (K, g, β 0 ) (mod 2). (P ···Ps )

1 Proof. The assumption (n(g), c dK/F

) = (1) implies that H(K, β) holds. We apply Proposition 12.6.4.7(v). 366

(P ···Ps )

1 (12.6.4.9) Proposition-Definition. Assume that (n(g), c dK/F

) = (1) and define

M (g, K; P1 , . . . , Ps ) := {v | p in F, v 6∈ {P1 , . . . , Ps }, v inert in K/F, ordv (n(g)) = 1} m(g) = m(g, K; P1 , . . . , Ps ) := |M (g, K; P1 , . . . , Ps )|.
(i) We have ηK/F n(g)(P1 ···Ps ) /n(g)(p) = (−1)m(g) . (P ···P )

(ii) If (n(g)St1 s , (p)) = (1) (e.g., if k 6= 2), then m(g) = 0. ∗ (iii) Let β : G(cP1∞ · · · Ps∞ ) −→ Lp be a character of finite order; then   g = (−1)[F :Q]+m(g) ηK/F n(g)(P1 ···Ps ) . ε(β) If (P1 · · · Ps )n | c(β) and n >> 0, then   ε(β) = (−1)[F :Q] ηK/F n(g)(P1 ···Ps ) .

e h1f (K, V ⊗ β) − h1f (K, V ⊗ β) = m(g),

Proof. (i), (ii) Let v | ((p), n(g)), v = 6 P1 , . . . , Ps be a prime of F ; then π(g)v is not supercuspidal and v is unramified in K/F , hence ηv (o(π(g)v )) = −1 ⇐⇒ v is inert in K/F, 2 - ordv (n(g)) ⇐⇒ v is inert in K/F, π(g)v = St(µ), µ unramified ⇐⇒ v is inert in K/F, ordv (n(g)) = 1, which proves both claims (above, η = ηK/F ).

g (the condition H(K, β) holds, as observed in the proof (iii) We use Proposition 12.6.3.14 to compute ε(β) of Corollary 12.6.4.8). Our assumptions imply that R(β)0 ∩ {v - p} = R(β)− ∩ {v | c(β), v - p} = ∅ R(1)− ∩ {v | pc(β)} = {v | p, v inert in K/F, ordv (n(g)) = 1}, hence   g (−1)[F :Q] = ηK/F n(g)(dK/F ) (−1)m(g) (−1)|{j=1,...s|Pj ε(β)   = ηK/F n(g)(P1 ···Ps ) (−1)m(g) .

inert in K/F, 2-ordPj (n(g))}|

=

Assume now that (P1 · · · Ps )n | c(β) and n >> 0. We use Corollary 12.5.9.3 to compute e h1f (K, V ⊗ β) − 1 2 hf (K, V ⊗ β). Let v | p be a prime of F satisfying π(g)v = St(µ) (µ = 1). Fix a prime w | v of K and put Nw = NKw /Fv . We must investigate the condition βw = µ ◦ Nw appearing in 12.5.9.3. 2 If v = Pi (i = 1, . . . , s) and n >> 0, then βw 6= 1, hence βw 6= µ ◦ Nw . ∗ ∗ ∗ Assume that v 6∈ {P1 , . . . , Ps }. By assumption, v - dK/F c(β), hence Nw (OK,w ) = OF,v and βw (OK,w ) = {1}; in particular, βw 6= µ ◦ Nw if µ is ramified. 2 If v splits in K/F and n >> 0, then βw (Fr(w)) 6= 1, hence βw 6= µ ◦ Nw . If v is inert in K/F (and n ≥ 0 is arbitrary), then βw = 1 and [βw = µ ◦ Nw ⇐⇒ µ is unramified

⇐⇒ ordv (n(g)) = 1] ,

by Proposition 12.6.1.6(i). Putting all the cases together, we deduce from Corollary 12.5.9.3 that (∀n >> 0)

e h1f (K, V ⊗ β) − h1f (K, V ⊗ β) = m(g). 367

Finally, Proposition 12.6.4.3 implies that   g (−1)eh1f (K,V ⊗β)−h1f (K,V ⊗β) = (−1)[F :Q] ηK/F n(g)(P1 ···Ps ) . ε(β) = ε(β)

(∀n >> 0)

(12.6.4.10) Trivial zeros in K[cP1∞ · · · Ps∞ ]. In the situation of Proposition 12.6.4.9, assume that K∞ ⊂ K[cP1∞ · · · Ps∞ ] is a subfield such that ∗

(∀n ≥ 1) ∃β : Γ = Gal(K∞ /K) −→ Lp of finite order with (P1 · · · Ps )n | c(β). ∼

Write Γ = Γ0 × ∆, Γ0 −→ Zrp (r ≥ 1), ∆ = Γtors (|∆| < ∞), and fix a character χ : ∆ −→ (O0 )∗ , where O0 is a discrete valuation ring, finite over O = OL,p . Set Λ = O[[Γ]] and, for each O[∆]-module M , M (χ) := M ⊗O[∆],χ O0 . Fix a GF -stable O-lattice T ⊂ V and put A = V /T , (∀v | p)

Tv+ = Vv+ ∩ T,

Tv− = T /Tv+ ,

± ± A± v = Vv /Tv .

e 1 (KS /K∞ , A) (cf. (9.6.5.1)): Consider the contribution of the trivial zeros to H f MM M Z= Zw , Zw = lim H 0 (Gwα , A− v ), −→ α

v|p w|v

wα |w

where Kα /K runs through the finite subextensions of K∞ /K, and wα | w | v are primes of Kα , K and F , respectively. (12.6.4.11) Proposition. In the situation of 12.6.4.10, we have, for each prime v | p of F , corkΛ(χ)

M

( (χ) Zw =

w|v

1,

v ∈ M (g, K; P1 , . . . , Ps )

0,

v 6∈ M (g, K; P1 , . . . , Ps ),

corkΛ(χ) Z (χ) = m(g, K; P1 , . . . , Ps ).

Proof. Fix a prime w | p of K and denote by v the induced prime of F . As Zw is a module of co-finite (χ) type over O[[Γ/Γw ]] (where Γw ⊂ Γ is the decomposition group of w in K∞ /K), we have corkΛ(χ) Zw = 0 whenever |Γw | = ∞. If v ∈ {P1 , . . . , Ps }, then |Γw | ≥ |I(Γw )| = ∞. Assume that v 6∈ {P1 , . . . , Ps }; then v is unramified in K/F . If v splits in K/F , then |Γw | = ∞. If v is inert in K/F , then Γw = 0, hence DΛ (Zw ) = DO (H 0 (Gw , A− v )) ⊗O Λ. Applying Proposition 12.5.8, we obtain, in this inert case, ( corkΛ(χ)

(χ) Zw

= corkO H

0

(Gw , A− v)

=

1,

ordv (n(g)) = 1

0,

otherwise.

Putting all cases together yields the claim. (12.6.4.12) Proposition. In the situation of 12.6.4.10, let K∞ = K[cP1∞ · · · Ps∞ ], Γ = Gal(K∞ /K) = G(cP1∞ · · · Ps∞ ). Enlarge S so that S ⊃ {v | c}; then KS ⊃ K∞ . Define δ ∈ {0, 1} by the formula   (−1)δ = (−1)[F :Q] ηK/F n(g)(P1 ···Ps ) . Assume that the following condition holds: ∗

(C(g, χ)) (∀n ≥ 1) ∃β : Γ −→ Lp of finite order with (P1 · · · Ps )n | c(β), β|∆ = χ−1 , h1f (K, V ⊗ β) = δ. Then: e j (K∞ /K, T )(χ±1 ) = cork (χ±1 ) H e j (KS /K∞ , A)(χ±1 ) = δ + m(g, K; P1 , . . . , Ps ). (i) (∀j = 1, 2) rkΛ(χ±1 ) H f,Iw f Λ ±1

str (ii) corkΛ(χ±1 ) SA (K∞ )(χ

)

= δ. 368

0 0 (iii) Let K ⊂ K∞ ⊂ K∞ be any intermediate field; set Γ0 = Gal(K∞ /K), Λ0 = O[[Γ0 ]]. If χ factors through 0 0 0 ∗ a character χ : Γtors −→ (O ) , then we have, for j = 1, 2, 0±1 0±1 0 0 e j (K∞ e j (KS /K∞ rkΛ0(χ0±1 ) H /K, T )(χ ) = corkΛ0(χ0±1 ) H , A)(χ ) ≥ δ + m(g, K; P1 , . . . , Ps ) f,Iw f

e j (KS /K 0 , A)(χ0±1 ) ≡ δ + m(g, K; P1 , . . . , Ps ) (mod 2). corkΛ0(χ0±1 ) H ∞ f Proof. (i), (ii) The equality of the various (co)ranks and their independence on the choices of S, Σ and T follow from Proposition 7.8.8, Theorem 10.7.17(ii) and Proposition 9.7.9 (the assumptions are satisfied, thanks to Proposition 12.4.8.4). Denote their common value by m and put m(g) := m(g, K; P1 , . . . , Ps ). The exact sequence (9.6.5.1) (in which Σ = {v | p}) together with Proposition 12.6.4.11 imply that ±1

str m(g) ≤ m = m(g) + corkΛ(χ±1 ) SA (K∞ )(χ

)

.

(12.6.4.12.1)

Fix n >> 0 and choose β as in the condition C(g, χ). Proposition 12.5.9.2 together with Proposition 12.6.4.9(iii) show that m(g) ≤ e h1f (K, V ⊗ β) = m(g) + h1f (K, V ⊗ β) = m(g) + δ ≤ m(g) + 1. Ker(β|

(12.6.4.12.2) ∗

)

Γ0 Γ0 On the other hand, set K0 = K∞ , K00 = K∞ , and denote by χ0 : Gal(K00 /K0 ) −→ Lp the character −1 0 induced by β |Γ0 (this has nothing to do with χ that appears in (iii)). After enlarging O0 if necessary, we can assume that Im(χ0 ) ⊂ O0 . The character β −1 coincides with the composite map

χ×χ0

∗

β −1 : Γ = ∆ ⊗ Γ0  ∆ ⊗ Gal(K00 /K0 )−−−−→Lp , hence 0

Hf1 (K00 , V )(χ×χ ) = Hf1 (K, V ⊗ (χ × χ0 )−1 ) = Hf1 (K, V ⊗ β) e 1 (K 0 , V )(χ×χ0 ) = H e 1 (K, V ⊗ (χ × χ0 )−1 ) = H e 1 (K, V ⊗ β), H f 0 f f by Proposition 8.8.7. Applying Theorem 10.7.17(iv), we deduce that m≤e h1f (K, V ⊗ β) = m(g) + δ,

m≡e h1f (K, V ⊗ β) ≡ m(g) + δ (mod 2),

which implies, thanks to (12.6.4.12.1-2), that m = m(g) + δ,

±1

str corkΛ(χ±1 ) SA (K∞ )(χ

)

= m − m(g) = δ.

(iii) The equality of the various (co)ranks follows as in the proof of (i); denote their common value by m0 . 0 Applying Theorem 10.7.17(iii) to both K∞ /K and K∞ /K, we see that m≡e h1f (K, V ⊗ χ−1 ) = e h1f (K, V ⊗ χ0−1 ) ≡ m0 (mod 2). The inequality m ≥ m0 is a consequence of Proposition 8.10.11(ii). (12.6.5) Hilbert modular forms with CM (12.6.5.1) Lemma. Let f ∈ Sk (n, ϕ) − {0} be a newform of level n and weight k ≥ 2. The following conditions are equivalent: ∼ (i) There exists a quadratic extension K 0 /F such that π(f ) −→ π(f ) ⊗ ηK 0 /F . ∼ (ii) There exists a quadratic extension K 0 /F such that V (f ) −→ V (f ) ⊗ ηK 0 /F (where ηK 0 /F : GF −→ {±1} is the quadratic character with kernel Ker(ηK 0 /F ) = GK 0 ). Proof. This follows from the compatibility (12.4.3.1) and the fact that, for each finite set S of primes of F , the representation π(f ) (resp., V (f )) is uniquely determined by the set of local representations {π(f )v | v 6∈ S} (resp., {V (f )v | v 6∈ S}). 369

(12.6.5.2) Proposition-Definition. If the canditions (i), (ii) of Lemma 12.6.5.1 are satisfied, then: (i) The field K := K 0 is totally imaginary (we say that f has CM by K). (ii) There exists a character λ : A∗K /K ∗ −→ C∗ satisfying ∗ (∀w | ∞ in K) (∀x ∈ Kw = C∗ )

(∀v prime of F ) (∀µ : Fv∗ −→ C∗ )

λw (x) = x1−k Y Lv (π(f )v ⊗ µ, s) = Lw (λw · (µ ◦ Nw ), s + (k − 1)/2). w|v

In other words, π(f ) ⊗ | · |(1−k)/2 is isomorphic to the automorphic induction IK/F (λ) and (∀v prime of F )

Lv (f, s) =

Y

Lw (λ, s).

w|v

(iii) n = dK/F NK/F (cond(λ)). (iv) If a non-archimedean prime v of F splits in K/F (vO ww0 ), then π(f )v = π(λw , λw0 ) ⊗ | · |(k−1)/2 .
K =k−1 (v) The central character of π(f ) is equal to ηK/F λ|A∗F | · | . (vi) If the central character of π(f ) is trivial and v is a non-archimedean prime of F which is unramified in K/F , then 2 | ordv (n). ∼

Proof. The statements (i) and (ii) are well-known: if V (f ) −→ V (f ) ⊗ ηK 0 /F , then we have (possibly after ∼ replacing Lp by a finite extension) V (f )|GK 0 −→ χ ⊕ χ ◦ ρ, for some character χ : GK 0 −→ L∗p (where Gal(K 0 /F ) = {1, ρ}). As V (f )|GK 0 is a semi-simple L-rational abelian representation of GK 0 , it follows from [He, Thm. 2] (see also [Se 3, III.2.3, Thm. 2]) that χ is the Galois representation attached to an algebraic ∼ F Hecke character ψ of K 0 . As V (f ) −→ IndG GK 0 (χ), we have L(f, s) = L(λ, s) (Euler factor by Euler factor), where λ : A∗K 0 /K 0∗ −→ C∗ is the id`ele class character attached to ψ and the fixed embedding ι∞ : Q ,→ C ([Sc], ch. 0, §6). Comparing the archimedean terms in the functional equations of L(f, s) and L(λ, s) (and 0 using the fact that P k − 1 6= 0), we deduce that K = K is totally imaginary and the infinity type of ψ is equal to (k − 1) σ∈Φ σ, for some CM type Φ of K. The formula (iii) follows from ([A-T], ch. 11, Thm. 18)
and (12.4.3.2), while (iv) is a consequence of (ii). The central character of IK/F (λ) is equal to ηK/F λ|A∗F , which proves (v). In order to prove (vi), we need to show that 2 | ordv (cond(λ)), by (iii). This divisibility is automatic if v is inert in K/F . If v splits in K/F , then π(f )v = π(µ, µ−1 ) (by (iv)), hence ordv (n) = o(π(f )v ) = 2 o(µ) is even. (12.7) Hida families of Hilbert modular forms In this section we apply the methods of 10.7.7-10 to big Galois representations attached to ordinary families of Hilbert modular forms. The parity results proved in 12.7.15 generalize, among others, Theorem A’ from [N-P]. (12.7.1) Fix an ideal n ⊂ OF prime to p. We shall be interested in ordinary eigenforms f ∈ Sk (npr , φ) for k ≥ 2, r ≥ 1. Such forms have central characters defined on Z = lim Inpr ∞ , ← r− where Ia∞ is the ray class group modulo a∞ (recall that ∞ denotes the sum of all archimedean primes of ∼ F ). The torsion subgroup Ztors is finite and Z/Ztors −→ Z1+δ p , where δ ≥ 0 is the default of the Leopoldt conjecture for F and p. In particular, δ = 0 if F is an abelian extension of Q. Let S0 be the set of all primes of F dividing np∞. Recall that we have normalized the isomorphism of class field theory by ∼

Z −→ Gab F,S0 q 7→ Fr(q)geom . With this normalization, the composite map 370

χ−1 cycl

∼

∗ Z −→ Gab F,S0  Gal(F (µp∞ )/F ) ,→ Gal(Q(µp∞ )/Q)−−→Zp

sends q to its norm N q ∈

Z∗p .

Putting ( q=

the torsion subgroup of

(12.7.1.1)

Z∗p

( (Z∗p )tors

=

p,

p 6= 2

4,

p = 2,

µp−1 (Zp ),

p 6= 2

{±1},

p=2

has a canonical complement 1 + qZp ⊂ Z∗p . The corresponding projections ω : Z∗p −→ (Z∗p )tors ,

κ : Z∗p −→ 1 + qZp

define a decomposition ∼

(ω, κ) : Z∗p −→ (Z∗p )tors × 1 + qZp . Let Q∞ /Q be the unique Zp -extension of Q, i.e. the fixed field of Gal(Q(µp∞ )/Q)tors ⊂ Gal(Q(µp∞ )/Q). Then the map ∼

κ ◦ χ−1 cycl : Gal(Q∞ /Q) −→ 1 + qZp is an isomorphism; denote by Γ ⊂ 1 + qZp the image of κ◦χ−1 cycl

Gal(F Q∞ /F ) ,→ Gal(Q∞ /Q)−−−−→1 + qZp (Γ is isomorphic to Zp ). (12.7.2) Let O be the ring of integers of a finite extension of Qp (e.g. of Lp ); put Λ = O[[Γ]]. The tautological representation χΓ : Γ ,→ Λ∗ defines a big Galois representation κ◦χ−1 cycl

χΓ : GF,S0  Gal(F Q∞ /F )−−−−→Γ ,→ Λ∗ , for which χΓ (Fr(q)geom ) = χΓ (κ(N q))

(q - np). e

Fix a topological generator γ ∈ Γ (one can take, for example, γ = (1 + q)p , where pe = (1 + qZp : Γ)). For each pair (k, ε), where k ∈ Z, k ≥ 2 and ε : Γ −→ µp∞ (O) ⊂ O∗ is a character of Γ of finite order, put Pk,ε = χΓ (γ) − ε(γ)γ k−2 ∈ Λ ∼

(where γ k−2 ∈ Γ ⊂ 1 + qZp ⊂ O∗ ). Then Λ/Pk,ε Λ −→ O and the representation χΓ (mod Pk,ε ) : GF,S0 −→ (Λ/Pk,ε Λ)∗ = O∗ sends Fr(q) = Fr(q)geom (for each q - np) to ε ◦ κ(N q) · κ(N q)k−2 , i.e. χΓ (mod Pk,ε ) = (ε ◦ κ ◦ N ) · (κ ◦ N )k−2 , ˇ by the Cebotarev density theorem. More generally, if O0 is the ring of integers of a finite extension of 0 Frac(O), k ≥ 2 and ε0 : Γ −→ (O0 )∗ a character of finite order, then 371

Pk0 ,ε0 Λ0 ∈ Spec(Λ0 )

(Λ0 = O0 [[Γ]])

induces a prime ideal Pk0 ,ε0 Λ0 ∩ Λ ∈ Spec(Λ) with residue field Frac(O)(ε0 (Γ)), and the representation χΓ (mod Pk0 ,ε0 Λ0 ∩ Λ) : GF,S0 −→ O(ε0 (Γ))∗ 0

is equal to (ε0 ◦ κ ◦ N ) · (κ ◦ N )k −2 . (12.7.3) For any level n0 ⊂ OF one defines Sk (n0 , Z) ⊂ Sk (n0 ) (k ≥ 2) to be the subgroup consisting of cusp forms f whose all Fourier coefficients c(a, f ) ∈ Z are rational integers. Then Sk (n0 , Z) ⊗Z C = Sk (n0 ) and one defines Sk (n0 , O) = Sk (n0 , Z) ⊗Z O. The Hecke algebra hk (n0 , O)

(k ≥ 2)

is the O-algebra generated by Tn0 (a) = T (a), Sn0 (b) (for all integral ideals a in OF and all integral ideals b prime to n0 ) acting on Sk (n0 , O). If np|n0 , then Up := Tnp (p) = Tn0 (p) and e = lim Upr! r−→∞

(Hida’s ordinary projector, e2 = e) act on Sk (n0 , O). The ordinary part of the Hecke algebra hk (n0 , O) is defined as 0 0 hord k (n , O) = e hk (n , O)

(k ≥ 2, np|n0 ).

For fixed k ≥ 2 and variable r ≥ 1, the Hecke algebras hk (npr , O) (as well as their ordinary parts) form a projective system under the transition maps T (a) 7→ T (a). According to ([Hi 2], Thm. 3.2), the projective limits lim hk (npr , O) ← r−

(k ≥ 2)

(= hk·t,(k−1)·t (np∞ ; O) in the notation of [Hi 2]) are canonically isomorphic to each other, i.e. do not depend on k ≥ 2. We denote by r hord (n; O) = e lim hk (npr , O) = lim hord k (np , O) ← ← r− r−

(k ≥ 2)

the ordinary part of this limit (= hord 0 (n; O) in the notation of [Hi 2]). The O-algebra hord (n; O) contains the images of all Hecke operators T (a) = Tnp (a) and S(b) = Snp (b) (for all a and b prime to np). The map b 7→ S(b) for k = 2 defines a homomorphism of O-algebras O[[Z]] −→ hord (n; O). Fix a splitting of the canonical projection Z −→ Z/Ztors =: W, 372

compatible via χ−1 cycl with the canonical splitting of Z∗p −→ Z∗p /(Z∗p )tors given by 1 + qZp . This induces an injection ∼

Λ(W ) := O[[W ]] ,→ O[[Z]] −→ O[Ztors ] ⊗O Λ(W ). The ring Λ(W ) is isomorphic to O[[T0 , . . . , Tδ ]] and hord (n; O) is a torsion-free Λ(W )-module of finite type, by ([Hi 2], Thm. 3.3). ∼ The fixed decomposition Z −→ Ztors ×W allows us to decompose any character φ : Z −→ O∗ as φ = φtame φW , where φtame (resp., φW ) denotes the restriction of φ to Ztors (resp., to W ). Fix also a splitting of the surjective homomorphism N : W  Γ (given by the norm map Fr(q) 7→ N q), i.e. ∼ a decomposition W −→ WnL × Γ, where WnL := Ker(N ) denotes the “non-Leopoldt” part of W (of course, WnL = 0 ⇐⇒ the Leopoldt Conjecture holds for (F, p)). For a fixed character of finite order φnL : WnL −→ µp∞ (O) ⊂ O∗ we define ord hord (n; O) ⊗O[[WnL ×Γ]],φnL ×id O[[Γ]] = hord (n; O) ⊗O[[WnL ]],φnL O; φnL (n; O) = h

this a finite Λ-algebra (Λ = O[[Γ]]). ord (12.7.4) An arithmetic point of hord φnL (n; O) is a prime ideal P ∈ Spec(hφnL (n; O)) whose restriction to 0 0 0 Λ = O[[Γ]] is equal to Pk0 ,ε0 Λ ∩ Λ ∈ Spec(Λ) for some k ≥ 2 and ε : Γ −→ (O0 )∗ as in 12.7.2. We define a φnL -arithmetic point of hord (n; O) to be the image in Spec(hord (n; O)) of an arithmetic point P ∈ Spec(hord φnL (n; O)) under the canonical map ord Spec(hord (n; O)). φnL (n; O)) ,→ Spec(h

For such P, the images of T (a) and S(b) in hord φnL (n; O)/P are equal to (the images under ιp of) the Hecke eigenvalues of an ordinary normalized Hecke eigenform fP ∈ Sk (npr(P) , φP ), where r(P) ≥ 1 and φP : Z −→ O0∗ is a character of finite order with “wild part” (φP )W = φnL · (ε ◦ κ ◦ N ) (see [Hi 2], Cor. 3.5, for v = 0). (12.7.5) Conversely, assume that f ∈ Sk (npr , φ)

(k ≥ 2)

is an ordinary normalized Hecke eigenform such that Frac(O) contains (the images under ιp of) all Hecke eigenvalues of f and all values of φ, and that the restriction of φ to WnL is equal to φnL . Assume, in addition, that f is a p-stabilized newform in the sense of ([Wi], p. 538). This means that r ≥ 1 and that the (ordinary) normalized newform f 0 attached to f has level n0 divisible by n. This implies that, for each prime ideal v|p, 1 − λf 0 (v)X + φ(v)(N v)k−1 X 2 = (1 − λf (v))(1 − φ(v)(N v)k−1 λf (v)−1 X) 1 − λf 0 (v)X = 1 − λf (v)X The “wild part” of φ is equal to φW = φnL · (ε ◦ κ ◦ N ) for some character of finite order 373

v - n0 v|n0 .

ε : Γ −→ µp∞ (O) ⊂ O∗ . By ([Hi 2], Cor. 3.5), there exists an arithmetic point P of hord φnL (n; O) above (Pk,ε ) ∈ Spec(Λ) such that T (v) (mod P) = λf 0 (v) = λf (v) 0

r for all prime ideals ideals v - p. This collection of Hecke eigenvalues occurs in hord k (np , O) only once, for 0 0 each r ≥ r. This follows from the fact that the level of f is divisible by n and the standard description of ordinary eigenforms in terms of newforms ([Wi], (1.2.2)). This implies that P is uniquelly determined by f and that

f = fP . One can say more: the multiplicity one statement alluded to above implies that Λ(Pk,ε ) ,→ hord φnL (n; O)P is an unramified extension of discrete valuation rings (we are grateful to A. Wiles for explaining this fact to us). In the case F = Q, this is proved in ([Hi 1], Cor 1.4). The general abstract principle is summarized in Lemma 12.7.6 below, which has to be applied to A = hord (n; O), B = O[[W ]], J = Ker(φnL × id : B = O[[W ]]  O[[Γ]] = Λ), B/J = Λ, A/JA = hord φnL (n; O), Q0 = (Pk,ε ). The assumption (a) (resp., (b)) holds by ([Hi 2], Thm. 3.3) (resp., by Hida’s “control theorem” ([Hi 2], Thm. 3.4) and the multiplicity one statement for f ). (12.7.6) Lemma. Let B ⊂ A be commutative noetherian rings, J ⊂ B an ideal, P0 ∈ Spec(A/JA). Let P ∈ Spec(A), Q = P ∩ B ∈ Spec(B) and Q0 ∈ Spec(B/J ) be induced by P0 . Assume that (a) As a B-module, A is of finite type and torsion-free. (b) Let M = (P/QA) ⊗B BQ /QBQ , N = (A/P ) ⊗B BQ /QBQ . In the canonical exact sequence of A/QA-modules 0 −→ M −→ (A/QA) ⊗B/Q BQ /QBQ −→ N −→ 0, N occurs in the middle term A ⊗B BQ /QBQ with multiplicity one in the sense that HomA/QA (M, N ) = 0. Then (i) QAP = P AP , Q0 (A/JA)P0 = P0 (A/JA)P0 . (ii) If (B/J )Q0 is a discrete valuation ring with prime element x (mod J) (x ∈ B), the same is true for (A/JA)P0 . Proof. (i) The assumption (a) implies that the map BQ −→ AP is injective. Both MP = P AP /QAP and NP = AP /P AP are finitely generated modules over the noetherian local ring S = AP /QAP , with NP = S/mS equal to the residue field of S. Localizing (b) at P , we obtain 0 = HomS (MP , NP ) = HomS/mS (MP /mS MP , S/mS ), hence MP /mS MP = 0. Nakayama’s Lemma implies that MP = 0, proving QAP = P AP . As (A/JA)P0 = AP ⊗B (B/J )Q0 = AP ⊗B (B/J ) = AP /J AP , we have Q0 (A/JA)P0 = QAP /J AP = P AP /J AP = P0 (A/JA)P0 , proving (i). The statement (ii) follows from (i). (12.7.7) Returning to the situation of 12.7.5, P contains a unique prime ideal Pmin ( P of hord φnL (n; O), necessarily minimal. Put R = hord (n; O)/P , P = P/P ∈ Spec(R). Then R is a local domain, finite min min φnL 374

over Λ, RP is a discrete valuation ring with prime element Pk,ε and Frac(R) = Frac(RP ) is a finite extension of Frac(Λ). The image of the canonical map Spec(R) −→ Spec(Λ) is closed (as R is a finitely generated Λ-module) and contains Spec(Λ(Pk,ε ) ), hence is equal to Spec(Λ). As a result, for each pair k 0 ≥ 2, ε0 : Γ −→ (O0 )∗ (with O0 as in 12.7.2), there is an arithmetic point P 0 ∈ Spec(hord φnL (n; O)) containing Pmin and lying above 0

Pk0 ,ε0 Λ0 ∩ Λ ∈ Spec(Λ); put P = P 0 /Pmin ∈ Spec(R). The homomorphism λ : hord (n; O)  hord φnL (n; O)/Pmin = R ,→ Frac(R) is minimal in the sense of ([Hi 2], p. 317), since f = fP is a p-stabilized newform. This implies, by ([Hi 2], Thm. 3.6, Cor. 3.5) that the form fP 0 attached to P 0 as in 12.7.4 is again a p-stabilized newform 0

fP 0 ∈ Sk0 (npr(P ) , φP 0 ), where (the images under ιp of) all Hecke eigenvalues of fP 0 are contained in O0 , (φP 0 )W = φnL ◦ (ε0 ◦ κ ◦ N ) and such that the character 0

k −2 φ0 := φtame = φtame (ω ◦ N )k−2 P 0 (ω ◦ N ) P

(12.7.7.1)

does not depend on P 0 . One can again apply Lemma 12.7.6 to show that ΛPk0 ,ε0 Λ0 ∩Λ ,→ hord φnL (n; O)


P0

= RP 0

is an unramified extension of discrete valuation rings. Without loss of generality, we can assume that the ring 0 0 0 O0 used in the definition of P coincides with the ring of integers of the residue field κ(P ) = RP 0 /P RP 0 = 0 κ(P 0 ) of P . (12.7.8) According to ([Wi], Thm. 2.2.1) there is a unique (up to equivalence) continuous Galois representation ρ : GF,S0 −→ GL2 (Frac(R)) satisfying det(1 − ρ(Fr(q)) X) = 1 − λ(T (q))X + (φ0 φnL )(q)χΓ (κ ◦ N (q))(N q)X 2 for all prime ideals q - np, where λ and φ0 were defined in 12.7.7 (more precisely, ρ is the dual of the representation constructed in [Wi], as we use Fr(q) = Fr(q)geom instead of Fr(q)arith ). This representation is (absolutely) irreducible (as its residual representation modulo P is (absolutely) irreducible; see 12.7.9 below) and is continuous in the sense that its representation space V (λ) is an admissible R[GF,S0 ]-module. According to ([Wi], Thm. 2.2.2), for each prime ideal v|p in OF there is an exact sequence of Frac(R)[Gv ]-modules − 0 −→ V (λ)+ v −→ V (λ) −→ V (λ)v −→ 0, + + such that each V (λ)± v is one-dimensional over Frac(R), Iv acts trivially on V (λ)v and Fr(v) acts on V (λ)v by λ(T (v)). These properties determine the subspace V (λ)+ uniquelly. v

(12.7.9) For each P 0 as in 12.7.7, RP 0 is a discrete valuation ring with fraction field Frac(R), which implies 0 that V (λ) contains a GF,S0 -invariant (RP 0 )-lattice T (P 0 ) ⊂ V (λ). Its reduction modulo P 0

T (P 0 )/P T (P 0 ) 0

is a two-dimensional representation of GF,S0 over the residue field κ(P ) = κ(P 0 ) satisfying 375

0

0

det(1 − ρ(Fr(q)) X | T (P 0 )/P T (P 0 )) = 1 − λfP0 (q)X + (φ0 φnL )(q)(ε0 ◦ κ ◦ N )(q)(κ ◦ N )k −2 (q)(N q)X 2 = = 1 − λfP0 (q)X + φP 0 (q)(N q)k−1 X 2 = det(1 − Fr(q) X | V (fP 0 )) ˇ for all prime ideals q - np. Absolute irreducibility of V (fP 0 ) and the Cebotarev density theorem imply that there is an isomorphism of κ(P 0 )[GF,S0 ]-modules 0

∼

T (P 0 )/P T (P 0 ) −→ V (fP 0 ), 0

(unique up to an element of κ(P 0 )∗ ) and that the only GF,S0 -invariant (RP 0 )-lattices in V (λ) are (P )i T (P 0 ) (i ∈ Z). For each prime ideal v|p in OF , put 0 + T (P 0 )+ v = T (P ) ∩ V (λ)v ,

0 0 + T (P 0 )− v = T (P )/T (P )v .

0 Both T (P 0 )± v are RP 0 [Gv ]-modules, free of rank one over RP 0 , and the exact sequence of κ(P )[Gv ]-modules 0

0

0

0 + 0 0 0 − 0 − 0 −→ T (P 0 )+ v /P T (P )v −→ T (P )/P T (P ) −→ T (P )v /P T (P )v −→ 0

is isomorphic to − 0 −→ V (fP 0 )+ v −→ V (fP 0 ) −→ V (fP 0 )v −→ 0

(by the uniqueness of V (fP 0 )+ v ). Strictly speaking, the notation V (fP 0 ) is ambiguous, as it does not specify the field of coefficients of the representation. Above, we have implicitly used κ(P 0 ) as the field of coefficients. Note that the determinant of the representation ρ is equal to Λ2 V (λ) = Frac(R)(−1) ⊗ χΓ φ0 φnL . We would like to define a suitable twist V = “V (λ)(1) ⊗ (χΓ φ0 φnL )−1/2 ” which would be self-dual in the sense that ∼

Λ2 V −→ Frac(R)(1), i.e. V would be isomorphic to ∼

V −→ V ∗ (1) = HomFrac(R) (V , Frac(R))(1) (cf. [N-P], 3.2.3). This is indeed possible (after a quadratic base change if p = 2) under the following assumption: (12.7.9.1) For f = fP from 12.7.5, k ∈ 2Z and there exists a (continuous) character χ : A∗F /F ∗ −→ C∗ such that φ := φP = χ−2 ; fix such χ and a finite set S ⊃ S0 of primes of F such that χ is unramified outside S. ∼ ∼ (12.7.10) Assume that (12.7.9.1) is satisfied and p 6= 2. As the groups W −→ Z1+δ and Γ −→ Zp are p ∗ ∗ ∗ ∗ uniquely 2-divisible, the representations φnL : WnL −→ O , φW : W −→ O , ε : Γ −→ O , κ : Zp −→ 1+qZp and χΓ : GF,S  Γ ,→ Λ∗ have canonical square roots 1/2

1

φnL

2 ∗ φnL : WnL −−→W nL −−→O ,

κ

1

1/2

1 2

φW

2 φW : W −−→W −−→O∗ ,

κ1/2 : Z∗p −−→1 + qZp −−→1 + qZp , satisfying 376

1/2

χΓ

1

ε

2 ε1/2 : Γ−−→Γ −−→O∗ , 1 2

: GF,S  Γ−−→Γ ,→ Λ∗

1/2

1/2

1/2

φW = φnL · (ε1/2 ◦ κ ◦ N ),

χΓ

∼

(mod Pk,ε ) −→ (ε1/2 ◦ κ ◦ N ) · (κ1/2 ◦ N )k−2 = (ε1/2 ◦ κ ◦ N ) · (κ ◦ N )k/2−1 .

Put (φ

1/2 −1

)

W ∗ χW : A∗F /F ∗  Z  W −−−−→O ,

χtame := χχ−1 W

2−k (=⇒ χ−2 ) tame = φtame = φ0 (ω ◦ N )

and define 1/2 1/2

1/2

V = V (λ)(1) ⊗ (χΓ φnL )−1 χtame (ω ◦ N )1−k/2 = V (λ)(1) ⊗ (χΓ )−1 (ε1/2 ◦ κ ◦ N )(ω ◦ N )1−k/2 χ. This is again a two-dimensional representation of GF,S over Frac(R) (admissible as an R[GF,S ]-module) satisfying −1 −1 Λ2 V = Λ2 V (λ)(2) ⊗ χ−1 Γ φnL φ0 = Frac(R)(1).

For each prime ideal v|p in OF , there is an exact sequence of Frac(R)[Gv ]-modules 0 −→ Vv+ −→ V −→ Vv− −→ 0, where 1/2 1/2

−1 Vv± = V (λ)± χtame (ω ◦ N )1−k/2 . v (1) ⊗ (χΓ φnL )

If we fix an isomorphism of Frac(R)[GF,S ]-modules ∼

V −→ V ∗ (1) = HomFrac(R) (V , Frac(R))(1) (it is unique up to a scalar in Frac(R)∗ ), it induces isomorphisms of Frac(R)[Gv ]-modules ∼

Vv± −→ (Vv∓ )∗ (1), by the uniqueness of V (λ)+ v. i

The only GF,S -invariant (RP )-lattices in V are P T (P) (i ∈ Z), where 1/2 1/2

T (P) = T (P)(1) ⊗ (χΓ φnL )−1 χtame (ω ◦ N )1−k/2 . The residual representation at P 1/2 1/2

T (P)/PT (P) = (T (P)/PT (P))(1) ⊗ (χΓ φnL )−1 χtame (ω ◦ N )1−k/2 is isomorphic to V (fP )(1) ⊗ N 1−k/2 χ = V (fP )(k/2) ⊗ χ = V (gP )(k/2), where the twisted cusp form gP := fP ⊗ χ ∈ Sk (n(gP ), 1) has trivial central character. For each v|p, the κ(P)[Gv ]-module ± T (P)± v /PT (P)v

is isomorphic to 377

± V (gP )(k/2)± v := V (fP )v (k/2) ⊗ χv . 0

Let P 0 be as in 12.7.7, with k 0 ∈ 2Z. The only GF,S -invariant (RP 0 )-lattices in V are (P )i T (P 0 ) (i ∈ Z), where 1/2 1/2

T (P 0 ) = T (P 0 )(1) ⊗ (χΓ φnL )−1 χtame (ω ◦ N )1−k/2 . Let 1

ε0

2 (ε0 )1/2 : Γ−−→Γ −−→(O0 )∗

0

be the canonical square root of ε0 . The residual representation of V at P is isomorphic to 0

∼

 0 ε1/2 ◦ κ ◦ N (ω ◦ N )1−k/2 χ(κ ◦ N )1−k /2 = 0 1/2 (ε ) 0 = V (fP 0 )(k /2) ⊗ χ0 = V (gP 0 )(k 0 /2),

T (P 0 )/P T (P 0 ) −→ V (fP 0 )(1) ⊗



where 0

(k0 −k)/2



χ = χ · (ω ◦ N )

ε1/2 ◦κ◦N (ε0 )1/2



: A∗F /F ∗ −→ C∗

is a character satisfying (χ0 )−2 = φ(ω ◦ N )k−k

0

ε ε0

 0 ◦ κ ◦ N = φnL φtame (ω ◦ N )k−k (ε0 ◦ κ ◦ N ) = φnL φ0tame (ε0 ◦ κ ◦ N ) = φP 0 =: φ0

and the twisted cusp form gP 0 := fP 0 ⊗ χ0 ∈ Sk0 (n(gP 0 ), 1) has trivial central character. Similarly, for each v|p, the κ(P 0 )[Gv ]-module 0

0 ± T (P 0 )± v /P T (P )v

is isomorphic to ± 0 0 V (gP 0 )(k 0 /2)± v := V (fP 0 )v (k /2) ⊗ χv .

For each archimedean prime v|∞ of F , we have 0

(χ0v /χv )(−1) = (−1)(k −k)/2 . If the character ε0 has a sufficiently large order, then (∀v | p)

ordv (cond(χ0 )) >> 0,

hence (∀v | p)

(χ0v )2 (Iv ) 6= 1,

which implies that (n(gP 0 )St , (p)) = (1). ∼

(12.7.11) Assume that (12.7.9.1) is satisfied and p = 2, hence Γ −→ Z2 . Multiplication by 2 378

2

Γ−→Γ induces an injective homomorphism of O-algebras i : Λ = O[[Γ]] −→ Λ, for which Λ = i(Λ) + χΓ (γ)i(Λ). We replace the assumption (12.7.9.1) by (12.7.11.1) For f = fP from 12.7.5, k ∈ 2Z and there exist (continuous) characters χ : A∗F /F ∗ −→ C∗ , εe : Γ −→ O∗ such that φ := φP = χ−2 and ε = εe2 ; fix such χ and εe, as well as a finite set S ⊃ S0 of primes of F such that χ is unramified outside S. 1/2 e so that “χ1/2 ”(γ) ∈ R. e We In order to define a square root “χΓ ”, we must extend the scalars R ⊂ R Γ distinguish two cases. e = R[r] = R + Rr ⊂ Frac(R). (A) There exists r ∈ Frac(R) satisfying r2 = χΓ (γ). We define R 2 e to be the image of R[X] in the field (B) There is no r ∈ Frac(R) satisfying r = χΓ (γ). We define R 2 e e = R + Rr. Frac(R)[X]/(X − χΓ (γ)) and r ∈ R to be the image of X; then R e is a local domain, finite over Λ, with Frac(R)/Frac(R) e In both cases R an extension of degree one (resp., e two) in the case (A) (resp., (B)). As Pk,ε factorizes in R as Pk,ε = (r − εe(γ)γ k/2−1 )(r + εe(γ)γ k/2−1 ), e above P ∈ Spec(R) such that there exists a unique prime ideal Pe ∈ Spec(R) e r − εe(γ)γ k/2−1 ∈ P. e of the tautological representation χΓ : GF,S  Γ ,→ Λ∗ has a square root (depending The base change to R e of χΓ (γ)) on the choice of the square root r ∈ R 1/2

χΓ

1/2

χΓ e∗ : GF,S  Γ−−→ R

given by 1/2

χΓ (γ x ) = rx

(x ∈ Z2 ).

Its reduction modulo Pe is isomorphic to 1/2

χΓ

∼ e −→ (mod P) (e ε ◦ κ ◦ N ) · (κ ◦ N )k/2−1 .

As in the case p 6= 2, we define   e (1) ⊗ (χ1/2 )−1 (e V = V (λ) ⊗Frac(R) Frac(R) ε ◦ κ ◦ N )(ω ◦ N )1−k/2 χ. Γ e (admissible as an R[G e F,S ]-module) satisfying This is a two-dimensional representation of GF,S over Frac(R) e Λ2 V = Frac(R)(1) e v ]-modules and sitting in exact sequences of Frac(R)[G 0 −→ Vv+ −→ V −→ Vv− −→ 0 defined analogously as in the case p 6= 2. Put 379

(v|p),

  e := T (P) ⊗R R e (1) ⊗ (χ1/2 )−1 (e T (P) ε ◦ κ ◦ N )(ω ◦ N )1−k/2 χ; Γ e P P e )-lattice in V , with T (P)/ e PT e (P) e a two-dimensional representation of GF,S over this is a GF,S -invariant (R e P e =R e /P ee κ(P) e RPe isomorphic to P V (fP )(1) ⊗ N 1−k/2 χ = V (fP )(k/2) ⊗ χ = V (gPe)(k/2), where the twisted cusp form gPe = fP ⊗ χ ∈ Sk (n(gPe), 1) has trivial central character. More generally, let P 0 be as in 12.7.7, and assume that the following condition is satisfied: (12.7.11.2) k 0 ∈ 2Z and there exists a character εe0 : Γ −→ (O0 )∗ such that ε0 = (e ε0 )2 ; fix such εe0 . e0 = P ea0 ∈ Spec(R) e above P 0 ∈ Spec(R) such that For each a ∈ Z/2Z there exists a unique prime ideal P 0 e0 . r − (−1)a εe0 (γ)γ k /2−1 ∈ P

1/2

The reduction of χΓ

e 0 is isomorphic to modulo P 1/2

χΓ

∼

e 0 ) −→ (e (mod P ε0 ◦ κ ◦ N ) · (κ ◦ N )k /2−1 χaF1 /F , 0

where ∼

χF1 /F : GF,S  Γ  Γ/2Γ −→ {±1} is the quadratic character attached to the first non-trivial layer F ⊂ F1 ⊂ F2 ⊂ · · · ⊂ F∞ = F Q∞ √ √ in the cyclotomic Z2 -extension of F (for example, if 2 6∈ F , then F1 = F ( 2)). As before,   e 0 ) := T (P 0 ) ⊗R 0 R e 0 (1) ⊗ (χ1/2 )−1 (e T (P ε ◦ κ ◦ N ) · (ω ◦ N )1−k/2 χ Γ e P P e 0 )-lattice in V , with T (P e 0 )/P e 0 T (P e 0 ) isomorphic to the two-dimensional representation is a GF,S -invariant (R e P e 0 )) (over κ(P

V

(fP0 )(1)

 ⊗

εe ◦κ◦N εe0



0

0 (κ ◦ N )1−k /2 (ω ◦ N )1−k/2 χaF1 /F χ = V (fP0 )(k 0 /2) ⊗ χ0 = V (gP )(k 0 /2),

where χ0 = χ0a =



εe ◦κ◦N εe0



0

(ω ◦ N )(k −k)/2 χaF1 /F χ : A∗F /F ∗ −→ C∗

is a character satisfying (χ0 )−2 = φP 0 =: φ0 and the twisted cusp form gPe0 := fP 0 ⊗ χ0 ∈ Sk0 (n(gPe0 ), 1) 380

has trivial central character. The rest of the discussion from 12.7.10 has obvious analogues in the present context; in particular, 0

(χ0v /χv )(−1) = (−1)(k −k)/2 .

(∀v|∞)

(12.7.12) We are almost ready to apply the results of 10.7.7-10 to Selmer complexes arising in the present situation under the assumption (12.7.9.1) (resp., (12.7.11.1-2)) if p 6= 2 (resp., if p = 2). In order to simplify the notation, we shall consider in Sect. 12.7.12-15 only the case p 6= 2. For p = 2, one has to replace (R, P, P 0 ) e P, e Pe0 ). Even though the condition (P ) from 5.1 is not satisfied, we shall consider in 12.7.13,15 only by (R, cohomology with coefficients in Q2 -vector spaces, which means that the contribution from the real places vanishes (see the discussion in 6.9). Fix a prime element $P (resp., $P 0 ) of RP (resp., of RP 0 ) and denote the fraction field of R by L = Frac(R). Fix an isomorphism of L [GF,S ]-modules ∼

Λ2 V −→ L (1)

(12.7.12.1)

∗

(it is unique up to an element of L ); under the induced isomorphism ∼

V −→ V ∗ (1) = HomL (V , L )(1), the GF,S -invariant lattices T (P) ⊂ V and T (P)∗ (1) = HomRP (T (P), RP )(1) ⊂ V ∗ (1) i differ by $P , for some i ∈ Z. Multiplying (12.7.12.1) by a suitable scalar in L ∗ , we obtain a skew-symmetric bilinear form

π : T (P) ⊗RP T (P) −→ RP (1)

(12.7.12.2)

such that ∼

adj(π) : T (P) −→ T (P)∗ (1) is an isomorphism. The form π being skew-symmetric (hence alternating, since L has characteristic zero), the map adj(π) induces, for each v|p, a morphism of exact sequences 0

−→

0

−→

T (P)+ v  α y

−→

∗ (T (P)− v ) (1) −→

T (P)  o y

T (P)− v  β y

−→

T (P)∗ (1) −→

−→

0

∗ (T (P)+ v ) (1) −→

0.

T (P)± v

As Ker(α) = Coker(β) = 0 and each is free of rank one over RP , both Ker(β) = Coker(α) ⊂ T (P)− v must vanish, hence adj(π) induces isomorphisms (∀v|p)

∼

∓ ∗ T (P)± v −→ (T (P)v ) (1).

In other words, if we replace RP on the R.H.S. of (12.7.12.2) by its canonical injective resolution [L −→ L /RP ], then (∀v|p)

+ T (P)+ v ⊥⊥π T (P)v .

(12.7.13) Selmer complexes in Hida families (12.7.13.1) As in 12.5.9.1, fix a subset Σ ⊂ Sf containing all primes dividing p and set Σ0 = Sf − Σ. In order to simplify the notation, we denote 381

V = T (P)/PT (P) = V (gP )(k/2),

0

V 0 = T (P 0 )/P T (P 0 ) = V (gP 0 )(k 0 /2).

Each of the R[GF,S ]-modules X = T (P), T (P 0 ), V, V 0 , V is admissible. More precisely, X is an RX [GF,S ]module, where RX = RP , RP 0 , κ(P), κ(P 0 ), L , respectively. We have morphisms of RX [GF,S ]-modules πX : X ⊗RX X −→ RX (1), 0

which are equal, respectively, to: π, π 0 , the reduction of π modulo P, the reduction of π 0 modulo P and the pairing induced by (12.7.12.1) (= π ⊗ 1). Define, for each X, Greenberg’s local conditions by the following formulas (∀v ∈ Σ): ( + ( − Xv , v | p, Xv , v | p, + − Xv = Xv = (12.7.13.1.1) 0, v - p, X, v - p. They satisfy, for each v ∈ Σ, ± ± T (P)± v /PT (P)v = Vv ,

0

0 ± 0 ± T (P 0 )± v /P T (P )v = (V )v ,

0 ± ± T (P)± v ⊗RP L = T (P )v ⊗RP 0 L = Vv (12.7.13.1.2)

and (

Xv+ ⊥πX Xv+

v-p

Xv+ ⊥⊥πX Xv+

v | p.

(12.7.13.1.3)

(12.7.13.2) Let E/F be a finite subextension of FS /F and β : GE,S = Gal(FS /E) −→ O∗ a (continuous) character of finite order. For each of the R[GF,S ]-modules X = T (P), T (P 0 ), V, V 0 , V we define Greenberg’s local conditions ∆Σ (X ⊗ β) for the admissible R[GE,S ]-module X ⊗ β as in 12.5.9.1: if w is a prime of E above v ∈ Sf , set  • Ccont (Gw , Xv+ ⊗ βw ), v|p    Uw+ (X ⊗ β) = 0, v ∈ Σ, v - p    • Ccont (Gw /Iw , (X ⊗ βw )Iw ), v ∈ Σ0 (above, Gw = Gal(F v /Ew )). For each X, the map πX induces a morphism of RX [GE,S ]-modules πX,β : (X ⊗ β) ⊗RX (X ⊗ β −1 ) −→ RX (1) satisfying, for each v ∈ Σ, (

−1 + (X ⊗ β)+ )v v ⊥πX,β (X ⊗ β

v-p

−1 + (X ⊗ β)+ )v v ⊥⊥πX,β (X ⊗ β

v | p.

(12.7.13.3) Proposition. Under the assumptions of 12.7.13.1-2, let X ∈ {T (P), T (P 0), V, V 0 , V }. • • (i) Let w - p∞ be a prime of E; then the complexes Ccont (Gw , X ⊗ βw ) and Ccont (Gw /Iw , (X ⊗ βw )Iw ) are acyclic. gf (GE,S , X ⊗ β; ∆Σ (X ⊗ β)) ∈ Db (R Mod) (ii) Up to a canonical isomorphism, the Selmer complex RΓ ft X gf (E, X ⊗ β) and its cohomology groups by does not depend on the choice of S and Σ; we denote it by RΓ e i (E, X ⊗ β). H f (iii) If X ∈ {V, V 0 }, then there is an exact sequence MM e 1 (E, X ⊗ β) −→ H 1 (E, X ⊗ β) −→ 0. 0 −→ H 0 (Ew , Xv− ⊗ βw ) −→ H f f v|p w|v

382

Proof. (i) For X = V, V 0 , the statement was proved in Proposition 12.5.9.2(i). As in the proof of loc. cit., it is enough to show that RΓcont (Gw , X ⊗ βw ) is acyclic, for X = T (P), T (P 0), V . This follows from Lemma 12.7.15.6 below applied to the exact triangle $

P RΓcont (Gw , T (P) ⊗ βw )−−→RΓ cont (Gw , T (P) ⊗ βw ) −→ RΓcont (Gw , V )

and its analogue for T (P 0 ), together with the isomorphism ∼

RΓcont (Gw , T (P) ⊗ βw ) ⊗RP L −→ RΓcont (Gw , V ⊗ βw ). (ii) The proof of 12.5.9.2(ii) applies. (iii) This was proved in 12.5.9.2(iii). (12.7.13.4) Proposition. Under the assumptions of 12.7.13.1-2, (i) There is an exact triangle in Dfb t (RP Mod) $P g gf (E, T (P) ⊗ β)−−→ gf (E, V ⊗ β) RΓ RΓf (E, T (P) ⊗ β) −→ RΓ

inducing short exact sequences e j (E, T (P) ⊗ β)/$P H e j (E, T (P) ⊗ β) −→ H e j (E, V ⊗ β) −→ H e j+1 (E, T (P) ⊗ β)[$P ] −→ 0 0 −→ H f f f f (and similarly for T (P 0 )). (ii) There are canonical isomorphisms in Dfb t (L Mod) ∼ g ∼ g 0 gf (E, T (P) ⊗ β) ⊗R L −→ RΓ RΓf (E, V ⊗ β) ←− RΓ f (E, T (P ) ⊗ β) ⊗RP 0 L . P

e 1 (E, T (P) ⊗ β) (resp., H e 1 (E, T (P 0 ) ⊗ β)) is a free (R )-module (resp., a free (R 0 )-module) of rank (iii) H f f P P e e f1 (E, V ⊗ β). h1f (E, V ⊗ β) := dimL H (iv) The duality morphism in Dfb t (RP Mod) gf (E, T (P) ⊗ β) −→ RHomR (RΓ gf (E, T (P) ⊗ β −1 ), R )[−3] adj(πT (P),β ) : RΓ P P is an isomorphism, which induces a non-degenerate (RP )-bilinear form e 2 (E, T (P) ⊗ β)tors × H e 2 (E, T (P) ⊗ β −1 )tors −→ Frac(R )/R = L /R , h, i:H f f P P P where the subscript “tors” refers to the (RP )-torsion. If β = β −1 , then this form is skew-symmetric. Similar statements hold for P 0 . (v) Fix a GF -stable R-lattice T ⊂ V (i.e., an R-submodule of finite type satisfying L T = V ). Then, for each prime w - p∞ of E, 0 Tamw (T ⊗ β ±1 , P) = Tamw (T ⊗ β ±1 , P ) = 0. gf (E, X ⊗ β) on Σ proved in Proposition 12.7.13.3(ii). Proof. We are going to use the independence of RΓ Throughout the proof of (i)-(iv) we let Σ = Sf . (i) The statement follows from the exact sequences $

P • • • 0 −→ Ccont (G, T (P) ⊗ β)−−→C cont (G, T (P) ⊗ β) −→ Ccont (G, V ⊗ β) −→ 0

(for G = GE,S , Gw ) and $

P • • • + + 0 −→ Ccont (Gw , T (P)+ v ⊗ β)−−→Ccont (Gw , T (P)v ⊗ β) −→ Ccont (Gw , Vv ⊗ β) −→ 0

(cf. (12.7.13.1.2)), which are compatible with respect to the ‘restriction’ maps resw . 383

(ii) This follows from the functorial isomorphisms of complexes (cf. Proposition 3.4.4) ∼

• • Ccont (G, M ) ⊗RP L −→ Ccont (G, M ⊗RP L )

for (G, M ) = (GE,S , T (P) ⊗ β), (Gw , T (P)+ v ⊗ β). (iii) Taking into account (i) and (ii), it is sufficient to show that e f1 (E, T (P) ⊗ β)[$P ] = 0, H which follows from (i) and the vanishing of e f0 (E, V ⊗ β) ⊆ H 0 (GE,S , V ⊗ β) = 0, H proved in Proposition 12.4.8.4 (and similarly for P 0 ). (iv) We apply the localized version of the duality Theorem 6.3.4. For w | v | p (resp., for w | v, v ∈ Sf , v - p) the error term Errw (T (P) ⊗ β, T (P) ⊗ β −1, πT (P),β ) vanishes in Dfb t (RP Mod) by Proposition 6.7.6(ii) (resp., by Proposition 6.7.6(iv) combined with Proposition 12.7.13.3(i)). This implies that the map adj(πT (P),β ) is, e 2 (−) indeed, an isomorphism. The induced generalized Cassels-Tate pairing on the torsion submodules of H f is non-degenerate, by Theorem 10.4.4, and skew-symmetric (in the case β = β −1 ), by Proposition 10.2.5 (as the pairing πT (P),β is skew-symmetric). (v) Let v be the prime of F below w. By 7.6.10.4, it is enough to treat the case v ∈ Sf . Consider the duality morphisms ± g gf (GE,S , T (P) ⊗ β ∓1 ; ∆Σ )), R )[−3] γΣ : RΓf (GE,S , T (P) ⊗ β ±1 ; ∆Σ ) −→ RHomRP (RΓ P

for arbitrary {v | p} ⊂ Σ ⊂ Sf . As observed in the proof of (iv), the map γS±f is an isomorphism. On the   ± other hand, Cone γ{v|p} is isomorphic to M M v∈Sf v-p

±1 Errur , T (P) ⊗ β ∓1 , πT (P),β ±1 ) = w (T (P) ⊗ β

MM v∈Sf

w|v

v-p

Errw .

w|v

gf (GE,S , T (P) ⊗ β ±1 ; ∆Σ ) on Σ, proved in Proposition 12.7.13.3(ii), implies that The independence of RΓ each term Errw is acyclic, hence Tamw (T ⊗ β ±1 , P) = `RP (H 1 (Errw )) = 0 0

(by 7.6.10.7). The same proof applies to P . (12.7.13.5) Corollary. In the situation of 12.7.13.1-2, assume that β = β −1 . Then: (i) There exist canonical decreasing filtrations e f1 (E, V ⊗ β) = F 1 ⊇ F 2 ⊇ · · · , H

e f1 (E, V 0 ⊗ β) = 0 F 1 ⊇ 0 F 2 ⊇ · · · H

by κ(P)-subspaces (resp., κ(P 0 )-subspaces) such that F ∞ :=

\

  e f1 (E, T (P) ⊗ β)/$P ,→ H e f1 (E, V ⊗ β) F j = Im H

j≥1 0

F

∞

:=

\

0

  e 1 (E, T (P 0 ) ⊗ β)/$P 0 ,→ H e 1 (E, V 0 ⊗ β) F j = Im H f f

j≥1

and

dimκ(P) F ∞ = dimκ(P 0 ) 0 F ∞ = e h1f (E, V ⊗ β). 384

(ii) For each j ≥ 1 there exist symplectic (= alternating and non-degenerate) pairings h , ij,π,$P : grFj × grFj −→ κ(P), h , ij,π0 ,$P 0 : gr0jF × gr0jF −→ κ(P 0 ) on grFj = F j /F j+1 and gr0jF = 0 F j /0 F j+1 . (iii) For each j ≥ 1 we have dimκ(P) F j ≡ dimκ(P 0 ) 0 F j ≡ e h1f (E, V ⊗ β) (mod 2). In particular, e h1f (E, V ⊗ β) = dimκ(P) F 1 ≡ e h1f (E, V ⊗ β) ≡ dimκ(P 0 ) 0 F 1 = e h1f (E, V 0 ⊗ β) (mod 2). Proof. Apply Lemma 10.6.5 (with ι = id and ε = 1) to the symplectic forms from Proposition 12.7.13.4(iv) (as the fields κ(P), κ(P 0 ) are of characteristic zero, there is no difference between “skew-symmetric” and “alternating”). e 1 (E, V ⊗ β) can be interpreted as the “generic subspace” of F 1 , (12.7.13.6) The subspace F ∞ ⊆ F 1 = H f corresponding to those cohomology classes that can be lifted to (the twist by β of) the whole Hida family λ. e 1 (E, V ⊗ β) The last statement of Corollary 12.7.13.5(iii) then says that the parity of the dimensions of H f is constant in the whole Hida family, and coincides with the parity of the rank of the common “generic subspace”, generalizing Theorem A’ from [N-P]. The dependence of the pairings from 12.7.13.5(ii) on $P (resp., on $P 0 ) is given by the formula from Proposition 10.7.9(ii). For example, if κ(P 0 ) = Frac(O) (e.g. if P 0 = P), then we can take $P 0 = Pk0 ,ε0 . This choice is not completely canonical, as it depends on a fixed topological generator γ ∈ Γ. If we replace γ by γ new = γ c (c ∈ Z∗p ), then 0

new 0 $P χΓ (γ)c − (ε0 (γ)γ k −2 )c 0 = ≡ c (ε0 (γ)γ k −2 )c−1 (mod $P 0 ), 0 −2 0 k $P 0 χΓ (γ) − ε (γ)γ

hence h , ij,π0 ,$new = u2−j h , ij,π0 ,$P 0 , 0 P

0

u = c (ε0 (γ)γ k −2 )c−1 ∈ Z∗p . ∼

There is another ambiguity in the construction of h , ij,π0 ,$P 0 , namely the choice of an isomorphism V −→ V ∗ (1) compatible with the chosen GF,S -invariant lattice T (P 0 ) ⊂ V . If the degree [F : Q] is odd, then this ambiguity can be avoided as follows: the representation V (λ) can be constructed geometrically, using the Tate modules of Shimura curves attached to a quaternion algebra over F which is ramified at all but one archimedean primes of F (and no finite primes). Such a geometric construction yields both a canonical GF,S -invariant R-lattice T ⊂ V (λ) and a canonical isomorphism Λ2 V (λ) = Frac(R)(−1) ⊗ χΓ φ0 φnL , well-behaved with respect to T . In [N-P], this geometric construction was summarized in the classical case F = Q (we would like to use this opportunity to remark that some of the results from ([N-P], Sect. 1.6) had earlier been obtained by Ohta [Oh 3]). As we now work consistently with (RP )-modules, the auxiliary assumptions used in [N-P], such as p > 3 and irreducibility of certain residual representations, are no longer necessary. (12.7.13.7) If E is an elliptic curve P over Q with good ordinary reduction at p, then L(E/Q, s) = L(f, s) for a p-ordinary eigenform f = n≥1 an q n ∈ S2 (Γ0 (N ), 1) with integral coefficients. Applying Corollary 12.7.13.5 to the p-stabilization of f (over Q, and to β = 1), we obtain the result stated in 0.15.2. 385

(12.7.13.8) Proposition (Dihedral case). In the situation of 12.7.13.1-2, assume that [E : F ] = 2 and β ◦ τ = β −1 , where τ is the non-trivial element of Gal(E/F ). Set Eβ = E Gal(Eβ /F ) (which will also be denoted by τ ). Then: (i) Adf (τ ) induces an isomorphism e f2 (E, T (P) ⊗ β) = H e f2 (Eβ , T (P))(β H

−1

)

(β)

∼

e f2 (Eβ , T (P)) −→ H

Ker(β)

and fix a lift of τ to

e f2 (E, T (P) ⊗ β −1 ) =H

such that the induced (RP )-bilinear form [x, y] = hx, Adf (τ )yi e 2 (E, T (P) ⊗ β)tors × H e 2 (E, T (P) ⊗ β)tors −→ Frac(R )/R = L /R [, ]:H f f P P P is non-degenerate and skew-symmetric. (ii) All conclusions of Corollary 12.7.13.5 hold. Proof. (i) This follows from Proposition 12.5.9.2(iv) and (10.3.2.2) (taking Σ = Sf and Σ0 = ∅, in order to ensure that all primes in Σ0 are unramified in E/F ). (ii) The proof of Corollary 12.7.13.5 applies to [ , ]. (12.7.13.9) The statements of Proposition 12.7.13.4,8 also hold, with minor modifications, if β : GE,S −→ O(β)∗ is a character of finite order with values in a discrete valuation ring O(β), finite over O. As R/P = O, the field Frac(O) is algebraically closed in Frac(R) = L , which implies that the ring R(β) := R ⊗O O(β) 0 0 is an integral domain. Fix P (β) ∈ Spec(R(β)) above P ∈ Spec(R); it follows from ([Mat], Thm. 23.7(ii)) that R(β)P 0 (β) is a discrete valuation ring, finite over RP 0 . We define an admissible R(β)P 0 (β) [GF,S ]-module T (P 0 ) ⊗ β := T (P 0 ) ⊗RP 0 R(β)P 0 (β) , 0 + with g ∈ GF,S acting as g ⊗ β(g) (and similarly for (T (P 0 ) ⊗ β)+ v := T (P )v ⊗ β, for v | p). The Galois representation 0

(T (P 0 ) ⊗ β)/P (β)(T (P 0 ) ⊗ β) resp.,

0

0 + (T (P 0 )+ v ⊗ β)/P (β)(T (P )v ⊗ β)

(v | p) 0

is isomorphic to V 0 ⊗ β (resp., (V 0 )+ v ⊗ β), with coefficients in κ(P (β)). The discussion in 12.7.13.2-8 goes 0 0 through, provided that one replaces R by R(β) and P by P (β). (12.7.14) ε-factors in Hida families We continue to use the notation of 12.7.13. (12.7.14.1) Proposition. Let v - p∞ be a prime of F . Assume that π(gP 0 )v = St(µ) (µ2 = 1). Then: (i) There is an exact sequence of L [Gv ]-modules 0 −→ L (1) ⊗ µ −→ Vv −→ L −→ 0. 00 (ii) For each arithmetic point P 00 ∈ Spec(hord φnL (n; O)) containing Pmin there is an exact sequence of κ(P )modules 0 −→ κ(P 00 )(1) ⊗ µ −→ Vv00 −→ κ(P 00 ) −→ 0

(in which V 00 = V (gP 00 )(k 00 /2)) and π(gP 00 )v = St(µ) = π(gP 0 )v . Proof. After multiplying χ by a suitable character A∗F /F ∗ −→ {±1}, we can assume that µ is unramified at v. In this case the wild inertia group Ivw acts through a finite quotient on T (P 0 ) and trivially on V 0 = 0 V (gP 0 )(k 0 /2) = T (P 0 )/P T (P 0 ) (cf. (12.4.4.2)); it follows that Ivw acts trivially on T (P 0 ) (hence on V ). Fix a topological generator t ∈ Ivt = Iv /Ivw of the tame inertia group and a lift f ∈ Gv of the geometric Frobenius element. The relation tf = f tN v implies that the set of eigenvalues of t acting on V is stable 2 under the map λ 7→ λN v , hence t0 = t(N v) −1 acts unipotently on V , (t0 − 1)2 = 0 on V . On the other hand, 386

0

it follows from (12.4.4.2) that (t − 1)2 = 0 6= t − 1 on T (P 0 )/P T (P 0 ), which implies that t0 − 1 6= 0 on V , hence T (P 0 )Iv = T (P 0 )t=1 is a free RP 0 -module of rank one. Counting the dimentions, we infer that the sequence 0

$

0

0

0

P 0 −→ T (P 0 )t =1 −−→T (P 0 )t =1 −→ (V 0 )t =1 −→ 0

is exact; taking invariants with respect to the finite cyclic group hti/ht0 i we obtain an exact sequence $

0

P 0 −→ T (P 0 )Iv −−→T (P 0 )Iv −→ (V 0 )Iv −→ 0.

(12.7.14.1.1)

In particular, dimL V t=1 = dimL V Iv = rkRP 0 T (P 0 )Iv = 1. ∼

As Λ2 V −→ L as a representation of Iv and Ker(t − 1 | V ) 6= 0, the operator t − 1 is nilpotent on V ; thus t − 1 induces an isomorphism of L [Gv /Iv ]-modules (both of which are one-dimensional L -vector spaces) ∼

V /V Iv = V /V t=1 −→ V (−1)t=1 = V (−1)Iv . ∼

Let β : Gv /Iv −→ L ∗ be the unramified character through which Gv acts on V /V Iv ; then V Iv −→ L (1)⊗β ∼ (as an L [Gv ]-module) and L (1) = Λ2 V −→ L (1) ⊗ β 2 , hence β 2 = 1. The exact sequence (12.7.14.1.1) ∼ implies that (V 0 )Iv −→ κ(P 0 )(1) ⊗ β, hence β = µ, thanks to (12.4.4.2); this finishes the proof of (i). (ii) As 00

V 00 = V (gP 00 )(k 00 /2) = T (P 00 )P T (P 00 ) is a subquotient of V , the group Ivw acts trivially on V 00 . The inclusion T (P 00 )Iv /$P 00 T (P 00 )Iv = κ(P 00 )(1) ⊗ µ ⊂ (T (P 00 )/$P 00 T (P 00 ))

Iv

= (V 00 )Iv

implies that dv := dimκ(P 00 ) (V 00 )Iv ≥ 1, hence  −1 Lv (π(gP 00 )v , s + 12 ) = Lv (V 00 , s) = (1 − µ(v)(N v)−1−s )(1 − a(N v)−s ) , where ( a=

0,

dv = 1

1,

dv = 2.

The purity result 12.4.8.2 implies that dv = 1 and π(gP 00 )v = St(µ), which concludes the proof. (12.7.14.2) Proposition. In the situation of 12.7.10, set π = π(gP ), π 0 = π(gP 0 ). Let v - p∞ be a prime of F . (i) If πv 6= St(µ), then Iv acts on Vv through a finite quotient. (ii) If πv = π(µ, µ−1 ), then πv0 = π(µ0 , (µ0 )−1 ), where µ0 /µ is an unramified character. (iii) If πv is supercuspidal, then πv0 = πv . Proof. (i) The same proof as in the classical case ([Se-Ta], App.) shows that there is an open normal subgroup U C Iv which acts unipotently on V , i.e. (∀u ∈ U ) (u − 1)2 = 0 on V . If U does not act trivially on V , then there exists an arithmetic point P 00 ∈ Spec(hord φnL (n; O)) containing Pmin and an element u ∈ U such that (u − 1)2 = 0 6= u − 1 on V 00 = V (gP 00 )(k 00 /2), thus π(gP 00 )v = St(µ), by 12.4.4. Applying Proposition 12.7.14.1 (with the pair P 00 , P playing the role of P 0 , P 00 ), we obtain πv = St(µ). This contradiction implies that Iv acts on V through the finite quotient Iv /U . (ii) For each g ∈ Iv /U , the trace Tr(g|V ) is contained in Qp ∩ L = Frac(O) = Lp , hence 387

Tr (g|V ) = Tr (g|T (P 0 )) = Tr (g|Vv0 ) ∈ Lp . This implies that there is, for a suitable Lp -embedding of fields κ(P 0 ) ,→ L , an isomorphism of L [Iv ]modules ∼

α : Vv0 ⊗κ(P 0 ) L −→ Vv ⊗L L . Replacing P 0 by P, we obtain an equality of traces Tr (g|Vv ) = Tr (g|Vv0 ) ∈ Lp ,

(∀g ∈ Iv /U )

(12.7.14.2.1)

hence an isomorphism of Qp [Iv ]-modules ∼

Vv ⊗Lp Qp −→ Vv0 ⊗κ(P 0 ) Qp

(12.7.14.2.2)

(for some Lp -embedding κ(P 0 ) ,→ Qp ). If πv = π(µ, µ−1 ), then (12.7.14.2.2) together with 12.4.4 imply that πv0 = π(µ0 , (µ0 )−1 ), with µ0 /µ unramified. (iii) If πv is supercuspidal, then Vv is an absolutely irreducible Iv -module; by (12.7.14.2.1-2), the same is true for Vv and Vv0 (hence πv0 is also supercuspidal). The action of a lift f ∈ Gv of the geometric Frobenius ∼ on Vv0 defines an isomorphism of κ(P 0 )[Iv ]-modules β : Vv0 −→ f Vv0 (where g ∈ Iv acts on f Vv0 by f gf −1|Vv0 ). It follows that
α ◦ (β ⊗ 1) ◦ α−1 ∈ IsomL [Iv ] Vv ⊗L L , f Vv ⊗L L , hence α ◦ (β ⊗ 1) ◦ α−1 = c · f |V

(12.7.14.2.3)

∗

for some c ∈ L . Comparing the determinants det(β) = det(f |Vv0 ) = det(f |V ) = (N v)−1 ∈ Q∗p , ∗

we see that c2 = 1, hence c = ±1 ∈ Q∗p ⊂ L . As the map κ(P 0 )[Iv ] −→ Endκ(P 0 )[Iv ] (Vv0 ) is surjective, we can assume (after replacing f by f g for suitable g ∈ Iv ) that u := Tr (f |Vv0 ) ∈ κ(P 0 ) is non-zero. It follows from (12.7.14.2.3) that u = cu, hence c = 1, which implies that α is an isomorphism of L [Gv ]-modules. Applying the same argument to Vv , we deduce that (12.7.14.2.2) is an isomorphism of ∼ Qp [Gv ]-modules, hence πv −→ πv0 . (12.7.14.3) Corollary. Let v - p∞ be a prime of F . (i) If π(gP )v = π(µ, µ−1 ) (resp., π(gP )v 6= π(µ, µ−1 )), then there is, for a suitable Lp -embedding κ(P 0 ) ,→ Qp , an isomorphism of Qp [Iv ]-modules (resp., of Qp [Gv ]-modules) ∼

Vv ⊗Lp Qp −→ Vv0 ⊗κ(P 0 ) Qp . (ii) ordv (n(gP )) = ordv (n(gP 0 )). (iii) εv (π(gP )v , 12 ) = εv (π(gP 0 )v , 12 ). Proof. (i) This follows from (the proof of) Proposition 12.7.14.1 and 12.7.14.2. (ii), (iii) If π(gP )v 6= π(µ, µ−1 ), then π(gP )v = π(gP 0 )v , hence ordv (n(gP )) = o(π(gP )v ) = o(π(gP 0 )v ) = ordv (n(gP 0 )) εv (π(gP )v , 12 ) = εv (π(gP 0 )v , 12 ). 388

If π(gP )v = π(µ, µ−1 ), then π(gP 0 )v = π(µ0 , (µ0 )−1 ) with µ0 /µ unramified, hence ordv (n(gP )) = 2 o(µ) = 2 o(µ0 ) = ordv (n(gP 0 ))

εv (π(gP )v , 12 ) = µ(−1) = µ0 (−1) = εv (π(gP 0 )v , 12 ). (12.7.14.4) Proposition. In the situation of 12.7.10, set g = gP , g 0 = gP 0 , V = V (g)(k/2), V 0 = V (g 0 )(k 0 /2). (i) εe(π(g), 12 ) = εe(π(g 0 ), 12 ). (ii) e h1f (F, V ) − h1f (F, V ) + ran (F, g) ≡ e h1f (F, V 0 ) − h1f (F, V 0 ) + ran (F, g 0 ) (mod 2). (iii) e h1 (F, V ) ≡ e h1 (F, V 0 ) (mod 2). f

f

(iv) h1f (F, V ) − ran (F, g) ≡ h1f (F, V 0 ) − ran (F, g 0 ) (mod 2).

Proof. (i) According to Proposition 12.5.9.4, (ii) is a reformulation of (i) and Y Y εe(π(g), 12 ) = χv (−1) εv (π(g)v , 12 ) χv (−1) (−1)k/2 v|∞

v-p∞ 0

εe(π(g ),

1 2)

=

Y

χ0v (−1) εv (π(g 0 )v , 12 )

Y

0

χ0v (−1) (−1)k /2 .

v|∞

v-p∞

Fix a prime v of F . If v - p∞, then χv (−1) = χ0v (−1) (as the character χ0 /χ is unramified at v) and 0 εv (π(g)v , 12 ) = εv (π(g 0 )v , 12 ) (by Corollary 12.7.14.3). If v | ∞, then χv (−1) (−1)k/2 = χ0v (−1) (−1)k /2 , as observed in 12.7.10. The statements (i), (ii) follow. (iii) This is a special case of Corollary 12.7.13.5 and (iv) is a combination of (ii) and (iii). (12.7.14.5) Proposition (Dihedral case). In the situation of 12.7.14.4, let K/F be a totally imaginary quadratic extension and β : A∗K /K ∗ A∗F −→ C∗ a ring class character of finite order. (i) If v is a prime of F , then εev (π × θ(β), 12 ) = εev (π 0 × θ(β), 12 ). (ii) e h1f (K, g, β) − h1f (K, g, β) + ran (K, g, β) ≡ e h1f (K, g 0 , β) − h1f (K, g 0 , β) + ran (K, g 0 , β) (mod 2). (iii) e h1 (K, g, β) ≡ e h1 (K, g 0 , β) (mod 2). f

f

(iv) h1f (K, g, β) − ran (K, g, β) ≡ h1f (K, g 0 , β) − ran (K, g 0 , β) (mod 2). (v) g has CM by K ⇐⇒ g 0 has CM by K.

Proof. (i) If v | ∞, then both sides are equal to 1. Assume that v - p∞. As πv = πv0 if πv 6= π(µ, µ−1 ) by Proposition 12.7.14.1-2, we have to treat only the case πv = π(µ, µ−1 ). In this case πv0 = π(µ0 , (µ0 )−1 ) by Proposition 12.7.14.2(ii); applying Proposition 12.6.2.4(i) we obtain εv (π × θ(β), 12 ) = ηv (−1) = εv (π 0 × θ(β), 12 ). If v | p, then neither πv nor πv0 is supercuspidal. Applying Proposition 12.6.2.4(i)-(ii) we obtain εev (π × θ(β), 12 ) = ηv (−1) = εev (π 0 × θ(β), 12 ). (ii) Multiply together the equalities (i) over all primes v of F and apply Proposition 12.6.4.3. (iii) This follows from 12.7.13.8(ii). (iv) Combine (ii) and (iii). (v) If g has CM by K, so does f , which implies that each prime of F above p splits in K/F . There exists a cyclotomic Hida family containing the p-stabilization f 0 of f , in which all arithmetic points correspond to forms with CM by K ([H-T], §4). As f 0 is a p-stabilized newform, the corresponding primitive family is unique, hence corresponds to the morphism λ; thus f 0 also has CM by K, and so does g 0 . The converse follows by the same argument. (12.7.15) Hida families and Iwasawa theory We continue to use the notation of 12.7.13. 389

(12.7.15.1) In the situation of 12.7.13.1-2, fix an abelian extension E∞ of E contained in FS , for which ∼ Γ := Gal(E∞ /E) = Γ0 × ∆, Γ0 −→ Zrp (r ≥ 1), |∆| < ∞. Assume that all primes of E above Σ0 are unramified in E∞ /E, which is automatic if ∆ = 0 or if Σ0 = ∅. (12.7.15.2) Set R = R[[Γ]] = R[∆][[Γ0 ]], Λ = O[[Γ]] = O[∆][[Γ0 ]]. As R/P = O, the field Frac(O) is algebraically closed in the field Frac(R) = L , which implies that the formula q 7→ q = qR defines a bijection ∼

{q ∈ Spec(Λ), ht(q) = 0} −→ {q ∈ Spec(R), ht(q) = 0}; fix such q and q = qR. Recall that we can (and will) assume that the discrete valuation ring O0 coincides with the normalization of 0 0 0 R/P ; thus Frac(O0 ) = κ(P ). Set Λ0 = O0 [[Γ]] = O0 [∆][[Γ0 ]] and denote by u0 : R[∆] −→ R/P [∆] −→ O0 [∆] the canonical map. The formulas q0 7→ q0 ∩ O0 [∆],

0

q0 7→ p0 = P R + u0−1 (q0 ∩ O0 [∆])R

(12.7.15.2.1)

define bijections between the fibre of Spec(Λ0 ) −→ Spec(Λ) above q and, respectively, the fibre of Spec(O0 [∆]) −→ Spec(O[∆]) above q ∩ O[∆] and 0

{p0 ∈ Spec(R) | ht(p0 ) = 1, p0 ⊃ q, p0 ⊃ P R}, (by Lemma 8.9.7.1). Fix q0 ∈ Spec(Λ0 ) above q and define p0 ∈ Spec(R) by the formula (12.7.15.2.1). The localization Rp0 is a discrete valuation ring with prime element $P 0 and residue field κ(p0 ) = Λ0q0 = κ(q0 ). ∗

More precisely, there exists a character χ : ∆ −→ Lp such that q0 = Ker(χ : O0 [∆] −→ Lp ). Fix such χ and denote by O0 (χ) the image of O0 [∆] under χ; it is a discrete valuation ring, finite over O0 . For each O0 [∆]-module M , denote M (χ) := M ⊗O0 [∆],χ O0 (χ) (as in 10.7.16). Using this notation, the normalization of R/p0 is equal to O0 (χ)[[Γ0 ]] = Λ0(χ) , hence κ(p0 ) = Frac(Λ0(χ) ) = Λ0q0 .

(12.7.15.2.2)

The fraction field of Rp0 is equal to Frac(Rp0 ) = Frac(Rq ) = κ(q) = Frac(R/q) = Frac((R ⊗O O(χ))[[Γ0 ]]). (12.7.15.3) Fix a GF,S -stable R-lattice (resp., O0 -lattice) T ⊂ V (resp., T 0 ⊂ V 0 ), and denote, for each v | p in F , Tv+ = T ∩ Vv+ ,

+ FΓ (T )+ v = FΓ (Tv ),

0 0 + (T 0 )+ v = T ∩ (V )v ,

0 + FΓ (T 0 )+ v = FΓ ((T )v ).

For each intermediate set {v | p} ⊂ Σ ⊂ Sf , we define Greenberg’s local conditions ∆Σ (X) for X = FΓ (T ) and FΓ (T 0 ) by the formulas from 12.7.13.2 (with β = 1). 0

b 0 0 (12.7.15.4) As TP 0 := RP 0 T = $P 0 T (P ) for some b ∈ Z, we have

390

(∀v | p)

Tv+


P

0

b 0 + := RP 0 Tv+ = $P 0 T (P )v

0

and 0

b 0 FΓ (T ) ⊗R Rp0 = T ⊗R Rp0 < −1 > = TP 0 ⊗RP 0 Rp0 < −1 > = $P 0 T (P ) ⊗R 0 Rp0 < −1 > P

(∀v | p)

0

b 0 + FΓ (Tv+ ) ⊗R Rp0 = $P 0 T (P )v ⊗R 0 Rp0 < −1 > . P

(12.7.15.4.1)

As T (P 0 )/$P 0 T (P 0 ) = V 0 and Rp0 /$P 0 Rp0 = Frac(Λ0(χ) ), the formulas (12.7.15.4.1) imply that multiplica−b0 tion by $P 0 , followed by reduction modulo $P 0 , induces isomorphisms ∼

FΓ (T )p0 /$P 0 FΓ (T )p0 −→ FΓ (T 0 ) ⊗Λ0 ,χ Frac(Λ0(χ) ) = FΓ (T 0 )q0 (∀v | p)

∼

0(χ) FΓ (Tv+ )p0 /$P 0 FΓ (Tv+ )p0 −→ FΓ ((T 0 )+ ) = FΓ ((T 0 )+ v ) ⊗Λ0 ,χ Frac(Λ v )q0

(12.7.15.4.2)

of Frac(Λ0(χ) )[GF ]-modules (resp., Frac(Λ0(χ) )[Gv ]-modules), which are compatible with respect to the in0 clusion maps Tv+ ⊂ T and (T 0 )+ v ⊂ T . The pairing  0   0  b 0 b 0 πT ,p0 : FΓ (T )p0 ⊗R 0 (FΓ (T )ι )p0 = $P ⊗ $ −→ 0 T (P ) ⊗R 0 Rp0 < −1 > 0 T (P ) ⊗R 0 Rp0 < 1 > P R 0 P P p

πT (P 0 ) ⊗id

2b0 −−−−−−→ $P 0

0

−2b $P 0

p

Rp0 (1) −−−−→ Rp0 (1)

is a perfect duality, satisfying (∀v | p)

FΓ (Tv+ )p0 ⊥⊥πT ,p0 FΓ (Tv+ )p0

(12.7.15.4.3)

(12.7.15.5) Proposition. (i) There exists an exact triangle in Dfb t (R 0 Mod) p

$P 0 gf (GE,S , FΓ (T ); ∆Σ ) 0 −−→ gf (GE,S , FΓ (T ); ∆Σ ) 0 −→ RΓ gf (GE,S , FΓ (T 0 ); ∆Σ )q0 , RΓ RΓ p p

which gives rise to exact sequences e j (GE,S , FΓ (T ); ∆Σ ) 0 /p0 −→ H e j (GE,S , FΓ (T 0 ); ∆Σ )q0 −→ H e j+1 (GE,S , FΓ (T ); ∆Σ ) 0 [p0 ] −→ 0, 0 −→ H p p f f f e j (GE,S , FΓ (T 0 ); ∆Σ )q0 = H e j (GE,S , FΓ (T 0 ); ∆Σ ) ⊗Λ0 ,χ Frac(Λ0(χ) ). where H f f gf (GE,S , FΓ (T 0 ); ∆Σ )q0 ∈ Db (Λ0 Mod) does not depend on the (ii) Up to a canonical isomorphism, RΓ f t q0 gf,Iw (E∞ /E, T 0 )q0 and its cohomology by H e j (E∞ /E, T 0)q0 . choice of S, Σ and T 0 ; we denote it by RΓ f,Iw gf (GE,S , FΓ (T ); ∆Σ ) 0 ∈ Db ( Mod) does not depend on the (iii) Up to a canonical isomorphism, RΓ p ft R 0 p

gf,Iw (E∞ /E, T ) 0 and its cohomology by H e j (E∞ /E, T ) 0 . choice of S and Σ; we denote it by RΓ p p f,Iw gf (GE,S , FΓ (T ); ∆Σ )q ∈ Db (κ(q) Mod) does not depend on the (iv) Up to a canonical isomorphism, RΓ ft gf,Iw (E∞ /E, T )q and its cohomology by H e j (E∞ /E, T )q . choice of S, Σ and T ; we denote it by RΓ f,Iw

gf,Iw (E∞ /E, T ) 0 in RΓ gf,Iw (E∞ /E, T )q depends only on T 0 = (v) For each j ∈ Z, the image of RΓ p P 0 $b 0 T (P 0 ). P

Proof. (i) The same argument as in the proof of Proposition 12.7.13.4(i) applies (using (12.7.15.4.2) instead of (12.7.13.1.2)). (ii) This follows from Proposition 9.7.9(ii). 391

(iii) Independence on S was proved in Proposition 7.8.8(ii). In order to prove independence on Σ, denote by gf (GE,S , FΓ (T ); ∆Σ ) 0 −→ RΓ gf (GE,S , FΓ (T ); ∆S ) 0 . Y ∈ Dfb t (R 0 Mod) the cone of the canonical map RΓ f p p p

L

It follows from (i) and (ii) that Y ⊗R 0 κ(p0 ) is acyclic; thus Y is also acyclic, by Lemma 12.7.15.6 below. p

(iv) This follows from (iii) and the fact that T ⊗R Rq does not depend on T , since gf (GE,S , FΓ (T ); ∆Σ )q = RΓ gf (GE,S , T ⊗R Rq < −1 >; ∆Σ ). RΓ (v) Similarly, we use gf (GE,S , FΓ (T ); ∆Σ ) 0 = RΓ gf (GE,S , T 0 ⊗R 0 R 0 < −1 >). RΓ p p P P (12.7.15.6) Lemma. Let R be a noetherian ring and X ∈ Df t (R Mod). Assume that r ∈ R in not a zero L

∼

∼

divisor, but is contained in the radical of R. If Y := X ⊗R R/rR −→ 0 in Df t (R/rR Mod), then X −→ 0 in Df t (R Mod). r

Proof. The free resolution [R−→R] of R/rR gives rise to injections H j (X)/rH j (X) ,→ H j (Y ). As each H j (Y ) vanishes by assumption, Nakayama’s Lemma implies that H j (X) = 0, too. (12.7.15.7) Proposition. (i) The duality morphism gf,Iw (E∞ /E, T ) 0 −→ RHom RΓ p R

p0

  gf,Iw (E∞ /E, T )ι ) 0 , R 0 [−3] (RΓ p p

is an isomorphism in Dfb t (R 0 Mod). p

(ii) The corresponding generalized Cassels-Tate pairing   e 2 (E∞ /E, T ) 0 h, i: H f,Iw p

(Rp0 )−tors

  e 2 (E∞ /E, T ) 0 × H f,Iw p

(Rp0 )−tors

−→ Frac(Rp0 )/Rp0

is a non-degenerate skew-hermitian form. e 1 (E∞ /E, T ) 0 is a free (R 0 )-module of rank equal to (iii) H p p f,Iw 1 1 e f,Iw e f,Iw rkR 0 H (E∞ /E, T )p0 = dimκ(q) H (E∞ /E, T )q . p

Proof. (i) Apply the localized version of the duality Theorem 6.3.4, for the local conditions ∆Sf . Let w be a prime of E above a prime v ∈ Sf . If v | p, then the error term Errw vanishes, thanks to (12.7.15.4.3). If v - p, then Errw = 0, by combining Proposition 8.9.7.7 with Proposition 12.7.13.3(i) for X = T (P 0 ). (ii) The pairing h , i is non-degenerate, by Theorem 10.4.4 and (i). It is skew-hermitian, by a localized version of Proposition 10.3.4.2. (iii) The proof of Proposition 12.7.13.4(iii) applies, using Proposition 12.7.15.5(i) instead of Proposition 12.7.13.4(i). (12.7.15.8) Proposition (Dihedral case). Assume that E = K, where K is as in 12.6, and E∞ = S K∞ ⊂ K[∞] = K[c] (in the notation of 12.6.1.5). Fix a lift τ ∈ Gal(K∞ /F ) of the non-trivial element of Gal(K/F ). Then: (i) The map uX from Lemma 10.3.5.4 (for X = T ) induces an isomorphism ∼

gf (GK,S , FΓ (T ); ∆Σ ) −→ RΓ gf (GK,S , FΓ (T ); ∆Σ )ι . uT : RΓ (ii) The formula [x, y] := hx, uT (y)i defines a non-degenerate, skew-symmetric (Rp0 )-bilinear pairing   e 2 (E∞ /E, T ) 0 H f,Iw p

(Rp0 )−tors

  e 2 (E∞ /E, T ) 0 × H f,Iw p 392

(Rp0 )−tors

−→ Frac(Rp0 )/Rp0 .

(iii) We have e 1 (K∞ /K, T 0)(χ) ≡ dimκ(q) H e 1 (E∞ /E, T )q rkΛ0(χ) H f,Iw f,Iw e 1 (K∞ /K, T 0 )(χ) H f,Iw

rkΛ0(χ)

≥

(mod 2)

e 1 (E∞ /E, T )q . dimκ(q) H f,Iw

Proof. (i), (ii) This follows from Lemma 10.3.5.4 and Proposition 10.3.5.8. (iii) Applying Lemma 10.6.5 to the pairing from (ii), we deduce that e 2 (E∞ /E, T ) 0 [p0 ] ≡ 0 (mod 2); dimκ(p0 ) H f,Iw p the statement then follows from the exact sequence in Proposition 12.7.15.5(i) (with j = 1). (12.8) Level raising In this section we assume that f ∈ Sk (n, ϕ) is a p-ordinary newform of level n and χ : A∗F /F ∗ −→ C∗ is a character (with values contained in ι∞ (L)) satisfying χ−2 = ϕ (=⇒ k is even). For simplicity, we shall denote O = OLp . (12.8.1) Congruences between newforms and ε-factors (12.8.1.1) As in 12.5.5, we put g = f ⊗ χ ∈ Sk (n(g), 1) (which is a newform of level dividing cond(χ)2 n) and V = V (g)(k/2) = V (f )(k/2) ⊗ χ, Vv± = V (f )± (k/2) ⊗ χv (∀v | p). Fix a GF -stable O-lattice T (f ) ⊂ V (f ) and put T (g) = T (f ) ⊗ χ ⊂ V (g), T = T (g)(k/2) = T (f )(k/2) ⊗ χ ⊂ V . The absolute irreducibility of V (f ) implies that there exists an integer c ≥ 0 such that Im (O[GF ] −→ EndO (T )) ⊇ pc EndO (T ). (12.8.1.2) Fix an integer M ≥ 1 and assume that we are given a newform g1 ∈ Sk (n(g)Q, 1) of level n(g1 ) divisible by Q such that (12.8.1.2.1) Q = q1 · · · qs is a product of s ≥ 1 distinct prime ideals not dividing (p)cond(χ)n(g). (12.8.1.2.2) (∀i = 1, . . . , s) λg1 (qi ) = −(N qi )k/2−1 ( ⇐⇒ π(g1 )qi = St(µ), µ2 = 1, µ unramified, µ(qi ) = −1, by Lemma 12.3.10(i)). (12.8.1.2.3) For all primes v - cond(χ)npQ contained in a set of density 1, λg1 (v) ≡ λg (v) (mod pM+6c ) (a congruence holding in O). (12.8.1.2.4) The newform f1 = g1 ⊗ χ−1 is p-ordinary (this is automatically true if M is large enough). (12.8.1.2.5) If v - Q is a non-archimedean prime of F , then [ordv (n(g)) = 1 ⇐⇒ ordv (n(g1 )) = 1]. ˇ (12.8.1.3) The congruence (12.8.1.2.3) implies, by Proposition 12.8.3.1 below and the Cebotarev density theorem, that there exists a GF -stable O-lattice T1 ⊂ V1 = V (g1 )(k/2) = V (f1 )(k/2)⊗χ and an isomorphism of O[GF ]-modules ∼

j : T /pM T −→ T1 /pM T1 .

(12.8.1.3.1)

As usual, we put A = V /T, A1 = V1 /T1 and, for each prime v | p, Tv+ = Vv+ ,

Tv− = T /Tv+ ,

± ± A± v = Vv /Tv ,

± M A[pM ]± v = Av [p ].

Thanks to the assumption (12.8.1.2.4), we can also define (for each v | p) ± (V1 )± v = V (f1 )v (k/2) ⊗ χv ,

+ (T1 )+ v = (V1 )v ,

+ (T1 )− v = T1 /(T1 )v ,

± ± (A1 )± v = (V1 )v /(T1 )v .

Let π(g) (resp., π(g1 )) be the automorphic representation of GL2 (AF ) attached to g (resp., to g1 ). 393

(12.8.1.4) Proposition. Let v be a prime of F . The local ε-factors εv (πv , 12 ) (π = π(g), π(g1 )) have the following properties: (i) If v - n(g)Q∞, then εv (π(g)v , 12 ) = εv (π(g1 )v , 12 ) = 1. (ii) If v | ∞, then εv (π(g)v , 12 ) = εv (π(g1 )v , 12 ) = (−1)k/2 . (iii) If v | Q, then εv (π(g)v , 12 ) = εv (π(g1 )v , 12 ) = 1. (iv) If v | n(g), v - p and M ≥ 1 + ordp (2) (+ ordp (N v + 1) if ordv (n(g)) = 1), then εv (π(g)v , 12 ) = εv (π(g1 )v , 12 ). Proof. We only have to prove (iii) and (iv). If v = qi (i = 1, . . . , s), then v - n(g), hence εv (π(g)v , 12 ) = 1. On the other hand, π(g1 )v = St(µ) with µ unramified and µ(v) = −1, hence εv (π(g1 )v , 12 ) = µ(−1) = 1 (by Lemma 12.3.13). It remains to prove (iv): let v | n(g), v - p. According to (12.4.3.1), we have εv (π(g)v , 12 ) = εv (V, ψv , dxv ) = ε0,v (V, ψv , dxv ) det(−Fr(v)geom , V Iv )−1 ,

(12.8.1.4.1)

where dxv is the self-dual Haar measure with respect to ψv (and similarly for g1 and V1 ). As ∼

T /2pT = T /p1+ordp (2) T −→ T1 /p1+ordp (2) T1 = T1 /2pT1

(12.8.1.4.2)

by (12.8.1.3.1), it follows from [De 2, Thm. 6.5] (resp., [Ya, Thm. 5.1]) in the case p 6= 2 (resp., p = 2) that ε0,v (V, ψv , dxv ) ≡ ε0,v (V1 , ψv , dxv ) (mod p1+ordp (2) ).

(12.8.1.4.3)

As both values εv (πv , 12 ) (π = π(g), π(g1 )) are equal to ±1, it is enough to show that they are congruent modulo 2p. In view of (12.8.1.4.1,3), it suffices to establish the following congruence (holding in O∗ ): ?

det(−Fr(v)geom , V Iv )−1 ≡ det(−Fr(v)geom , V1Iv )−1 (mod p1+ordp (2) ). Combining the assumption (12.8.1.2.5) with Lemma 12.4.5(ii), we see that V Iv 6= 0 ⇐⇒ V1Iv 6= 0; if this is the case, then π(g)v = St(µ) (µ2 = 1), π(g1 )v = St(µ1 ) (µ21 = 1), the characters µ and µ1 are unramified and there are exact sequences of O[Gv ]-modules 0 −→ O(1) ⊗ µ −→ T −→ O ⊗ µ −→ 0,

0 −→ O(1) ⊗ µ1 −→ T1 −→ O ⊗ µ1 −→ 0.

It follows that 
 O/pM ⊆ (T /pM T )(−1) Gv  2(N v + 1) · (T /pM T )(−1)
Gv = 0

if µ(v) = 1 if µ(v) = −1

and 
 O/pM ⊆ (T1 /pM T1 )(−1) Gv  2(N v + 1) · (T /pM T )(−1)
Gv = 0 1 1

if µ1 (v) = 1 if µ1 (v) = −1.

The isomorphism (12.8.1.4.2) together with the assumption M > ordp (2(N v + 1)) then imply that µ(v) = µ1 (v) (= ±1), hence det(−Fr(v)geom , V Iv )−1 = −µ(v)N v = −µ1 (v)N v = det(−Fr(v)geom , V1Iv )−1 , which concludes the proof. 394

(12.8.1.5) Corollary. If M ≥ 1 + ordp (2) + max{ordp (N v + 1) | v - p, ordv (n(g)) = 1}, then εe(π(g), 12 ) = εe(π(g1 ), 12 ) e h1f (F, g) − h1f (F, g) + ran (F, g) ≡ e h1f (F, g1 ) − h1f (F, g1 ) + ran (F, g1 ) (mod 2) Proof. Combining Proposition 12.5.9.4 and 12.8.1.4, we obtain Y Y 1 1 (−1)ehf (F,g)−hf (F,g)+ran (F,g) = εe(π(g), 12 ) = χv (−1) εv (π(g)v , 12 ) χv (−1)(−1)k/2 = =

Y

χv (−1) εv (π(g1 )v , 12 )

Y

v-p∞

v|∞

χv (−1)(−1)k/2 = εe(π(g1 ), 12 ) = (−1)e

h1f (F,g1 )−h1f (F,g1 )+ran (F,g1 )

.

v|∞

v-p∞

(12.8.2) Linear algebra Fix a prime element $ of O = OLp . (12.8.2.1) Definition. Let α, β ≥ 0 be integers and f : X −→ Y a homomorphism of O-modules. We say that f is an (α, β)-morphism if pα · Ker(f ) = 0 and pβ · Coker(f ) = 0. (12.8.2.2) Lemma. (i) If f : X −→ Y (resp., f 0 : Y −→ Z) is an (α, β)-morphism (resp., an (α0 , β 0 )morphism), then f 0 f : X −→ Z is an (α + α0 , β + β 0 )-morphism. (ii) If f : X −→ Y is an (α, β)-morphism, then there exists a homomorphism g : Y −→ X satisfying f g = $α+β · idY , gf = $α+β · idX (=⇒ g is an (α + β, α + β)-morphism). (iii) If f : X −→ Y is an (α, β)-morphism, so is the induced morphism fn : pn X −→ pn Y (for each n ≥ 0). Proof. Elementary exercise. (12.8.2.3) Proposition. Let k, M ≥ 1 and r, r1 , α, β ≥ 0 be integers, W and W1 two O/pM -modules of finite length and f : (O/pM )⊕r ⊕ W ⊕k −→ (O/pM )⊕r1 ⊕ W1⊕k an (α, β)-morphism. Write W = O/pn1 ⊕ · · · ⊕ O/png (g ≥ 0, n1 ≥ n2 ≥ · · · ≥ ng ). Put n0 = M , ng+1 = 0 and assume that there exists an index i ∈ {1, . . . , g} such that ni > ni+1 + (α + β) (this is automatically true if M > (α + β)(g + 1) or if pM−1−α−β W = 0). Then r ≡ r1 (mod k). Proof. If M is an O-module of finite length and n ≥ 1, put sn (M ) := max{t ≥ 0 | (O/pn )⊕t ⊆ M }. Using Lemma 12.8.2.2(iii), we can replace f by fni+1 , hence assume that ng > α + β. The surjections (O/pM )⊕r ⊕ W ⊕k  Im(f )  ((O/pM−α )⊕r ⊕ (pα W )⊕k resp., the inclusions pβ (O/pM )⊕r1 ⊕ (pβ W1 )⊕k ⊆ Im(f ) ⊆ (O/pM )⊕r1 ⊕ W1⊕k imply that s1 (Im(f )) = sβ+1 (Im(f )) = r + k s1 (W ) resp., r1 + k sβ+1 (W1 ) = r1 + k s1 (pβ W1 )⊕k ≤ s1 (Im(f )) = sβ+1 (Im(f )) ≤ r1 + k sβ+1 (W1 ). 395

It follows that r + k s1 (W ) = s1 (Im(f )) = r1 + k sβ+1 (W1 ) =⇒ r ≡ r1 (mod k). (12.8.2.4) The following example shows that the assumption ni − ni+1 > α + β is necessary: the map f : (O/pM ) ⊕ (O/pM−1 )⊕2 ⊕ · · · ⊕ (O/p)⊕2 −→ (O/pM−1 )⊕2 ⊕ · · · ⊕ (O/p)⊕2 f (y0 , x1 , y1 , . . . , xM−1 , yM−1 ) = (y 0 , x1 , y1 , x2 , . . . , yM−2 , xM−1 ) (where y i = yi (mod pM−i−1 )) is a (1, 0)-morphism f : O/pM ⊕ W ⊕2 −→ W1⊕2 . (12.8.2.5) Lemma. If G is a topological group and f : X −→ Y an (α, β)-morphism of ind-admissible O[G]-modules, then the induced maps i i Hcont (G, X) −→ Hcont (G, Y )

are all (α + β, α + β)-morphisms. Proof. Combine the cohomology sequences attached to 0 −→ Ker(f ) −→ X −→ Im(f ) −→ 0,

0 −→ Im(f ) −→ Y −→ Coker(f ) −→ 0.

(12.8.3) Congruences between traces and congruences between representations (12.8.3.1) Proposition. Let Φ be a finite extension of Qp , O ⊂ Φ its ring of integers, $ ∈ O a uniformizer of O and G a group. Let T1 be an O[G]-module, free of rank n ≥ 1 as an O-module. (i) If T1 ⊗O Φ is an absolutely simple Φ[G]-module, then Im (O[G] −→ EndO (T1 )) ⊃ $c EndO (T1 ).

(∃c ≥ 0)

(ii) Assume that N ≥ 1 an integer and T2 is an O/$ N +5c O[G]-module, free of rank n as an O/$ N +5c Omodule, satisfying (∀g ∈ G) TrO (g|T1 ) ≡ TrO (g|T2 ) (mod $N +5c ), where c ≥ 0 is as in (i). Then there exists a homomorphism of O[G]-modules α : T1 /$ N T1 −→ T2 /$ N T2 satisfying $c Ker(α) = $c Coker(α) = 0. If N ≥ c, then there is an O[G]-submodule $c T2 ⊂ T20 ⊂ T2 and ∼ an isomorphism of O-modules T1 /$ N −c T1 −→ T20 /$ N −c T20 . Proof. (i) As V1 = T1 ⊗O Φ is an absolutely simple Φ[G]-module, we have ([Cu-Re], 3.43, 3.32) EndΦ[G] (V1 ) = Φ,

Im (O[G] −→ EndO (T1 )) ⊗O Φ = Im (Φ[G] −→ EndΦ (V1 )) = EndΦ (V1 ),

which proves the claim. (ii) Our assumptions imply that (∀a ∈ O[G])

TrO (a|T1 ) ≡ TrO (a|T2 ) (mod $N +5c ).

∼

(12.8.3.1.1)

∼

∼

Fix O-module isomorphisms O⊕n −→ T1 , (O/$ N +5c O)⊕n −→ T2 ; they induce isomorphisms EndO (T1 ) −→ ∼ Mn (O), EndO (T2 ) −→ Mn (O/$ N +5c O). Put
B = Im O[G] −→ EndO (T1 /$ N +5c T1 ) ⊕ EndO (T2 /$ N +5c T2 ) ⊂ Mn (O/$ N +5c O)⊕2 396

and denote by ij : B −→ Mn (O/$ N +5c O) (j = 1, 2) the projections of B on the two factors. Their kernels Ij = Ker(ij ) are bilateral ideals of B satisfying I1 ∩ I2 = 0. We claim that $2c i1 (I2 ) = 0: indeed, fix x ∈ i1 (I2 ). As i1 (I2 ) is a bilateral ideal in i1 (B) ⊃ $c Mn (O/$ N +5c O), i1 (I2 ) contains the bilateral ideal J = $c Mn (O/$ N +5c O)x$c Mn (O/$ N +5c O) of Mn (O/$ N +5c O), which is necessarily of the form J = $k Mn (O/$ N +5c O) (2c ≤ k ≤ N + 5c). It follows from (12.8.3.1.1) that (∀A ∈ i1 (I2 ))

Tr(A) = 0 ∈ O/$ N +5c O;

taking A = diag($k , 0, . . . , 0), we obtain $k ≡ 0 (mod $N +5c ), hence J = 0. As $2c x ∈ J, we have $2c i1 (I2 ) = 0, hence $2c I2 ⊂ I1 ∩ I2 = 0. In the special case c = 0 we can conclude as follows: as I2 = 0, the inequalities `O (Mn (O/$ N O)) ≥ `O (i2 (B)) = `O (B) = `O (i1 (B)) + `O (I1 ) = `O (Mn (O/$ N O)) + `O (I1 ) ∼

imply that I1 = 0 and i2 (B) = Mn (O/$ N O), hence both maps i1 , i2 : B −→ Mn (O/$ N O) are isomorphisms ∼ N N of O-algebras. The isomorphism i2 i−1 1 : Mn (O/$ O) −→ Mn (O/$ O) is necessarily inner (cf. Lemma 12.8.3.2 below), which implies that B = {(A, gAg −1 ) | A ∈ Mn (O/$ N O)} for some g ∈ GLn (O/$ N O). Multiplication by g then induces the desired isomorphism of O[G]-modules g

∼

∼

T1 /$ N T1 −→ (O/$ N O)−−→(O/$ N O) −→ T2 /$ N T2 . ∼

If c ≥ 1, then the multivalued ‘map’ i1 (b) 7→ i2 (b) induces an isomorphism of O-algebras i1 (B)/i1 (I2 ) −→ i2 (B)/i2 (I1 ), hence a homomorphism of O-algebras ∼

∼

f : i1 (B) −→ i1 (B)/i1 (I2 ) −→ i2 (B)/i2 (I1 ) −→ Mn (O/$ N +5c O)/Mn (O/$ N +5c O)[$2c ] −→ ∼

−→ Mn (O/$ N +3c O).

As $c Mn (O/$ N +5c O) ⊂ i1 (B), Lemma 12.8.3.2 below implies that there exists a matrix g ∈ Mn (O) ∩ GLn (Φ) such that $c g −1 ∈ Mn (O) and (∀A ∈ Mn (O/$ N +5c O))

f ($c A) ≡ gA$c g −1 (mod $N +c ).

In particular, for A ∈ i1 (B) we obtain a congruence f (A)g ≡ gA (mod $N Mn (O/$ N +5c O)g),

(∀A ∈ i1 (B))

which implies that multiplication by g ∈ Mn (O) induces a morphism of O[G]-modules ∼

g

∼

α : T1 /$ N T1 −→ (O/$ N O)−−→(O/$ N O) −→ T2 /$ N T2 . Multiplication by $c g −1 ∈ Mn (O) induces a morphism of O[G]-modules β : T2 /$ N T2 −→ T1 /$ N T1 satisfying βα = αβ = $c ; thus $c Ker(α) = $c Coker(α) = 0. If, in addition, N ≥ c, let T20 ⊂ T2 be the inverse image of Im(α) under the canonical projection T2 −→ T2 /$ N T2 ; this is an O[G]-submodule of T2 containing $c T2 . The surjective homomorphism of O[G]-modules (induced by α) T1 /$ N T1  T20 /$ N T2  T20 /$ N −c T20 factors through α0 : T1 /$ N −c T1  T20 /$ N −c T20 ; as `O (T1 /$ N −c T1 ) = n(N − c) = `O (T20 /$ N −c T20 ), it follows that α0 is an isomorphism. 397

(12.8.3.2) Lemma. Let O be a discrete valuation ring with a uniformizer $ ∈ O. Assume that n ≥ 1, c ≥ 0, N ≥ 2c + 1 are integers and f : (O/$ N +c O) · I + $c Mn (O/$ N +c O) −→ Mn (O/$ N O) an O-algebra homomorphism. Then there exists a non-singular matrix g ∈ Mn (O) such that $c g −1 ∈ Mn (O) and (∀A ∈ Mn (O/$ N +c O))

f ($c A) ≡ gA $c g −1 (mod $N −2c Mn (O/$ N O)).

Proof. If c = 0, then f : Mn (O/$ N O) −→ Mn (O/$ N O) is an O-algebra homomorphism, in fact an isomorphism (as `O (Mn (O/$ N O)) < ∞, it is enough to check that Ker(f ) = 0; but Ker(f ) is a bilateral ideal in Mn (O/$ N O) not containing $N −1 I, hence Ker(f ) ⊆ $N Mn (O/$ N O) = 0). If N = 1, we conclude by the Skolem-Noether theorem ([Cu-Re], 3.63). If N > 1, we can assume, by induction, that f (A) ≡ A (mod $N −1 ) for all A ∈ Mn (O/$ N O) (after replacing f (A) by g −1 f (A)g for suitable g ∈ GLn (O)). Writing f (A) = A + $N −1 h(A), the reduction modulo $ A 7→ h(A) (mod $) defines an O-linear map h : Mn (O/$O) −→ Mn (O/$O),

h(I) = 0,

h(AB) = h(A)B + Ah(B).

Substituting for A, B various elementary matrices, an easy calculation shows that there exists a matrix H ∈ Mn (O/$O) such that h(A) = [H, A] = HA − AH for all A. Choosing a lift H ∈ Mn (O/$ N O) of H, we obtain f (A) = (I + $N −1 H)A(I + $N −1 H)−1 , as required. If c ≥ 1, consider Mn (O/$ N O) acting on T /$ N T = T = (O/$ N O)⊕n , where T = O⊕n . Let U ⊂ T be the O/$ N O-submodule of T generated by f ($c A)T , for all A ∈ Mn (O/$ N +c O). It follows from f ($c A)f ($c A0 ) = f ($2c AA0 ) = $c f ($c AA0 )

(A, A0 ∈ Mn (O/$ N +c O))

that $c T = f ($c I)T ⊂ U ,

f ($c A)U ⊂ $c U

(A ∈ Mn (O/$ N +c O)).

Let U ⊂ T be the inverse image of U under the projection T −→ T /$ N T . Fix a matrix s ∈ Mn (O) such that s(T ) = U ; then $c s−1 ∈ Mn (O), as $c T ⊂ U . For each A ∈ Mn (O/$ N +c O), the O-linear map s

f ($ c A)

$ c s−1

$ 2c

u(A) : T −−−−→U −−−−→$c U −−−−→$2c T ←−−−−T /T [$2c ] = (O/$ N −2c O)⊕n satisfies u(A)u(A0 ) = u(AA0 ), hence the map u : Mn (O/$ N +c O) −→ Mn (O/$ N −2c O), A 7→ u(A), is a homomorphism of O-algebras, factoring through u : Mn (O/$ N −2c O) −→ Mn (O/$ N −2c O). The case c = 0 treated above implies that there exists t ∈ GLn (O/$ N −2c O) such that u(A) = tAt−1 for all A ∈ Mn (O/$ N −2c O), hence
$c s−1 f ($c A)s ≡ $2c tAt−1 (mod $N −2c ) ,


$2c f ($c A) ≡ $2c gA π c g −1 (mod $N −2c )

holds for all A ∈ Mn (O/$ N O), if we put g = st. (12.8.4) Comparison of Selmer groups (12.8.4.1) Let S be the set of primes of F dividing pn(g)Q∞, Σ = {v | p} and Σ0 = Sf − Σ. We shall consider the groups e fi (Z) := H e fi (GF,S , Z; ∆(Z)) H for Z = T, V, A, T1 , V1 , A1 , A[pk ] = T /pk T, A1 [pk ] = T1 /pk T1

(k ≥ 1)

and Greenberg’s local conditions ( Uv+ (Z)

=

• Ccont (Gv , Zv+ ),

v|p

• Ccont (Gv /Iv , Z Iv ),

v ∈ Σ0

398

(12.8.4.1.1)

(if Z = T, V, A, A[pk ], then we obtain the same groups if we replace S by S − {q1 , . . . , qs }, by Corollary 7.8.9). Set e f1 (V ), r := e h1f (F, V ) = dimLp H

e f1 (V1 ); r1 := e h1f (F, V1 ) = dimLp H

the goal of Sect. 12.8.4 is to show that r ≡ r1 (mod 2), provided that M is large enough. Here is a sketch of the argument: we have ∼ ⊕r e f1 (A) −→ f H (Lp /O) ⊕ X(A),

∼ ⊕r e f1 (A1 ) −→ f 1 ), H (Lp /O) 1 ⊕ X(A

where f e f1 (A)/H e f1 (A) , X(A) =H div

f 1) = H e f1 (A1 )/H e f1 (A1 ) X(A div

e 1 (A)[pM ] and H e 1 (A1 )[pM ] are almost the are O-modules of finite length. We show that the O-modules H f f f f 1 )) is close to being isomorphic same and, using the generalized Cassels-Tate pairing, that X(A) (resp., X(A to W ⊕ W (resp., W1 ⊕ W1 ). The result then follows from Proposition 12.8.2.3. (12.8.4.2) Proposition. For each prime v | p of F , put   2 ∗ av = min{ordp (NFk−1 · χ )(u) − 1 | u ∈ OF,v } v /Q v p and define a = maxv|p (av ), j = $a j : A[pM ] −→ A1 [pM ]. Then we have, for each v | p,


M + j A[pM ]+ v ⊆ A1 [p ]v

M + − M − M − and the map jv+ : A[pM ]+ v −→ A1 [p ]v (resp., jv : A[p ]v −→ A1 [p ]v ) induced by j is an (a, 2a)morphism (resp., an (2a, a)-morphism).

Proof. In the commutative diagram with exact rows 0 −→

A[pM ]+ v

0 −→ A1 [pM ]+ v

i+

v −−→

A[pM ]   yj

−−→

−−→

A1 [pM ]

v −−→

(i1 )−

A[pM ]− v

−→

0

A1 [pM ]− v

−→

0,

the Gv -modules Zv+ (−k/2) ⊗ χ−1 Zv− (k/2 − 1) are unramified (Z = A[pM ], A1 [pM ]). This implies that v and 
k−1 + 2 ∗ Im (i1 )− v ◦ j ◦ iv is killed by χcycl χv (gv ) − 1, for each gv ∈ Iv (where χcycl : Gv −→ Zp is the cyclotomic   ∗ character), hence by NFk−1 · χ2v (u) − 1, for each u ∈ OF,v (recall that the isomorphism of class field v /Qp ∼

−1 ∗ theory I(Gal(Qab p /Qp )) −→ Zp is given by χcycl , not χcycl , as we are using geometric Frobenius elements). M + ± This shows that j(A[pM ]+ v ) ⊆ A1 [p ]v , as claimed. The statements about jv follow from the exact sequence ∂

0 −→ Ker(jv+ ) −→ Ker(j) −→ Ker(jv− )−→Coker(jv+ ) −→ Coker(j) −→ Coker(jv− ) −→ 0, as j is an (a, a)-morphism and pa · Im(∂) = 0. e 1 (A[pM ]) −→ H e 1 (A1 [pM ]). (12.8.4.3) Corollary. The maps j and jv+ (v | p) induce a (3a, 3a)-morphism H f f Proof. Combine Proposition 12.8.4.2 with Lemma 12.8.2.5. 399

(12.8.4.4) Proposition. (i) (∀v ∈ Σ0 ) (∀Z = A, A1 ) H 0 (Gv , Z) is finite. (ii) Put b = maxv∈Σ0 (bv ) and b1 = maxv∈Σ0 (b1,v ), where bv = Tamv (T, p), b1,v = Tamv (T1 , p). If M ≥ max{b + 1, ordp (2), 1 + max{ordp (2(N v + 1)) | v - p, ordv (n(g)) = 1}}, then

(∀v ∈ Σ0 − {q1 , . . . , qs }) bv = bv,1 , (∀i = 1, . . . , s) bqi = 0, b1,qi = ordp (2).

0 1 1 (iii) (∀v ∈ Σ ) `O Hur (Gv , A) = bv , `O Hur (Gv , A1 ) = b1,v . 0 bv i Iv M Iv (iv) (∀v ∈ Σ ) p · H (Gv /Iv , A /p A ) = 0, pb1,v · H i (Gv /Iv , AI1v /pM AI1v ) = 0. Proof. (i) This follows from the vanishing H 0 (Gv , V ) = H 0 (Gv , V1 ) = 0 (Proposition 12.4.8.4). (ii) Fix i ∈ {1, . . . , s}. As T is unramified at qi , we have bqi = 0. In order to compute b1,qi , we use the exact sequence of O[Gqi ]-modules 0 −→ O(1) ⊗ µ1 −→ T1 −→ O ⊗ µ1 −→ 0, ∼

in which µ1 is an unramified character satisfying µ1 (qi ) = −1. As Iqi acts trivially on T1 /pM T1 −→ T /pM T , we have ∼

0

1 Hcont (Iqi , T1 )tors −→ O/pM ⊗ µ1

for some M 0 ≥ M ≥ ordp (2), hence Fr(qi )=1

1 Hcont (Iqi , T1 )tors

∼

−→ O/2O =⇒ b1,qi = ordp (2).

If v ∈ Σ0 − {q1 , . . . , qs }, then

  
Fr(v)=1  Fr(v)=1 bv = `O H 1 (Iv , T )tors = `O AIv /(AIv )div ,

and similarly for A1 and b1,v . If ordv (n(g)) 6= 1, then V Iv = V1Iv = 0 (by Lemma 12.4.5(ii)), hence (AIv )div = (AI1v )div = 0, which implies that

bv = `O H 0 (Gv , A) = `O H 0 (Gv , A1 ) = b1,v (using the fact that pbv kills H 0 (Gv , A) and A[pbv +1 ] = A1 [pbv +1 ]). If ordv (n(g)) = 1, then the proof of Proposition 12.8.1.4 shows that there are exact sequences of O[Gv ]modules 0 −→ O(1) ⊗ µ −→ T −→ O ⊗ µ −→ 0,

0 −→ O(1) ⊗ µ −→ T1 −→ O ⊗ µ −→ 0

(where µ : Gv −→ {±1} is an unramified character). Consider the corresponding (non-trivial) extension classes ∼

1 ∗ b [T ], [T1] ∈ Hcont (Gv , O(1)) = OF,v ⊗O −→ O

(the last isomorphism being induced by the valuation ordv ). It follows from the exact sequence [T ]⊗1

1 0 −→ O(1) ⊗ µ −→ T Iv −→ O ⊗ µ−−−−→O ⊗ µ −→ Hcont (Iv , T ) −→ O(−1) ⊗ µ −→ 0

that ( bv =

`O (O/[T ]O) ,

if µ(v) = 1

`O (O/([T ], 2)O) ,

if µ(v) = −1

(and similarly for T1 and b1,v ). If µ(v) = 1, then 400

`O



T /pk T


Iv 

= k + min(k, bv ),

`O



T1 /pk T1


Iv 

= k + min(k, b1,v )

(∀k ≥ 0).

The assumption M > bv then implies that 2M > M + bv = `O



T /pM T


Iv 

= `O



T1 /pM T1


Iv 

= M + min(M, b1,v ) =⇒ bv = b1,v .

If µ(v) = −1, then either `O (O/[T ]O) ≥ ordp (2) =⇒ bv = ordp (2) or `O (O/[T ]O) < ordp (2) =⇒ bv = `O (O/[T ]O) = `O ((T /pM T )Iv ) − M (and similarly for T1 and b1,v ), hence

bv = min ordp (2), `O ((T /pM T )Iv ) − M = min ordp (2), `O ((T1 /pM T1 )Iv ) − M = b1,v . (iii) It is enough to treat A: the first term in the exact sequence

1 (AIv )div /(Fr(v) − 1) −→ Hur (Gv , A) = (AIv )/(Fr(v) − 1) −→ AIv /(AIv )div /(Fr(v) − 1) −→ 0 1 1 is a quotient of Hur (Gv , V ) = (V Iv )/(Fr(v) − 1) = 0 (as dim Hur (Gv , V ) = dim H 0 (Gv , V ) = 0), while the the third term has the same length as

AIv /(AIv )div


Fr(v)=1

Fr(v)=1

1 = Hcont (Iv , T )tors

.

(iv) As X := AIv /pM AIv is finite, we have


`O H 0 (Gv /Iv , X) = `O H 1 (Gv /Iv , X) = `O (X/(Fr(v) − 1)X) ≤ `O (AIv )/(Fr(v) − 1) =
1 = `O Hur (Gv , A) = bv . The argument for A1 is the same. e 0 (A) = 0. If M > c, then pc · H e 0 (A1 ) = 0. (12.8.4.5) Lemma. pc · H f f e 0 (A) ⊂ H 0 (GF , A) ⊂ A[pc ]. If M > c, then A1 [pc+1 ] = A[pc+1 ], hence Proof. By definition of c, we have H f e 0 (A1 ) ⊂ H 0 (GF , A1 ) ⊂ A1 [pc ]. H f e 1 (A[pM ]) −→ H e 1 (A)[pM ] (resp., H e 1 (A1 [pM ]) −→ (12.8.4.6) Proposition. The canonical morphism H f f f e 1 (A1 )[pM ]) is a (c, b)-morphism (resp., a (c, b1 )-morphism). H f Proof. The natural exact sequences of complexes $M

• • • • 0 −→ Cur (Gv , A[pM ]) −→ Cur (Gv , A)−−→Cur (Gv , A) −→ Ccont (Gv /Iv , AIv /pM AIv ) −→ 0

$M

• • • + + 0 −→ Ccont (Gv , A[pM ]+ v ) −→ Ccont (Gv , Av )−−→Ccont (Gv , Av ) −→ 0

give rise to exact triangles u gf (A[pM ]) −→ RΓ gf (A)−→Y, RΓ

0

u g Y −→RΓ f (A) −→

M v∈Σ0

401

• Ccont (Gv /Iv , AIv /pM AIv )

(v ∈ Σ0 ) (v | p)

such that u0 ◦ u = $M · id. It follows from Proposition 12.8.4.4(iv) and Lemma 12.8.2.5 that the maps e i (A) are (b, b)-morphisms, hence the O-modules H i (u0 ) : H i (Y ) −→ H f   e i (A[pM ]) −→ H e i (A)[pM ] ⊆ Ker(H i (u0 )) Coker H f f are all killed by pb . According to Lemma 12.8.4.5, the kernel   e f1 (A[pM ]) −→ H e f1 (A)[pM ] ⊆ H 0 (X) ⊆ H e f0 (A) Ker H is killed by pc . The argument for A1 is the same (if M ≤ c, then the statement about the kernel is trivially true). L

L

gf (T )⊗O Lp /O −→ RΓ gf (A) (resp., RΓ gf (T1 )⊗O Lp /O −→ (12.8.4.7) Proposition. The canonical map RΓ i+1 gf (A1 )) defines, for each i, a (b, b)-morphism H e (T ) e i (A)/H e i (A) (resp., a (b1 , b1 )-morphism RΓ −→ H f f f div tors i+1 i i e (T1 ) e (A1 )/H e (A1 ) ). H −→ H f

f

tors

f

div

Proof. In the exact triangle L

gf (T )⊗O Lp /O −→ RΓ gf (A) −→ Z = RΓ

M

Errv (T ),

v∈Σ0

the cohomology groups H i (Errv (T )) vanish for i 6= 0, 1 (resp., are killed by pbv for i = 0, 1), by 7.6.9. Combining the corresponding cohomology sequence with the Snake Lemma applied to 0

−→

e i (T ) ⊗O Lp /O H f   y

−→

H i (Z)   y

−→

e i+1 (T ) H f  tors  y

−→

0

0

−→

e i (A) H f div

−→

e i (A) H f

−→

e i (A)/H e i (A) H f f div

−→

0

yields the result for T, A (as the left vertical map is surjective). The argument for T1 , A1 is the same. (12.8.4.8) Corollary. The maps from 12.8.4.7 induce a (b, 2b)-morphism (resp., a (b1 , 2b1 )-morphism)   2 M 1 1 M 2 M 1 1 e e e e e e Hf (T )[p ] −→ Hf (A)/Hf (A)div [p ] (resp., Hf (T1 )[p ] −→ Hf (A1 )/Hf (A1 )div [pM ]). (12.8.4.9) Proposition. (i) There exists an injective morphism of O[GF ]-modules ν : T −→ T ∗ (1) satisfying ν(T ) 6⊂ pT ∗ (1) and pc · Coker(ν) = 0. This morphism is skew-symmetric (i.e. ν ∗ (1) = −ν) and unique up to a scalar multiple by an element of O∗ . For each prime v | p of F , ν(Tv+ ) ⊗O Lp = (Vv− )∗ (1). (ii) If M > c, then there exists an injective morphism of O[GF ]-modules ν1 : T1 −→ T1∗ (1) satisfying ν1 (T1 ) 6⊂ pT1∗(1) and pc · Coker(ν1 ) = 0. It is skew-symmetric (i.e. ν1∗ (1) = −ν1 ) and unique up to a scalar − ∗ multiple by an element of O∗ . For each prime v | p of F , ν1 ((T1 )+ v ) ⊗O Lp = ((V1 )v ) (1). Proof. (i) We know that there exists a skew-symmetric (i.e. such that νV∗ (1) = −νV ) isomorphism of ∼ Lp [GF ]-modules νV : V −→ V ∗ (1). Let k ∈ Z be the smallest integer such that $k νV (T ) ⊆ T ∗ (1) and put ∼ ν = $k νV . As ν(T ) ⊂ T ∗ (1) are GF -stable O-lattices in V ∗ (1) −→ V satisfying ν(T ) 6⊂ pT ∗ (1), we must ∼ c ∗ ∗ have p T (1) ⊆ ν(T ). The absolute irreducibility of V −→ V (1) implies that νV is unique up to a scalar multiple in L∗p , hence ν is unique up to an element of O∗ . If v | p, then the vanishing of HomO[Gv ] (Vv+ , (Vv+ )∗ (1)) = HomO[Gv ] (Vv+ , Vv− ) = 0 implies that ν(Vv+ ) = (Vv− )∗ (1). ∼ (ii) The isomorphism of O[GF ]-modules T /pc+1 T −→ T1 /pc+1 T1 implies that the image of the subring R1 := Im(O[GF ] −→ EndO (T1 )) ⊆ EndO (T1 ) = E in EndO (T1 /pc+1 T1 ) = E/pc+1 E contains pc EndO (T1 /pc+1 T1 ) = pc E/pc+1 E; thus (R1 ∩ pc E) + pc+1 E = pc E. Nakayama’s Lemma then implies that R1 ∩ pc E = pc E, hence R1 ⊇ pc EndO (T1 ). We apply the argument from (i) to T1 . 402

(12.8.4.10) Corollary. Consider Greenberg’s local conditions (12.8.4.1.1) for Z = T ∗ (1) (with T ∗ (1)+ v = T ∗ (1) ∩ (Vv− )∗ (1) for v | p), and similarly for T1∗ (1). The map ν induces, for each i, a (c, c)-morphism e i (T ) −→ H e i (T ∗ (1)). If M > c, then ν1 induces, for each i, a (c, c)-morphism H e i (T1 ) −→ H e i (T1∗ (1)). H f f f f Proof. It follows from Proposition 12.8.4.9(i) that ν defines an injective morphism of complexes ν∗ : e • (T ) −→ C e• (T ∗ (1)). As pc kills T ∗ (1)+ /ν(T + ) (resp., T ∗ (1)Iv /ν(T Iv )) for v | p (resp., for v ∈ Σ0 ), it C v v f f also kills Coker(ν∗ ). We conclude by Lemma 12.8.2.5 (and similarly for T1 and ν1 ). (12.8.4.11) Proposition. (i) There exists an O-module W 0 of finite length and a (b + c + 2 ordp (2), 0)e 2 (T ) morphism H −→ W 0 ⊕ W 0 . f tors (ii) If M > c, then there exists an O-module W10 of finite length and a (b + c + 2 ordp (2), 0)-morphism e 2 (T1 ) −→ W 0 ⊕ W 0 . H 1 1 f Proof. (i) The generalized Cassels-Tate pairing e f2 (T ) e f2 (T ∗ (1)) H ×H −→ Lp /O tors tors defined in 10.2.2 has left kernel killed by 2pb , according to Theorem 10.2.3 (the factor 2 comes from the contribution of the archimedean primes). The induced pairing id×2ν∗ e 2 e 2 (T ) e 2 (T ) e 2 (T ∗ (1)) h, i:H ×H −−−−→ Hf (T )tors × H −→ Lp /O f f f tors tors tors

is alternating, by Proposition 10.2.5. As 2pc · Coker(2ν∗ ) = 0, by Corollary 12.8.4.10, it follows that the e 2 (T ) kernel of h , i is killed by 4pb+c . The quotient of H by the kernel of h , i then admits a symplectic f tors pairing with values in Lp /O, hence is isomorphic to W 0 ⊕ W 0 for some O-module of finite length W 0 . (ii) The same argument applies to T1 . (12.8.4.12) Corollary. (i) Put W = W 0 [pM ]. Then there is a (b + c+ 2 ordp (2), b + c+ 2 ordp (2))-morphism e 2 (T )[pM ] −→ W ⊕ W . H f e 2 (T1 )[pM ] −→ (ii) If M > c, put W1 = W10 [pM ]. Then there is a (b+c+2 ordp (2), b+c+2 ordp (2))-morphism H f W1 ⊕ W1 . (12.8.4.13) Putting together 12.8.4.6-12, we obtain, for M ≥ max(b + 1, c + 1, ordp (2), 1 + max{ordp (2(N v + 1)) | v - p, ordv (n(g)) = 1}),

(12.8.4.13.1)

a chain of (αi , βi )-morphisms fi
⊕r

f1


⊕r

f2

f3

f4

e f2 (T )[pM ]−−→H e f1 (A)[pM ]−−→H e f1 (A[pM ])−−→ ⊕H
⊕r1
⊕r1 f4 f5 f6 f7 e 1 (A1 [pM ])−−→ e 1 (A1 )[pM ]−−→ e 2 (T1 )[pM ]−−→ −−→H H O/pM ⊕H O/pM ⊕ W1 ⊕ W1 f f f O/pM

⊕ W ⊕ W −−→ O/pM

with (α1 , β1 ) = (2b + 2c + 4 ordp (2), 2b + 2c + 4 ordp (2)) (α2 , β2 ) = (b, 2b) (α3 , β3 ) = (b + c, b + c) (α4 , β4 ) = (3a, 3a) (α5 , β5 ) = (b1 + c, b1 + c) (α6 , β6 ) = (3b1 , 3b1 ) (α7 , β7 ) = (b1 + c + 2 ordp (2), b1 + c + 2 ordp (2)) Their composition is an (α, β)-morphism 403

O/pM


⊕r

⊕ W ⊕ W −→ O/pM


⊕r1

⊕ W1 ⊕ W1 ,

(12.8.4.13.2)

where α = 3a + 4b + 5b1 + 5c + 6 ordp (2) ≤ 3a + 9b + 5c + 10 ordp (2) β = 3a + 5b + 5b1 + 5c + 6 ordp (2) ≤ 3a + 10b + 5c + 11 ordp (2) (as b1 ≤ b + ordp (2)). e 2 (T ) (12.8.4.14) Proposition. Let t ≥ 0 be an integer such that pt · H = 0. If M satisfies (12.8.4.13.1) f tors and M > t + 6a + 19b + 10c + 21 ordp (2), then (i) e h1f (F, V ) = r ≡ r1 = e h1f (F, V1 ) (mod 2). (ii) h1f (F, V ) − ran (F, g) ≡ h1f (F, V1 ) − ran (F, g1 ) (mod 2). e 2 (T ) Proof. (i) The assumption pt · H = 0 implies that pt W = 0. Applying Proposition 12.8.2.3 to the f tors morphism (12.8.4.13.2) then gives r ≡ r1 (mod 2) (as M > t + α + β). The statement (ii) is a consequence of (i) and Corollary 12.8.1.5. (12.8.4.15) Proposition. Assume that p 6= 2, the residual representation of V is absolutely irreducible,   k−1 ∗ 2 (∀v | n(g), v - p) Tamv (T, p) = 0 and (∀v | p) (∃u ∈ OF,v ) p - (NFv /Qp · χv )(u) − 1 (the last condition holds with u = −1 if [Fv : Qp ] is odd for all v | p). Then the conclusions of 12.8.4.14 hold if M ≥ 1 and M ≥ 1 + max{ordp (N v + 1) | v - p, ordv (n(g)) = 1}. Proof. The assumptions imply that a = b = c = ordp (2) = 0, hence (12.8.4.13.2) is an isomorphism, which proves the congruence r ≡ r1 (mod 2). We conclude as in 12.8.4.14. (12.9) Parity results in the dihedral case In this section we assume that f ∈ Sk (n, ϕ), g = f ⊗ χ ∈ Sk (n(g), 1) and V = V (g)(k/2) = V (f )(k/2) ⊗ χ are as in 12.6.4.1 (i.e. f is p-ordinary and ϕ = χ−2 ). (12.9.1) Fix a prime P | p of F . Throughout 12.9, we assume that the following condition is satisfied: (12.9.1.1) If 2 | [F : Q], then

(∃q 6= P ) 2 - ordq (n(g)).

If 2 | [F : Q], then fix such a prime q. It follows from the remarks at the end of 12.5.5 that if k 6= 2, then q - p. (12.9.2) Fix a totally imaginary quadratic extension K/F satisfying the following two conditions (in which η = ηK/F : A∗F /F ∗ −→ {±1} denotes the quadratic character attached to K/F ):
(12.9.2.1) dK/F , n(g)(P ) = (1).
(12.9.2.2) η n(g)(P ) = (−1)[F :Q]−1 . For example, any K in which all primes of F dividing n(g)(P ) split (resp., in which q is inert and all primes v 6= q dividing n(g)(P ) split) will do if 2 - [F : Q] (resp., if 2 | [F : Q]). (12.9.3) Similarly as in 12.6.4.6, we set K[P ∞ ] =

[

K[P n ],

G(P ∞ ) = Gal(K[P ∞ ]/K),

n≥1

K[p∞ ] =

[

K[pn ],

G(p∞ ) = Gal(K[p∞ ]/K).

n≥1

As K[P ∞ ] ⊂ K[p∞ ], there is a canonical epimorphism G(p∞ ) −→ G(P ∞ ). The torsion subgroup G(P ∞ )tors of G(P ∞ ) is finite and the quotient G(P ∞ )/G(P ∞ )tors is isomorphic to ZrpP , where rP = [FP : Qp ]. Fix a ∗ character β0 : G(P ∞ )tors −→ Lp . 404

(12.9.4) Lemma. Assume that K/F is as in 12.9.2 and β : G(P ∞ ) −→ C∗ is a ring class character of conductor c(β) = P n (n ≥ 0). If P splits in K/F or if n >> 0, then ran (K, g, β) ≡ 1 (mod 2). Proof. We apply Proposition 12.6.3.8. If P splits in K/F , then η(P ) = 1 and n(g)(dK/F P ) = (1), which implies that R(β)0 = R(β)0 ∩ {P } = ∅ = R(β)− ∩ {P } for any ring class character β, hence   ε(β) = (−1)[F :Q] η(n(g)) = (−1)[F :Q] η n(g)(P ) = −1, by (12.9.2.2). Similarly, if c(β) = P n with n >> 0, then R(β)0 = R(β)0 ∩ {P } = ∅ = R(β)− ∩ {P }. If P is ramified in K/F , then n(g)(dK/F ) = n(g)(P ) and R(1)− ∩ {P } = ∅, hence   ε(β) = (−1)[F :Q] η n(g)(P ) = −1. If P is inert in K/F , then −ε(β) = (−1)[F :Q]−1 η(n(g)) (−1)|R(1)

−

∩{P }|

= η(P )ordP (n(g)) (−1)|R(1)

−

∩{P }|

= 1,

where the last equality follows from the formulas used in the proof of Corollary 12.6.3.13 (recall that π(g)P is not supercuspidal, since P | p and f is p-ordinary). ∗

(12.9.5) Theorem. Let g = f ⊗ χ ∈ Sk (n(g), 1), P | p, K/F and β0 : G(P ∞ )tors −→ Lp be as in 12.9.1-3. Assume that the form g does not have CM by any totally imaginary quadratic extension K 0 of F contained (P ) in K[P ∞ ]Ker(β0 ) . If k 6= 2, assume that (dK/F , (p)) = 1. Then, for any ring class field character of finite order ∗

β : G(p∞ ) −→ Lp such that β|G(p∞ )tors is induced by β0 via the canonical map G(p∞ )tors −→ G(P ∞ )tors , we have ran (K, g, β) ≡ h1f (K, V ⊗ β) (mod 2). If c(β) = P n and (P splits in K/F or n >> 0), then ran (K, g, β) ≡ h1f (K, V ⊗ β) ≡ 1 (mod 2). Proof. The proof consists of three steps, the last of which is inspired by the proof of Theorem B in [Ne 3]. (12.9.5.1) Reduction to the case k = 2 If k = 2, then we go directly to (12.9.5.2). If k 6= 2, then we can embed (the p-stabilization f 0 of) f into a Hida family, as in 12.7.5: there exists an arithmetic point P above (Pk,ε ) such that f 0 = fP . If p 6= 2 (resp., if p = 2), then the assumption (12.7.9.1) (resp., (12.7.11.1)) is satisfied – possibly after enlarging Lp if p = 2 – and we have g = gP (resp., g = gPe). As in 12.7.9-11, choose a character ε0 (resp., εe0 ) of sufficiently large order and an arithmetic point P 0 (resp., Pe0 ) above P2,ε0 Λ0 ∩ Λ. We obtain a p-ordinary cuspidal eigenform f 0 = fP 0 of weight (2, . . . , 2) and its twist g 0 = f 0 ⊗ χ0 ∈ S2 (n(g 0 ), 1). We must check that the form g 0 also satisfies (12.9.1.1) and (12.9.2.1-2). We use repeatedly the equality n(g)(p) = n(g 0 )(p) proved in Corollary 12.7.14.3(ii). Firstly, if 2 | [F : Q] and if 2 - ordq (n(g)), then q - p (as (P ) k 6= 2), hence 2 - ordq (n(g 0 )); thus (12.9.1.1) holds. Secondly, the assumption (dK/F , (p)) = 1 implies that (dK/F , n(g 0 )(P ) ) = (1), proving (12.9.2.1). Thirdly, we have assumed that the order of ε0 was large. This implies that (n(g 0 )St , (p)) = (1), hence π(g 0 )v is in the principal series (=⇒ 2 | ordv (n(g 0 ))), for each prime v | p. The same holds for g (as k 6= 2), hence ηK/F (n(g 0 )(P ) ) = ηK/F (n(g 0 )(p) ) = ηK/F (n(g)(p) ) = ηK/F (n(g)(P ) ), 405

proving (12.9.2.2). Finally, the form g 0 has CM by a quadratic extension K 0 of F iff g has (by 12.7.14.5(v)), which implies that the assumptions of Theorem 12.9.5 are also satisfied by g 0 . The statement of Proposition 12.7.14.5(iv) then shows that Theorem 12.9.5 holds for g 0 iff it holds for g. (12.9.5.2) Passage to a Shimura curve Thanks to 12.9.5.1, we can assume that k = 2. Put R = {v | n(g)(P ) , 2 - ordv (n(g)), v is inert in K/F }; the condition (12.9.2.2) implies that |R| ≡ [F : Q]−1 (mod 2). Fix an archimedean prime τ1 of F and denote by B the quaternion algebra B over F ramified at the set Ram(B) = {v|∞, v 6= τ1 } ∪ R. By construction, (∀v ∈ R)

π(g)v is not in the principal series

(using (12.3.9.2-3)), which implies (cf. 12.4.7) that π(g) is attached by the Jacquet-Langlands correspondence ∗ to an irreducible automorphic representation π 0 of BA with trivial central character. More precisely, we have (∀v 6∈ Ram(B))

∼

πv0 −→ π(g)v

∼

(as representations of Bv∗ −→ GL2 (Fv )). Fix an F -embedding K ,→ B (it exists, since K ⊗F Fv is a field for each prime v ∈ Ram(B)). One can describe an explicit level subgroup of π 0 as follows (see [Zh 2], 1.2; [Cor-Va], (6)): write n(g) = n(g)(P ) P δ (δ ≥ 0) and fix an Eichler order R0 ⊂ B of level P δ such that the conductor of the OF -order O = OK ∩ R0 ⊂ OK is a power of P . By construction, there exists an ideal J ⊂ OK such that Y NK/F (J) λ = n(g)(P ) . λ∈R

Fix such an ideal J and put R = O + (J ∩ O) · R0 . Then we have, for each finite prime v of F , ∗

dimC (πv0 )Rv = 1

(12.9.5.2.1)

([Zh 1], Thm. 3.2.2). In what follows, we are going to use the notation from [Ne 4] (see also [Cor-Va]). Denote by NH the Shimura b ∗ , where H = R b∗ . The complex points of NH ⊗F,τ1 C are curve over F attached to the subgroup H Fb ∗ ⊂ B ∼ ∗ ∗ ∗ ∗ b b equal to B \((C − R) × B /H F ), with B acting on C − R via a fixed isomorphism B ⊗F,τ1 R −→ M2 (R). ∗ If F = Q and R = ∅, then NH is a classical modular curve and we put NH = NH ∪{cusps}. In all other cases ∗ ∗ NH is a proper curve over Spec(F ) and we put NH = NH . In general, the curve NH is not geometrically ◦ ∗ irreducible; one defines its “Jacobian” as J(NH ) = PicN ∗ /F . H

The multiplicity one result (12.9.5.2.1) implies that the representation π 0 is generated by an automorphic form Φ (unique up to a scalar multiple) of level H. In geometric terms, such a form defines a one-dimensional subspace ∗ ∗ C · Φ ⊂ Γ(NH , Ω1N ∗ /F ) ⊗F,τ1 C = Γ(J(NH ), Ω1J(N ∗ )/F ) ⊗F,τ1 C, H

H

b ∗ acts with the same eigenvalues as it does on the one-dimensional on which the level H Hecke algebra TH of B space ∗

(π 0∞ )H = ⊗0v-∞ (πv0 )Rv . The quotient abelian variety ∗ ∗ A1 = J(NH )/AnnTH (C · Φ) · J(NH )

is an abelian variety of GL2 -type defined over F . More precisely, the number field L1 generated by the ∼ Hecke eigenvalues of TH acting on C · Φ is totally real, it is equipped with a natural isomorphism L1 −→ 406

EndF (A1 ) ⊗ Q, and its degree is equal to [L1 : Q] = dim(A1 ). Denoting by p1 the prime of L1 induced by ιp , then the generalized Eichler-Shimura relation implies that there is an isomorphism of GF,S -modules ∼

Vp (A1 ) ⊗(L1 )p1 Lp −→ V (g)(1) = V (cf. the discussion in [Ne 4], Sect. 1). (12.9.5.3) CM points The set CMH of CM points by K inside b ∗ /H Fb ∗ ) (NH ⊗F,τ1 C)(C) = B ∗ \((C − R) × B b ∗ /H Fb ∗ ), where z ∈ C − R is the unique fixed point of is equal to K ∗ \({z} × B ∼

K ∗ ,→ B ∗ ,→ (B ⊗F,τ1 R)∗ −→ GL2 (R) b ∗ , the CM point x = [z, g] ∈ CMH represented by (z, gH Fb ∗) is defined over with Im(z) > 0. For each g ∈ B the ring class field K[c], whose conductor c ⊂ OF is determined by bc∗ = K b ∗ ∩ gH Fb ∗ g −1 = K b ∗ ∩ gR b∗ Fb ∗ g −1 . O We say (1) that c is the conductor of x and we write x ∈ CMH (c). The next step is to map CM points to A1 . In the classical case F = Q, R = ∅ this is done using a suitable multiple m(∞) of the cusp ∞; in the remaining cases, there is a morphism ∗ ∗ ι = ιH Fb∗ : NH −→ J(NH ) ∗ defined using a suitable multiple of the so-called “Hodge class” ([Zh 1], p. 30; [Cor-Va], 3.5, for MH instead ∗ of NH , [Ne 4], 1.19). Denote by ι1 the composite map ι

∗ ∗ ι1 : CMH ,→ NH ,→ NH −−→J(NH ) −→ A1 . ∼

By construction, the quaternion algebra B is unramified at P and RP ⊂ BP −→ M2 (FP ) is an Eichler order of level P δ . This implies that for each n >> 0 there exists a “good” CM point x ∈ CMH (P n ) in the sense of ([Cor-Va], Def. 1.6). The main result of [Cor-Va] in the indefinite case (Thm. 4.1) in the case of a trivial central character ω = 1 states that, for each n >> 0 and every “good” x ∈ CMH (P n ), there exists a ring class character β : G(P ∞ ) −→ L∗p (after enlarging Lp if necessary) satisfying c(β) = P n and β|G(P ∞ )tors = β0 , for which X eβ (ι1 (x)) = β −1 (σ) σ(ι1 (x)) 6= 0 ∈ A1 (K[P n ]) ⊗ C. (12.9.5.3.1) σ∈Gal(K[P n ]/K)

Fix such n >> 0, x ∈ CMH (P n ) and β. As H(P n )Ker(β) ⊂ H(P ∞ )Ker(β0 ) , our assumptions imply that the form g does not have CM by any totally imaginary quadratic extension K 0 /F contained in H(P n )Ker(β) (hence the abelian variety A1 does not acquire CM over any such extension K 0 ). This fact together with the non-triviality statement (12.9.5.3.1) imply, by an Euler system argument ([Ne 4], Thm. 3.2), that h1f (K, V ⊗ β) = 1. On the other hand, we can also assume that n >> 0 is big enough in order to apply Lemma 12.9.4, which then yields (1)

Note that the conventions concerning ring class fields used by Cornut and Vatsal in [Cor-Va] are not the same as ours, as they treat Hilbert modular forms with a possibly non-trivial (unramified) character. 407

ran (K, g, β) ≡ 1 ≡ h1f (K, V ⊗ β) (mod 2) for the chosen ring class character β : G(P ∞ ) −→ L∗p . Applying Corollary 12.6.4.8, we deduce the desired congruence ran (K, g, β 0 ) ≡ h1f (K, V ⊗ β 0 ) (mod 2) for every ring class character of finite order β 0 : G(p∞ ) = Gal(K[p∞ ]/K) −→ L∗p for which β 0 |G(p∞ )tors is induced by β0 via the canonical map G(p∞ )tors −→ G(P ∞ )tors . If, in addition, c(β 0 ) = P n and (P splits in K/F or n >> 0), then ran (K, g, β 0 ) ≡ 1 (mod 2), by Lemma 12.9.4. Theorem is proved. (12.9.6) Proposition. Let g = f ⊗ χ ∈ Sk (n(g), 1), P | p and K/F be as in 12.9.1-2. We say that a totally imaginary quadratic extension K 0 /F is exceptional, if K 0 ⊂ K[P ∞ ] and the form g has CM by K 0 . (i) If K 0 /F is exceptional, then dK 0 /F = P m (m ≥ 0). (ii) If 2 | [F : Q], then there is no exceptional extension K 0 /F . ∞ (iii) If K 0 /F is exceptional, p 6= 2 and K 0 ⊂ K[P ∞ ]G(P )tors , then K 0 = K and dK/F = P m (m ≥ 0). ∞ (iv) If 2 - [F : Q] and if P splits in K/F , then there is no exceptional K 0 ⊂ K[P ∞ ]G(P )tors . Proof. (i) We have dK 0 /F | n(g) (since g has CM by K 0 ). As K[P ∞ ]/F is ramified only at primes dividing dK/F P ∞, any finite prime ramified in K 0 /F must divide ( (1), if P - n(g)
n(g), dK/F P = P, if P | n(g). (ii) Assume that K 0 /F is exceptional. As dK 0 /F = P m by (i), each prime v | n(g)(P ) is unramified in K 0 /F , hence 2 | ordv (n(g)), by Proposition 12.6.5.2(vi). This means that the ideal n(g)(P ) is a square, which contradicts (12.9.2.2). ∞ ∼ (iii) As Gal(K[P ∞ ]G(P )tors /K) = G(P ∞ )/G(P ∞ )tors −→ ZrpP is a pro-p-group and p 6= 2, the only ∞ quadratic subextension K 0 /F of K[P ∞ ]G(P )tors /K is K 0 = K; we apply (i). (iv) Class field theory gives and exact sequence ∗ 0 −→ OK /OF∗ −→ G0 −→ Gal(K[P ∞ ]/K[1]) −→ 0, ∗ ∗ ∗ ∗ in which G0 = (OF,P × OF,P )/∆(OF,P ), where ∆ is the diagonal map. Put µ = OF,P 0 0 group and we have Gtors = Im(µ × {1} −→ G ). It follows that the reciprocity map

A∗K /K ∗ A∗F −→ Gal(K[P ∞ ]G(P

∞

)tors


tors

; this is a finite

/K)

factors through A∗K /K ∗ A∗F ((µ × {1}) ×

Y

∗ OK,w ).

(12.9.6.1)

w-P ∞ ∞

0

If K /F is any totally imaginary quadratic extension contained in K[P ∞ ]G(P )tors with dK 0 /F = P m (m ≥ 0), applying the norm NK 0 /F to (12.9.6.1) shows that the quadratic character η 0 = ηK 0 /F : A∗F /F ∗ −→ {±1} attached to K 0 /F factors through Y ∗ A∗F /F ∗ A∗2 OF,v ). F (µ × v-P ∞

As −1 ∈ µ, it follows that the id`ele x = (xv ) defined by ( 1, xv = −1, 0

is contained in Ker(η ), which contradicts the fact that 408

v|∞ v-∞

η 0 (−x) =

Y

ηv (−1) = (−1)[F :Q] = −1.

v|∞

Combined with (i), this contradiction proves the statement (iv). (12.9.7) Theorem. Let g = f ⊗ χ ∈ Sk (n(g), 1), P | p and K/F be as in 12.9.1-2. Denote by K∞ = ∞ [F :Q] K[p∞ ]G(p )tors the unique Zp -extension of K contained in K[p∞ ]. If 2 - [F : Q], assume that, either (i) P splits in K/F , or (ii) p 6= 2 and (g does not have CM by K or there exists a finite prime v 6= P ramified in K 0 /F ). ∗ Then, for each character of finite order β : Gal(K∞ /K) −→ Lp , we have ran (K, g, β) ≡ h1f (K, V ⊗ β) (mod 2). If c(β) = P n and (P splits in K/F or n >> 0), then ran (K, g, β) ≡ h1f (K, V ⊗ β) ≡ 1 (mod 2). Proof. Thanks to the assumptions (i) or (ii), Proposition 12.9.6 implies that g does not have CM by any ∞ totally imaginary quadratic extension K 0 /F contained in K[P ∞ ]G(P )tors ; we apply Theorem 12.9.5 with β0 = 1. ∗

(12.9.8) Theorem. Let g = f ⊗ χ ∈ S2 (n(g), 1), P | p, K/F and β0 : G(P ∞ )tors −→ Lp be as in 12.9.1-3. Set O = OL,p and Λ = O[[G(P ∞ )]]. Fix a GF -stable O-lattice T ⊂ V and put A = V /T . Assume that g does not have CM by any totally imaginary quadratic extension K 0 /F contained in K[P ∞ ]Ker(β0 ) . Then we have, using the notation from 10.7.16 and 12.6.4.9-10: e j (K[P ∞ ]/K, T )(β0±1 ) = cork (β±1 ) H e j (KS /K[P ∞ ], A)(β0±1 ) = m(g, K; P ) + 1. (i) (∀j = 1, 2) rk (β±1 ) H f,Iw f Λ

(ii) cork

±1 (β ) Λ 0

0

±1

str SA (K[P ∞ ])(β0

)

Λ

0

= 1.

(iii) If R, g = gP and g 0 = gP are as in 12.7.10 (resp. 12.7.11), denote R = R[[G(P ∞ )]] and q = R · Iβ0 ∈ Spec(R) (ht(q) = 0), where Iβ0 = Ker(β0 : O[G(P ∞ )tors ] −→ Lp ). Fix T ⊂ V as in 12.7.15.3. If (12.9.1.1) and (12.9.2.1-2) hold for n(g 0 ) and if m(g 0 , K; P ) = 0, then (∀j = 1, 2)

e j (K[P ∞ ]/K, T )q = rk rkR H f,Iw R q

 q

e j (K[P ∞ ]/K, T )ι H f,Iw

 q

= 1.

Proof. (i), (ii) The proof of Theorem 12.9.5 shows that the condition C(g, β0 ) from Proposition 12.6.4.12 is satisfied (with c = (1), s = 1, P1 = P , δ = 1); we apply Proposition 12.6.4.12(i)-(ii). (iii) The four ranks coincide, by Proposition 12.7.15.7(i) and 12.7.15.8(i); denote their common value by h, and set 1 e f,Iw m0 = rkΛ0(β0 ) H (K[P ∞ ]/K, T 0 )(β0 ) ,

using the notation of 12.7.15. According to Proposition 12.7.15.8(iii), we have h ≤ m0 ,

h ≡ m0 (mod 2);

on the other hand, m0 = m(g 0 , K; P ) + 1 = 1, by (i) applied to g 0 (note that g 0 does not have CM by any totally imaginary quadratic extension K 0 /F contained in K[P ∞ ]Ker(β0 ) , by Proposition 12.7.14.5), hence h = 1. (12.9.9) Corollary. Let g = f ⊗ χ ∈ S2 (n(g), 1), P | p and K/F be as in 12.9.1-2; set Λ = O[[G(P ∞ )]]. If g does not have CM by any totally imaginary quadratic extension K 0 /F contained in K[P ∞ ] (which is 409

automatic if 2 | [F : Q]), then: (i) For each q ∈ Spec(Λ) with ht(q) = 0, (∀j = 1, 2)

  e j (KS /K[P ∞ ], A)) = m(g, K; P ) + 1 dimκ(q) DΛ (H f q
str ∞ dimκ(q) DΛ (SA (K[P ])) q = 1.

(ii) If g = gP and g 0 = gP are as in 12.7.10 (resp. 12.7.11), set R = R[[G(P ∞ )]] and choose T ⊂ V as in 12.7.15.3. Assume that (12.9.1.1) and (12.9.2.1-2) hold for n(g 0 ) and m(g 0 , K; P ) = 0. Then, for each q ∈ Spec(R) with ht(q) = 0, e j (K[P ∞ ]/K, T )q = 1. dimκ(q) H f,Iw

(∀j = 1, 2)

(12.9.10) Howard [Ho 2] showed (combine his Thm. B with ([Cor-Va], Thm. 4.1) that if g = f ∈ S2 (n(g), 1) is p-ordinary, P = (p), ηK/F (n(g)) = (−1)[F :Q]−1 and a few more assumptions hold, then str corkO[[Γ]] SA (K∞ ) = 1,

where K∞ = K[P ∞ ]G(P

∞

)tors

and Γ = Gal(K∞ /K).

(12.9.11) Theorem. Let g = f ⊗ χ ∈ Sk (n(g), 1), P | p and K/F be as in 12.9.1. Fix a character ∗ (P ) β0 : G(P ∞ )tors −→ Lp . Assume that ((p)n(g), dK/F ) = (1), ηK/F (n(g)(p) ) = (−1)[F :Q]−1 and g does not

have CM by any totally imaginary quadratic extension K 0 /F contained in K[P ∞ ]Ker(β0 ) . Then we have, using the notation from 12.9.8:   e j (K[P ∞ ]/K, T )q = rk e j (K[P ∞ ]/K, T )ι = 1. (i) (∀j = 1, 2) rkR H H f,Iw f,Iw R q

q

q

(ii) For all but finitely many arithmetic points P 0 of R, all (co)ranks appearing in Proposition 12.9.8(i) for T 0 ⊂ V 0 = V (gP 0 ) and A0 = V 0 /T 0 , are equal to 1. Proof. The statement (ii) is an immediate consequence of (i) and the exact sequence in Proposition 12.7.15.5(i). In order to prove (i), choose an arithmetic point P 0 as in 12.9.5.1 and set g 0 = gP 0 . As (P ) observed in 12.9.5.1, we have m(g 0 , K; P ) = 0 and n(g 0 )(p) = n(g)(p) , hence ((p)n(g 0 ), dK/F ) = (1) and 0

ηK/F (n(g 0 )(P ) ) = (−1)m(g ,K;P ) ηK/F (n(g 0 )(p) ) = ηK/F (n(g)(p) ) = (−1)[F :Q]−1 , where the first equality follows from Proposition 12.6.4.9(i). Furthermore, g 0 does not have CM by any totally imaginary quadratic extension K 0 /F contained in K[P ∞ ]Ker(β0 ) , thanks to Proposition 12.7.14.5. It follows that Theorem 12.9.8(iii) applies to g 0 , which proves (i). (12.9.12) Corollary. Let g = f ⊗ χ ∈ Sk (n(g), 1), P | p and K/F be as in 12.9.1. Assume that (P ) ((p)n(g), dK/F ) = (1), ηK/F (n(g)(p) ) = (−1)[F :Q]−1 and g does not have CM by any totally imaginary quadratic extension K 0 /F contained in K[P ∞ ] (which is automatic if 2 | [F : Q]). Write g = gP , put R = R[[G(P ∞ )]] and choose T ⊂ V as in 12.7.15.3. Then (∀q ∈ Spec(R), ht(q) = 0) (∀j = 1, 2)

e j (K[P ∞ ]/K, T )q = 1. dimκ(q) H f,Iw

(12.9.13) Proposition. In the situation of 12.6.4.9, assume that g = f ⊗ χ ∈ Sk (n(g), 1) and ((p)n(g), ∗ (P1 ···Ps ) c dK/F ) = (1). Set K∞ = K[cP1∞ · · · Ps∞ ], Γ = Gal(K∞ /K) and fix a character β0 : Γtors −→ Lp . Assume that there exist infinitely many arithmetic points P 0 of R for which the condition C(gP 0 , β0 ) from 12.6.4.12 holds. Then   e j (K∞ /K, T )p = rk e j (K[P ∞ ]/K, T )ι = δ, (∀j = 1, 2) rkR H H f,Iw f,Iw R p

p

410

p


where δ ∈ {0, 1} is characterized by the formula ηK/F n(g)(p) = (−1)[F :Q]+δ . Proof. As m(gP 0 ) := m(gP 0 , K; P1 , . . . , Ps ) = 0 for all but finitely many arithmetic points P 0 of R, we choose P 0 such that g 0 := gP 0 satisfies C(g 0 , β0 ) and m(g 0 ) = 0. As in the proof of Theorem 12.9.11, we have (P1 ···Ps ) ((p)n(g 0 ), c dK/F ) = (1) and       0 ηK/F n(g 0 )(P1 ···Ps ) = (−1)m(g ) ηK/F n(g 0 )(p) = ηK/F n(g)(p) = (−1)[F :Q]+δ , hence we deduce from Proposition 12.6.4.12 that the value of e j (K∞ /K, V (g 0 )) m := rkΛ0(β0 ) H f,Iw is equal to m = m(g 0 ) + δ = δ. On the other hand, the rank 1 e f,Iw h := rkR H (K∞ /K, T )p p

satisfies h ≤ m, h ≡ m (mod 2), by Proposition 12.7.15.8(iii). It follows that h = δ, as claimed. (12.10) Proof of Theorem 12.2.3 In this section we prove Theorem 12.2.3. Recall that f ∈ Sk (n, χ−2 ) is p-ordinary, g = f ⊗ χ ∈ Sk (n(g), 1) ∼ and V = V (g)(k/2) = V (f )(k/2) ⊗ χ −→ V ∗ (1). (12.10.1) Descent properties. Assume that F0 /F is a finite solvable extension (not necessarily totally real) and F1 /F0 a finite abelian extension. Denote by g0 and g1 the base change of g to F0 and F1 , respectively (strictly speaking, we are abusing the language, as we should speak about the base change b be the character group of ∆ = Gal(F1 /F0 ). of the corresponding automorphic representations). Let ∆ b have values in L∗ . We are going to use Enlarging L if necessary, we can assume that all characters α ∈ ∆ the relations L(g1 , s) =

Y

Hf1 (F1 , V ) =

L(g0 ⊗ α, s),

b α∈∆

M

Hf1 (F0 , V ⊗ α),

e f1 (F1 , V ) = H

b α∈∆

M

e f1 (F0 , V ⊗ α) H

b α∈∆

(12.10.1.1) b α2 6= 1, then the functional equation relating L(g0 ⊗ α, s) and (see Proposition 12.5.9.2(iv)). If α ∈ ∆, L(g0 ⊗ α−1 , s) yields ran (F0 , g0 ⊗ α) = ran (F0 , g0 ⊗ α−1 ) (the archimedean L-factors take non-zero finite values at the central point). Similarly, Proposition 7.8.11 implies that e h1f (F0 , V ⊗ α) = e h1f (F0 , V ⊗ α−1 ). The H 0 -terms for α and α−1 in 12.5.9.2(iii) have the same dimension, hence h1f (F0 , V ⊗ α) = h1f (F0 , V ⊗ α−1 ). The preceding discussion yields ran (F1 , g1 ) ≡

X

ran (F0 , g0 ⊗ α) (mod 2)

α:∆−→{±1}

h1f (F1 , V ) ≡

X

h1f (F0 , V ⊗ α) (mod 2),

α:∆−→{±1}

which implies that 411

X

ran (F1 , g1 ) − h1f (F1 , V ) ≡


ran (F0 , g0 ⊗ α) − h1f (F0 , V ⊗ α)

(mod 2).

(12.10.1.2)

α:∆−→{±1}

(12.10.2) Reduction to the case F 00 = F 0 = F for the twists g ⊗ α (α : Gal(F 0 /F ) −→ {±1}) As each finite group of odd order is solvable, there exists a tower of fields F 0 = F0 ⊂ F1 ⊂ · · · ⊂ Fn = F 00 in which each layer Fi+1 /Fi is an abelian extension of odd degree. Applying (12.10.1.2) to each Fi+1 /Fi , as well as to F 0 /F , we deduce that X

ran (F 00 , g) − h1f (F 00 , V ) ≡


ran (F, g ⊗ α) − h1f (F, V ⊗ α)

(mod 2),

α:Gal(F 0 /F )−→{±1}

which means that it suffices to prove the congruences ?

ran (F, g ⊗ α) ≡ h1f (F, V ⊗ α) (mod 2)

(12.10.2.1)

for all α : Gal(F 0 /F ) −→ {±1}. (12.10.3) Reduction to the assumptions 12.2.3(1)-(3) Assume that, for each α : Gal(F 0 /F ) −→ {±1}, there exist distinct prime ideals qα,j - (p)n(f ⊗ α)cond(χ) (j = 1, . . . , rα , rα ≥ 1) of F and a p-ordinary newform fα ∈ Sk (n(f ⊗ α)Qα , χ−2 ) (Qα = qα,1 · · · qα,rα ) such that (∀j = 1, . . . , rα )

λfα (qα,j ) = −χ−1 (qα,j )(N qα,j )k/2−1

and λfα (v) ≡ λf ⊗α (v) (mod pM )

(12.10.3.1)

for a set of primes v - (p)n(f ⊗ α)Qα of density 1, where M satisfies (12.8.4.13.1) for g ⊗ α and M > t + 6a + 19b + 10c + 21 ordp (2) (again for g ⊗ α), in the notation of 12.8.4. We put gα = fα ⊗ χ ∈ Sk (n(gα ), 1) and Vα = V (gα )(k/2) = V (fα )(k/2) ⊗ χ. Proposition 12.8.4.14(ii) applies to g ⊗ α and gα , yielding ran (F, g ⊗ α) − h1f (F, V ⊗ α) ≡ ran (F, gα ) − h1f (F, Vα ) (mod 2), which means that in order to prove (12.10.2.1) it is enough to establish the congruences ?

ran (F, gα ) ≡ h1f (F, Vα ) (mod 2) for the forms gα . These forms satisfy the assumption 12.2.3(2), as ordq1 (n(gα )) = ordq1 (n(fα )) = 1. (12.10.4) Reduction to the assumptions 12.2.3(1)-(2) Consider the case when g satisfies the condition 12.2.3(3), but not 12.2.3(1) nor 12.2.3(2). This implies, in particular, that p > 3, 2 | [F : Q] and π(g)v 6= St(µ) for all primes v of F . As n(g) is prime to p, it follows from the remarks at the end of 12.5.5 that χ is unramified at all v | p, hence the form g itself is p-ordinary. 412

(12.10.4.1) Lemma. Fix a GF -stable O = OL,p -lattice T ⊂ V and denote by ρp : GF −→ AutO (T ) the Galois action on T . Then −1 ∈ Im(ρp ). ∼

Proof. According to 12.2.3(3iii), if g does not have CM, then there exists a choice of a basis T −→ O⊕2 ∼ such that the image of the reduction ρp = ρp (mod p) of ρp : GF −→ AutO (T ) −→ GL2 (O) satisfies Im(ρp ) ⊇ SL2 (Fp ). As p > 3, it follows from ([Se 3], IV.3.4, Lemma 3) that Im(ρp ) ⊇ SL2 (Zp ), hence −1 ∈ Im(ρp ). If g has CM by a totally imaginary quadratic extension K/F , then the restriction of ρp to GK is at∗ 0 ∗ 0 tached ([Sc], ch. 0, §5) to
an algebraic Hecke character ψ : AK −→ (L ) ([L : L] = 2) of infinity type P k k σ∈Φ ( 2 − 1)σ − 2 c ◦ σ , where Φ is a CM type of K and c the complex conjugation. As n(g) is prime to p, so is the conductor of ψ, hence the composite map Y recK ∗ ∗ ab ρp |GK 0 ϕ: OF,v −→ A∗F /F ∗ −→ A∗K /K ∗ −−→G K −−−−→ (L ⊗L Lp ) ⊂ GL2 (Lp ) v|p

Q is given by ϕ((uv )v|p ) = v|p uv (provided we normalize the reciprocity map recK via the geometric Frobenius elements). In particular, −1 = ϕ((−1, 1, . . . , 1)) ∈ Im(ρp | GK ) ⊂ Im(ρp ). (12.10.4.2) Corollary. For each character α : Gal(F 0 /F ) −→ {±1}, −1 ∈ Im(ρp ⊗ α). Proof. The condition 12.2.3(3) for g implies that the same condition is satisfied by g ⊗ α (for example, n(g ⊗ α) | n(g) cond(α)2 is prime to p, since cond(α) | dF 0 /F and p - dF 0 /Q ), hence Lemma 12.10.4.1 applies to g ⊗ α. (12.10.4.3) Level raising for g ⊗ α Fix α : Gal(F 0 /F ) −→ {±1} and let Eg⊗α ⊂ OL(g) be the non-zero ideal attached to g ⊗ α in ([Tay 1], Thm. 1). Set m = ordp (Eg⊗α OL ) and denote by E the fixed field of the kernel of ∼

ρp ⊗ α (mod pm+1 ) : GF −→ AutO (T /pm+1 T ) −→ GL2 (O/pm+1 O).
As Gal(E/F ) = Im ρp ⊗ α (mod pm+1 ) ⊂ GL2 (O/pm+1 O) contains −1 (by Corollary 12.10.4.2), the ˇ Cebotarev density theorem implies that there exists a prime ideal q - (p) n(g ⊗ α) cond(χ) of F such that (ρp ⊗ α)(Fr(q)) ≡ −I (mod pm+1 )

(Fr(q) = Fr(q)geom ).

Fix such q; then 1 − (N q)−k/2 λg⊗α (q)X + (N q)X 2 = det (1 − Fr(q)X | V ⊗ α) ≡ det (1 + X | V ⊗ α) = (1+ X)2 (mod pm+1 ), hence N q − 1 ≡ λg⊗α (q) + 2 ≡ 0 (mod pm+1 ),


ordp λg⊗α (q)2 − (N q + 1)2 − ordp (Eg⊗α OL ) ≥ 1.

Applying ([Tay 1], Thm. 1) and ([De-Se], Lemma 6.11), we deduce that there exists g1 ∈ Sk (n(g ⊗ α)q, 1), which is an eigenform for all T (v) (v 6= q), its Hecke eigenvalues are contained in a number field L0 ⊃ L and satisfy (∀v 6= q)

λg1 (v) ≡ λg⊗α (v) (mod p0 ),

(12.10.4.3.1)

where p0 is a prime of L0 above p. Moreover, ordq (n(g1 )) = 1, hence g1 is also an eigenform for T (q). Choose a GF -stable O0 -lattice T1 ⊂ V1 = V (g1 )(k/2) (O0 = OL0 ,p0 ) and set T 0 = T ⊗O O0 . The congruence (12.10.4.3.1) implies that (∀σ ∈ GF )

Tr (σ | T1 /p0 T1 ) = Tr (σ | T 0 /p0 T 0 ⊗ α) . 413

As T /pT is an absolutely irreducible representation of GF (it is irreducible, by 12.2.3(3ii), and the complex conjugation acts on it with two distinct eigenvalues ±1 (recall that p = 6 2) contained in the residue field O/p), it follows that ∼

T1 /p0 T1 −→ T 0 /p0 T 0

(12.10.4.3.2)

are isomorphic, absolutely irreducible representations of GF . We are going to verify that Proposition 12.8.4.15 applies (with M = 1) to the pair of congruent forms g ⊗ α and g1 (with (O, p) replaced by (O0 , p0 )). (12.10.4.4) Proposition. (i) π(g1 )q = St(µ), where µ is unramified, µ(q) = −1, and λg1 (q) = −(N q)k/2−1 . (ii) The form g1 is p-ordinary and its level n(g1 ) is prime to p. (iii) (∀v 6= q) ordv (n(g1 )) 6= 1. Proof. (i) As ordq (n(g1 )) = 1, we have π(g1 )q = St(µ), where µ is unramified and µ(q) = ±1. As I

I

∼

I

T1 q  T1 q /p0 T1 q ⊂ T1 /p0 T1 −→ T 0 /p0 T 0 I

with Fr(q) = Fr(q)geom acting on T1 q (resp., on T 0 /p0 T 0 ) by the scalar (N q)−k/2 λg1 (q) = µ(q)(N q)−1 (resp., by −1), it follows that µ(q) ≡ −N q ≡ −1 (mod p0 ), hence (since p 6= 2) µ(q) = −1,

λg1 (q) = (N q)k/2−1 µ(q) = −(N q)k/2−1 .

(ii) Firstly, n(g1 ) | n(g ⊗ α)q | n(g) cond(α)2 q | n(g) d2F 0 /F q, which means that n(g1 ) is prime to p. Fix a prime v | p of F . According to Proposition 12.4.9.2, V (g1 )v is a crystalline representation of Gv with Hodge-Tate weights equal to 1 − k and 0. Denote by T (g1 ) ⊂ V (g1 ) the lattice T1 (−k/2). The isomorphism (12.10.4.3.2) combined with (12.5.3.1) for g ⊗ α implies that (T (g1 )/p0 T (g1 ))Iv is a one-dimensional subspace of T (g1 )/p0 T (g1 ). An easy exercise in Fontaine-Laffaille theory (which applies to T (g1 )v , as v is unramified in F/Q and k − 1 < p − 1) shows that T (g1 )Iv 6= 0. T. Saito’s comparison result ([Sa], Thm. 1) then implies that λg1 (v) ∈ O0∗ , hence the form g1 is p-ordinary. (iii) As n(g1 ) | n(g ⊗ α)q, it is enough to consider primes v dividing n(g ⊗ α) (hence prime to p). As π(g)v 6= St(µ) by assumption, it follows from Proposition 12.4.10.3 and the assumption p - w2 (L(g)) that ((T /pT ) ⊗ α)Iv = 0. The isomorphism (12.10.4.3.2) implies that (T1 /p0 T1 )Iv = 0, hence V1Iv = 0, which proves the claim (by Lemma 12.4.5(ii)). (12.10.4.5) Proposition. (i) The integers a, b, c ≥ 0 attached to g ⊗ α (over O0 ) in 12.8.4.2, 12.8.4.4(ii) and 12.8.1.1, respectively, are equal to a = b = c = 0, and the condition (12.8.4.13.1) is satisfied for M = 1. (ii) ran (F, g ⊗ α) − h1f (F, V ⊗ α) ≡ ran (F, g1 ) − h1f (F, V1 ) (mod 2). Proof. (i) As T 0 /p0 T 0 ⊗ α is an absolutely irreducible representation of GF , we have c = 0. Fix v | p in F . By assumption, Fv /Qp and χv are unramified, and α2 = 1, hence ∗ NFk−1 · (χv α)2 (OF,v ) = (Z∗p )k−1 6⊃ 1 + pZp v /Qp

(as k − 1 < p − 1), which implies that av = 0 (see 12.8.4.2) and a = 0. Fix a prime v | n(g ⊗ α). By assumptions, v - p and π(g ⊗ α)v is in the ramified principal series, or a supercuspidal representation. As shown in Proposition 12.4.10.3, in either case the assumption p - w2 (L(g)) implies that ((V /T ) ⊗ α)Iv = 0, hence Tamv (T ⊗ α, p) = 0 (=⇒ Tamv (T 0 ⊗ α, p0 ) = 0). This implies that bv = 0 (see 12.8.4.4(ii)) and b = 0. The condition (12.8.4.13.1) is satisfied with M = 1, thanks to Proposition 12.10.4.4(iii) and the fact that b = c = ordp0 (2) = 0. 414

(ii) This follows from Proposition 12.8.4.15, which applies to the forms g ⊗ α ≡ g1 (mod p0 ) (over O0 ) with M = 1, thanks to (i). (12.10.4.6) To sum up, the congruences (12.10.2.1) follow from ?

ran (F, g1 ) ≡ h1f (F, V1 ) (mod 2), for the forms g1 attached to all g ⊗ α (α : Gal(F 0 /F ) −→ {±1}) as in 12.10.4.3. As ordq (g1 ) = 1, these forms satisfy the assumption 12.2.3(2). In other words, thanks to 12.10.1-4, it is enough to prove the congruence ?

ran (F, g) ≡ h1f (F, V ) (mod 2),

(12.10.4.6.1)

in the case when at least one of the assumptions 12.2.3(1)-(2) holds. (12.10.5) Reduction of the assumption 12.2.3(2) to the case when n(g) is not a square Assume that n(g) is a square and that there exists a quadratic extension F1 /F and a prime q of F for which 2 - ordq (n(g ⊗ µ)), where µ = ηF1 /F is the quadratic character attached to F1 /F . As ordq (n(g ⊗ µ)) depends only on the local representation π(g)q ⊗ µq , we can assume that the field F1 is totally real. Denote by f1 = BCF1 /F (f ) (resp., g1 = BCF1 /F (g)) the base change of f (resp., g) to F1 . The form f1 is again p-ordinary and g1 = f1 ⊗ χ1 , where χ1 = χ ⊗ NF1 /F . Our assumptions imply that ordq (n(g)) ≡ 0 6≡ 1 ≡ ordq (n(g ⊗ µ)) (mod 2), hence q is ramified in F1 /F ; denote by q1 the only prime of F1 above q. If W is any Lp -representation of GF , denote by W1 its restriction to GF1 . The formula dim(W )

dF1 /F

NF1 /F (cond(W1 )) = cond(W ) cond(W ⊗ µ)

([A-T], ch. 11, Thm. 18) for W = V (g) together with (12.4.3.2) imply that ordq1 (n(g1 )) ≡ ordq (n(g)) + ordq (n(g ⊗ µ)) ≡ 1 (mod 2). It follows that neither n(g1 ), nor n(g ⊗ µ) is a square. The formulas (12.10.1.1) for F1 /F imply that ran (F, g) − h1f (F, V ) = (ran (F1 , g) − h1f (F1 , V )) − (ran (F, g ⊗ µ) − h1f (F, V ⊗ µ)), which means that it is enough to prove (12.10.4.6.1) for the forms g ⊗ µ (over F ) and g1 (over F1 ), whose levels are not squares. (12.10.6) Reduction of the assumption 12.2.3(2) to the case when there exists a prime P | p such that n(g)(P ) is not a square Assume that F has only one prime P above p, that n(g)(P ) is a square and 2 - ordP (n(g)). Thanks to ([A-T], ch. 10, Thm. 5), there exists a cyclic extension F2 /F of degree [F2 : F ] = 3 (hence F2 is totally real) in which P splits: P OF2 = Q1 Q2 Q3 . Let g2 = BCF2 /F (g) be the base change of g to F2 . For each j = 1, 2, 3, we have π(g2 )Qj = π(g)P , hence 2 - ordQj (n(g2 )). The congruence (12.10.1.2) for F2 /F reads as ran (F2 , g2 ) − h1f (F2 , V ) ≡ ran (F, g) − h1f (F, V ) (mod 2), hence it is enough to prove (12.10.4.6.1) for the form g2 over F2 , for which n(g2 )(Q1 ) is not a square. (12.10.7) Reduction to the case when (12.9.1.1) holds and F 00 = F To sum up the results of 12.10.1-6, we have reduced Theorem 12.2.3 to the following statement: if the condition (12.9.1.1) holds, then ?

ran (F, g) ≡ h1f (F, V ) (mod 2). (12.10.8) Reduction to the case when (12.9.1.1) holds, k = 2 and F 00 = F 415

(12.10.7.1)

We assume that g = f ⊗ χ ∈ Sk (n(g), 1), f is p-ordinary, k 6= 2 and that P | p is a prime of F for which (12.9.1.1) holds (in the case when [F : Q] is even). As in 12.9.5.1, we embed (the p-stabilization of) f into a Hida family as fP for a suitable arithmetic points P and apply the discussion from 12.7.9-11 (possibly after slightly enlarging Lp if p = 2) to obtain a p-ordinary form f 0 = fP 0 of weight (2, . . . , 2) in the same family, the corresponding twist g 0 = f 0 ⊗ χ0 ∈ S2 (n(g 0 ), 1) (with χ0 = χ · (ω ◦ N )1−k/2 ) and the Galois representation V 0 = V (g 0 )(1). As remarked in 12.9.5.1, g 0 also satisfies (12.9.1.1). According to Proposition 12.7.14.4(iv), we have ran (F, g) − h1f (F, V ) ≡ ran (F, g 0 ) − h1f (F, V 0 ) (mod 2). In particular, the desired congruence (12.10.7.1) for g follows from the analogous result for g 0 . (12.10.9) Proof in the case F 00 = F , k = 2, (12.9.1.1) holds We assume that g = f ⊗ χ ∈ S2 (n(g), 1), f is p-ordinary and that P | p is a prime of F for which (12.9.1.1) holds (if [F : Q] is even). Our goal is to prove (12.10.7.1). If K/F is any totally imaginary quadratic extension satisfying (12.9.2.1-2) in which P splits, then we have ran (K, g) ≡ h1f (K, V ) ≡ 1 (mod 2),

(12.10.9.1)

by Theorem 12.9.7 applied to β = 1. Denoting by ηK/F the quadratic character over F attached to K/F , then we have ran (K, g) = ran (F, g) + ran (F, g ⊗ ηK/F ) h1f (K, V ) = h1f (F, V ) + h1f (F, V ⊗ ηK/F )

(12.10.9.2)

(by (12.10.1.1) for K/F ). We now apply the argument used in [Ne 3] to deduce Thm. A from Thm. B. Case (I): 2 - ran (F, g) By ([Wa 2], Thm. 4; [F-H], Thm. B (1)) there exists K/F satisfying (12.9.2.1-2) in which P splits and for which ran (F, g ⊗ ηK/F ) = 0. Put g0 = g ⊗ ηK/F and V0 = V ⊗ ηK/F = V (g0 )(1). As (dK/F , n(g)) = (1), we have ordv (n(g)) = ordv (n(g0 )) for any prime v | n(g). In particular, the form also g0 satisfies the condition (12.9.1.1). If K 0 /F is any totally imaginary quadratic extension satisfying (dK 0 /F , n(g0 )) = (1),

ηK 0 /F (n(g0 )) = (−1)[F :Q]−1

in which P splits, then we have ran (F, g0 ⊗ ηK 0 /F ) = ran (K 0 , g0 ) − ran (F, g0 ) = ran (K 0 , g0 ) ≡ 1 (mod 2), by Corollary 12.6.3.7 applied to g0 and K 0 /F . As g0 satisfies (12.9.1.1), such fields K 0 exist; ([F-H], Thm. B (2)) implies that there exists K 0 for which ran (F, g0 ⊗ ηK 0 /F ) = 1 (and such that g0 does not have CM by K 0 ), hence ran (K 0 , g0 ) = ran (F, g0 ⊗ ηK 0 /F ) + ran (F, g0 ) = 1 + 0 = 1.

(12.10.9.3)

Applying the discussion from 12.9.5.2 to g0 instead to g, we obtain an abelian variety A0 . According to Zhang’s generalization of the Gross-Zagier formula ([Zh 1], Thm. C; [Zh 2], Thm. 1.2.1), it follows from (12.10.9.3) that a certain Heegner point y ∈ A0 (K 0 ) is non-torsion. An Euler system argument ([Ne 4], Thm. 3.2) together with the assumption that g0 does not have CM by K 0 imply that Hf1 (K 0 , V0 ) = Lp · δy, where δ : A0 (K 0 ) −→ Hf1 (K 0 , V0 ) is the standard Kummer map. Moreover, the non-trivial element τ ∈ Gal(K 0 /F ) acts on δy ∈ Hf1 (K 0 , V0 ) as τ (δy) = −ε(π(g0 ), 12 ) δy = −δy 416

(by [Ti], Lemma 9.1). As Hf1 (K 0 , V0 )τ =1 = Hf1 (F, V0 ),

Hf1 (K 0 , V0 )τ =−1 = Hf1 (F, V0 ⊗ ηK 0 /F ),

it follows that Hf1 (F, V0 ) = 0, i.e. that h1f (F, V0 ) = h1f (F, V ⊗ ηK/F ) = 0. Combining ran (F, g ⊗ ηK/F ) = h1f (F, V ⊗ ηK/F ) = 0 with (12.10.9.1-2), we obtain the desired congruence (12.10.7.1). Case (II): 2 | ran (F, g) Let K/F be any totally imaginary quadratic extension satisfying (12.9.2.1-2) in which P splits. It follows from (12.10.9.1-2) that 2 - ran (F, g ⊗ ηK/F ). The same argument as in Case (I) shows that the form g ⊗ ηK/F satisfies (12.9.1.1), hence Case (I) applies to it: ran (F, g ⊗ ηK/F ) ≡ h1f (F, V ⊗ ηK/F ) ≡ 1 (mod 2). The relations (12.10.9.1-2) then imply ran (F, g) = ran (K, g) − ran (F, g ⊗ ηK/F ) ≡ h1f (K, V ) − h1f (F, V ⊗ ηK/F ) = h1f (F, V ) (mod 2). This completes the proof of Theorem 12.2.3. (12.11) Proof of Theorem 12.2.8 In this section we prove Theorem 12.2.8. The following base change result, which does not require any ordinarity assumption, will not be used in the proof of 12.2.8. However, we record it here for future reference. (12.11.1) Proposition. Let f ∈ S2 (n, 1) be a newform over a totally real number field F such that π(f )v0 = St(µ0 ) (µ20 = 1) at some prime v0 of F . Let L be the number field generated by the Hecke eigenvalues of f and p a non-archimedean prime of L; set V = Vp (f )(1). Then there exists a finite solvable totally real extension F 0 /F such that the base change form f 0 = BCF 0 /F (f ) satisies the following properties: (i) The level n(f 0 ) of f 0 is square-free. Equivalently, each local representation π(f 0 )v0 (where v 0 is a nonarchimedean prime of F 0 ) is either in the unramified principal series, or is an unramified twist of the Steinberg representation. (ii) h1f (F 0 , V ) − ran (F 0 , f 0 ) ≡ h1f (F, V ) − ran (F, f ) (mod 2). Proof. Let λ be a non-archimedean prime of L relatively prime to N n. For each prime v | n of F , there exists a finite Galois extension Ew /Fv such that the restriction of Vλ (f ) to Gw = Gal(F v /Ew ) is semistable in the following sense: either the inertia group Iw acts trivially on Vλ (f ), or there is an exact sequence of Lp [Gw ]-modules 0 −→ Lp (1) ⊗ µ −→ Vλ (f )w −→ Lp ⊗ µ −→ 0, where µ : Gw −→ {±1} is unramified and Iw acts on Vλ (f )w through an infinite quotient. More precisely, it follows from 12.4.4.3 that one can choose Ew in such a way that the Galois group Hv := Gal(Ew /Fv ) is of the following form: ∗ If π(f )v = π(µ, µ−1 ), then Hv ic cyclic (isomorphic to µ(OF,v )). 2 If π(f )v = St(µ) (µ = 1), then Hv is trivial (resp., cyclic of order 2) if µ is unramified (resp., ramified). ∼ If π(f ) is supercuspidal and not exceptional, then Hv −→ Dn = Z/nZ o {±1} (n ≥ 2) is dihedral. If π(f ) is supercuspidal exceptional (=⇒ v | 2), then Hv is an extension of A3 or S3 by Dn (n ≥ 2).

This implies that 417

If 2 - |Hv |, then Hv is cyclic. If 2 | |Hv |, then either there exists an epimorphism Hv  Z/2Z, or there exists an epimorphism Hv  Z/3Z with kernel Dn (n ≥ 2). Assume that at least one of the groups Hv is not trivial (otherwise we can take F 0 = F ). Step 1. If the set Σ(F ) = {v | n, ∃ Hv  {±1}} is not empty, choose, for each v ∈ Σ(F ), an epimorphism Hv  {±1}; denote by ϕ(v) : Gv  Hv  {±1} the corresponding (ramified) quadratic character. According to ([A-T], ch. 10, Thm. 5), there exists a totally real global character ϕ : A∗F /F ∗ −→ {±1} such that (∀v ∈ Σ(F )) ϕv = ϕ(v) . By construction, π(f ⊗ ϕ)v0 = St(µ0 ϕ(v0 ) ) is an unramified twist of the Steinberg representation, hence εv0 (f ⊗ ϕ, 12 ) = −(µ0 ϕ(v0 ) )(v0 ).

We are going to define another character ϕ0 : A∗F /F ∗ −→ {±1}. If ε(π(f ) ⊗ ϕ, 12 ) = 1, set ϕ0 = 1. If ε(π(f ) ⊗ ϕ, 12 ) = −1, then there exists ϕ0 : A∗F /F ∗ −→ {±1} of conductor prime to n cond(ϕ), which is totally real and satisfies ( −1, v = v0 0 (∀v | n cond(ϕ)) ϕ (v) = 1, v 6= v0 . The local formulas from Proposition 12.6.2.4 imply that ε(π(f ) ⊗ ϕ, 12 ) ε(π(f ) ⊗ ϕϕ0 , 12 ) = −1, hence ε(π(f )⊗ϕϕ0 , 12 ) = 1, in either case. It follows from ([F-H], Thm. B (1)) that there exists a (non-trivial) character χ : A∗F /F ∗ −→ {±1} such that (∀v | ∞ n cond(ϕ))

χv = (ϕϕ0 )v ,

ran (F, f ⊗ χ) = 0.

The first property implies that χ = ηF1 /F is attached to a totally real quadratic extension of F ; the second that h1f (F, V ⊗ χ) = 0 (as in 12.10.9), hence h1f (F1 , V ) − ran (F1 , f ) = (h1f (F, V ) − ran (F, f )) + (h1f (F, V ⊗ χ) − ran (F, f ⊗ χ)) = h1f (F, V ) − ran (F, f ). This means that, in order to prove the Proposition, we can replace F by F1 . Repeating this process, if necessary, we reduce to the case when each group Hv (if non-trivial) is either cyclic of odd order, or an extension of A3 by Dn (n ≥ 2). Step 2. By Step 1, we can assume that, for each v with Hv non-trivial, there exists an epimorphism Hv  Z/n(v)Z (2 - n(v)), whose kernel is either trivial, or is isomorphic to Dn (n ≥ 2). Denote by χ(v) : Gv  Hv  Z/n(v)Z the corresponding character of Gv . Again, ([A-T], ch. 10, Thm. 5) implies that there exists a global character χ : A∗F /F ∗ −→ Z/nZ (n = lcm(n(v)), 2 - n)such that χv = χ(v) for each Ker(χ)

prime v with Hv 6= {1}. As n is odd, the field F2 := F is a totally real cyclic extension of F satisfying Hw = {1} or Hw = Dn (n ≥ 2), for each prime w - ∞ of F2 . As in 12.10.1-2, we have h1f (F2 , V ) − ran (F2 , f ) ≡ h1f (F, V ) − ran (F, f ) (mod 2), which means that we can replace F by F2 . Step 3. By Step 2, we can assume that all non-trivial groups Hv are isomorphic to Dn (n ≥ 2). Repeating again Step 1 (r times, if 2r k2n) and then Step 2, we obtain the desired field F 0 . (12.11.2) Lemma. Let A, F , L and p be as in Theorem 12.2.8; set V = Vp (A). Let E/F be a finite extension and E 0 /E a finite Galois extension with Galois group ∆ = Gal(E 0 /E). (i) Hf1 (E 0 , V ) = Hf1 (E, V ⊗Lp Lp [∆]), with GE acting on V ⊗Lp Lp [∆] as in 8.1.3. (ii) If α : ∆ −→ Lp (α)∗ is a character with values in a finite extension Lp (α) of Lp , then
∆
(α−1 ) Hf1 (E, V ⊗ α) = Hf1 (E 0 , V ) ⊗ α = Hf1 (E 0 , V ) ⊗Lp Lp [∆] . 418

Denote h1f (E, V ⊗ α) = dimLp (α) Hf1 (E, V ⊗ α). (iii) If ∆ is abelian and all characters of ∆ have values in a finite extension M of Lp , then M X ∼ Hf1 (E 0 , V ) ⊗Lp M −→ Hf1 (E, V ⊗ α), h1f (E 0 , V ) = h1f (E, V ⊗ α). α:∆−→M ∗

α:∆−→M ∗

If α2 = 1, then V ⊗ α = Vp (A) ⊗ α = Vp (Aα ), where Aα is the abelian variety over F obtained by twisting A with α ∈ Homcont (GF , {±1}) = H 1 (GF , {±1}). (iv) For each α : ∆ −→ Lp (α)∗ , h1f (E, V ⊗ α) = h1f (E, V ⊗ α−1 ). (v) If ∆ is abelian, then h1f (E 0 , V ) ≡

X

X

h1f (E, V ⊗ α) ≡

α:∆−→{±1}

h1f (E, Vp (Aα )) (mod 2).

α:∆−→{±1}

(vi) If E ⊂ E1 ⊂ E 0 , where E1 /E is a 2-abelian extension and 2 - [E 0 : E1 ], then X h1f (E 0 , V ) ≡ h1f (E, V ⊗ α) (mod 2). α:Gal(E1 /E)−→{±1}

Proof. The statement (i) is a consequence of Shapiro’s Lemma; (ii) and (iii) follow from the ∆-equivariance of the isomorphism in (i), and (iv) was proved in Proposition 12.5.9.5(iv). Finally, (v) and (vi) are deduced from (iii) and (iv) as in the proof of Prop. 8.8.8(ii) (cf. 12.10.1). (12.11.3) Proposition-Definition. Let F and L be totally real number fields and A an abelian variety over F with OL ⊂ EndF (A) and dim(A) = [L : Q]. (1) We say that A is potentially modular if, for each finite set S of non-archimedean primes of F , there exists a totally real finite Galois extension F 0 /F such that: (i) F 0 /F is unramified at S. (ii) A ⊗F F 0 is modular in the sense that there exists a Hilbert modular eigenform g 0 ∈ S2 (n(g 0 ), 1) over F 0 with Hecke eigenvalues in L such that we have, for each embedding ι : L ,→ R, an equality of L-functions (Euler factor by Euler factor) L(ιA/F 0 , s) = L(ιg 0 , s). Equivalently, for each non-archimedean prime λ of L, the λ-adic Galois representations Vλ (A)|GF 0 and Vλ (g 0 )(1) of GF 0 are isomorphic. (2) ([Tay 3], proof of Thm. 2.4) If (1)(i)-(ii) hold for F 0 , then, for each intermediate field F ⊂ F 00 ⊂ F 0 with Gal(F 0 /F 00 ) solvable, there exists a newform g 00 ∈ S2 (n(g 00 ), 1) over F 00 such that BCF 0 /F 00 (g 00 ) = g 0 and, for each embedding ι : L ,→ R, L(ιA/F 00 , s) = L(ιg 00 , s) (Euler factor by Euler factor). Equivalently, Vλ (A)|GF 00 is isomorphic to Vλ (g 00 )(1), for each λ as in (ii). (12.11.4) Potential modularity of any A as in 12.11.3 seems to be well-known to the experts [Tay 5]; a proof is expected to appear in a forthcoming thesis of a student of R. Taylor. (12.11.5) Proposition. In the situation of 12.11.3(2), (i) If w | p is a prime of F 00 such that A⊗F F 00 has totally multiplicative reduction at w, then π(g 00 )w = St(µ), µ2 = 1, µ is unramified and λg00 (w) = µ(w) = ±1. (ii) If w | p is a prime of F 00 such that A ⊗F F 00 has good ordinary reduction at w, then ordp (λg00 (w)) = 0. (iii) If A ⊗F F 00 has ordinary (= good ordinary or totally multiplicative) reduction at all primes of F 00 above p, then g 00 is p-ordinary (with respect to each embedding L ,→ Qp ). (iv) If A has potentially ordinary (= potentially good ordinary or potentially totally multiplicative) reduction ∗ at all primes of F above p, then there exists a character of finite order χ : A∗F 00 /F 00∗ −→ Q such that g 00 ⊗χ−1 is p-ordinary (with respect to each embedding Q ,→ Qp ). Proof. Fix a non-archimedean prime u - p of L. 419

(i) It follows from the analytic uniformization of A⊗F Fw00 that there is an exact sequence of Lu [Gw ]-modules 0 −→ Lu (1) ⊗ χ1 −→ Vu (A)w −→ Lu ⊗ χ2 −→ 0, where χ1 , χ2 : Fw00∗ −→ L∗u are unramified characters of finite order, and that the inertia group Iw acts on Vu (A)w through an infinite quotient. The latter condition implies that χ1 = χ2 ; on the other hand, χ1 χ2 = 1, since Λ2 Vu (A) = Lu (1). We conclude by 12.4.4.3. (ii) The assumption of good ordinary reduction implies that Iw acts trivially on Vu (A) and det (1 − Fr(w)arith X | Vu (A)) = (1 − αw X)(1 − βw X), where αw , βw are contained in a quadratic extension L0 of L and, for each prime p0 | p in L0 , exactly one of the valuations ordp0 (αw ), ordp0 (βw ) is zero, while the other is positive. It follows that ordp (λg00 (w)) = ordp (αw + βw ) = 0. (iii) This is a consequence of (i) and (ii). (iv) For each prime v | p in F 00 there exists a finite Galois extension of Fv00 over which A ⊗F Fv00 has ordinary reduction. A repeated application of ([A-T], ch. 10, Thm. 5) implies that there exists a finite totally real solvable extension K/F 00 such that A ⊗F K has ordinary reduction at all primes above p. According to (iii) applied to A ⊗F K, the form gK = BCF 00 /K (g 00 ) over K is p-ordinary. The existence of χ then follows from Proposition 12.5.10. (12.11.6) Application of Brauer’s Theorem ([Tay 4], §6) Let A, F , L and p | p be as in Theorem 12.2.8; set V = Vp (A). Assume that E/F is a finite solvable ∗ extension and α : GE −→ Q a character of finite order. Fixing an embedding Q ,→ Qp , we view α as a character with values in a finite extension of Lp . Let F 0 /F be as in 12.11.3. According to a version of Brauer’s Theorem due to L. Solomon ([Cu-Re], Thm. 15.10), there exist intermediate fields E ∩ F 0 ⊂ Fj ⊂ F 0 and integers nj ∈ Z (for j running through a finite set J(E)) such that each Galois group Gal(F 0 /Fj ) is solvable and the trivial representation of Gal(F 0 /E ∩F 0 ) is equal, as a virtual representation, to X

1Gal(F 0 /E∩F 0 ) =

Gal(F 0 /E∩F 0 )

nj IndGal(F 0 /Fj )

(1Gal(F 0 /Fj ) ).

(12.11.6.1)

j∈J(E)

As ∼

Gal(EF 0 /E) −→ Gal(F 0 /E ∩ F 0 ),

∼

Gal(EF 0 /EFj ) −→ Gal(F 0 /Fj ),

we deduce from (12.11.6.1) and the projection formula an equality of virtual representations of GE X

V |GE ⊗ α =

E nj IndG GEF ((V ⊗ α)|GEFj ).

(12.11.6.2)

j

j∈J(E)

∼

On the other hand, Proposition 12.11.3(2) implies that V |GFj −→ Vp (gj )(1) for some newform gj ∈ S2 (n(gj ), 1) over Fj , hence
∼ (V ⊗ α)|GEFj −→ Vp BCEFj /Fj (gj ) (1) ⊗ α|GEFj .

(12.11.6.3)

Combining (12.11.6.2-3), we obtain L(ιA/E, α, s) =

Y

L BCEFj /Fj (gj ) ⊗ α, s


nj

.

(12.11.6.4)

j∈J(E)

This formula gives rise to a meromorphic continuation of the L.H.S. to C and a functional equation relating L(ιA/E, α, s) to L(ιA/E, α−1 , 2−s) (as in [Tay 4], Thm. 6.6). Putting ran (ιA/F, α) := ords=1 L(ιA/E, α, s), we deduce from (12.11.6.2-4) the following formulas: 420

X

ran (ιA/E, α) =

nj ran (EFj , BCEFj /Fj (gj ) ⊗ α)

(12.11.6.5)

nj h1f (EFj , V ⊗ α).

(12.11.6.6)

j∈J(E)

h1f (E, V

X

⊗ α) =

j∈J(E)

(12.11.7) Descent In the situation of 12.11.6, assume that E 0 /E is a finite abelian extension with Galois group ∆ = Gal(E 0 /E). Applying (12.11.6.5) to each term in the product Y L(ιA/E 0 , s) = L(ιA/E, α, s) b α∈∆ b denotes the group of characters of ∆) and using the fact that, for each α ∈ ∆, b (where ∆ ran (EFj , BCEFj /Fj (gj ) ⊗ α) = ran (EFj , BCEFj /Fj (gj ) ⊗ α−1 ) by the functional equation, we obtain X

ran (ιA/E 0 ) ≡

ran (ιA/E, α) (mod 2).

(12.11.7.1)

α:∆−→{±1}

Similarly, Lemma 12.11.2(v)-(vi) yields X

h1f (E 0 , V ) ≡

h1f (E, V ⊗ α) (mod 2).

(12.11.7.2)

α:∆−→{±1}

The following special cases of (12.11.7.1-2) are of particular interest: (A) If 2 - [E 0 : E], then ran (ιA/E 0 ) ≡ ran (ιA/E) (mod 2),

h1f (E 0 , V ) ≡ h1f (E, V ) (mod 2).

(12.11.7.3)

(B) If E = F , then we have, for each character α : ∆ −→ {±1}, L(ιA/F, α, s) = L(ιAα /F, s) (in the notation from 12.11.2(iii)). In particular, the congruences (12.11.7.1-2) read as follows: X ran (ιA/E 0 ) ≡ ran (ιAα /F ) (mod 2) α:∆−→{±1}

h1f (E 0 , Vp (A))

≡

X

h1f (F, Vp (Aα )) (mod 2).

(12.11.7.4)

α:∆−→{±1}

(12.11.8) Reducing Theorem 12.2.8 to the case F1 = F for Aα Returning to the proof of Theorem 12.2.8, there exists a tower of fields F ⊂ F0 = E0 ⊂ · · · ⊂ En = F1 in which each Galois group Gal(Ei+1 /Ei ) is abelian, of odd degree. Applying (12.11.7.3) (resp., (12.11.7.4)) to each Ei+1 /Ei (resp., to F0 /F ), we obtain X
h1f (F1 , Vp (A)) − ran (ιA/F1 ) ≡ h1f (F, Vp (Aα )) − ran (ιAα /F ) (mod 2) α:∆−→{±1}

(where ∆ = Gal(F0 /F )). It is enough, therefore, to prove Theorem 12.2.8 for each Aα over F1 = F . (12.11.9) The case 12.2.8(1) 421

In this case one can take F 0 = F in 12.11.3(1): A is attached to a newform g ∈ S2 (n, 1) in the sense that Vλ (A) = Vλ (g)(1), for all non-archimedean primes λ of L. Moreover, Proposition 12.11.5(iv) (for F 00 = F ) implies that g is of the form g = f ⊗ χ, where f ∈ S2 (n(f ), χ−2 ) is p-ordinary. The result then follows from Theorem 12.2.3(1) applied to g. (12.11.10) The case 12.2.8(2) Applying the discussion in 12.11.6 to E = F , we obtain, for each j ∈ J(F ), a newform gj ∈ S2 (n(gj ), 1) over a totally real extension Fj /F such that V |GFj = Vp (A)|GFj = Vp (gj )(1). As A does not have potentially good reduction everywhere, there exists a prime vj of Fj such that π(gj )vj = St(µj ). Thanks to (12.11.6.5-6), we have, for each α : Gal(F0 /F ) −→ {±1}, h1f (F, Vp (Aα )) − ran (ιAα /F ) =

X


nj h1f (Fj , V ⊗ α) − ran (Fj , gj ⊗ α) .

(12.11.10.1)

j∈J(F )

According to Proposition 12.11.5(iv), for each j ∈ J(F ) there exists a character of finite order χj : ∗ A∗Fj /Fj∗ −→ Q such that fj := gj ⊗ χ−1 is p-ordinary. Applying Theorem 12.2.3(2) to the forms j gj ⊗ α = fj ⊗ αχj over Fj , we obtain h1f (Fj , V ⊗ α) ≡ ran (Fj , gj ⊗ α) (mod 2).

(12.11.10.2)

Combining (12.11.10.1-2), we obtain the desired congruences h1f (F, Vp (Aα )) ≡ ran (ιAα /F ) (mod 2),

(12.11.10.3)

which prove (thanks to 12.11.8) Theorem 12.2.8 in the case when A does not have potentially good reduction everywhere. (12.11.11) The case 12.2.8(3) If A is modular over F , i.e. if Vp (A) = Vp (f )(1) for some newform f ∈ S2 (n(f ), 1) over F , then the result follows from Theorem 12.2.3(3). In general, choose F 0 /F as in Proposition 12.11.3(1), unramified at all primes above p. As in 12.11.10, we obtain intermediate fields F ⊂ Fj ⊂ F 0 and newforms gj ∈ S2 (n(gj ), 1) over Fj , for which (12.11.10.1) holds (for each α : Gal(F0 /F ) −→ {±1}). According to Proposition 12.11.5(iii), the forms gj are p-ordinary. We are going to check that the assumptions 12.2.3(3)(i)-(iii) are satisfied for each form gj ⊗ α over Fj (with the field F0 playing the role of the field F 0 in 12.2.3). Firstly, 12.2.3(3)(i) follows automatically from the assumptions 12.2.8(ii). As regards 12.2.3(3)(ii), this will follow from 12.2.3(3)(iii) if A does not have CM. If A has CM, so does gj ⊗ α (over a totally imaginary quadratic extension Kj of Fj ). Our assumptions imply that the level of gj ⊗ α is prime to p, hence the restriction ofP the Galois representation Vp (gj ⊗ α)(1) to GKj is attached to an algebraic Hecke character of infinity type σ∈Φ −σ (for some CM type Φ of Kj ) of conductor prime to p. This implies that, in a suitable basis of T /pT (where we have denoted T = Tp (A)), the image of the residual representation (ρp ⊗ α)(GFj ) ⊂ Aut(T /pT ) contains the normalizer of a Cartan subgroup of SL2 (Fp ), hence acts irreducibly on T /pT . Finally, if A does not have CM, nor does gj ; we must check that, for a suitable choice of a basis of T /pT , (ρp ⊗ α)(GFj ) ⊃ SL2 (Fp ). Let K 0 be the fixed field of Ker(ρp ⊗ α : GF −→ Aut(T /pT )); set G = Im(ρp ⊗ α) = Gal(K 0 /F ), K := Fj ∩ E 0 . Let c ∈ G be the image of the complex conjugation (for a fixed embedding F ,→ C). As Fj is totally real, Gal(K 0 /K) contains all conjugates gcg −1 , g ∈ G. By assumption, G contains a group conjugate to SL2 (Fp ); Lemma 12.11.12 below implies that Gal(K 0 /K) also contains a group conjugate to SL2 (Fp ), which is precisely the condition 12.2.3(3)(iii) for the form gj ⊗ α over Fj . To sum up, Theorem 12.2.3(3) applies to gj ⊗ α, yielding h1f (Fj , V ⊗ α) ≡ ran (Fj , gj ⊗ α) (mod 2). Together with (12.11.10.1), this implies (12.11.10.3), finishing the proof of Theorem 12.2.8. 422

(12.11.12) Lemma. Let k be a finite extension of Fp (p 6= 2) and G a subgroup of GL2 (k) containing an element c with eigenvalues ±1. Denote by H the subgroup of GL2 (k) generated by {gcg −1 | g ∈ G}. If G contains a subgroup conjugate to SL2 (Fp ) (“G is large enough”), the same is true for H. Proof. According to the list of subgroups of P GL2 (k) given in [Dic, § 260], the statement will follow from the following claims: (i) H acts irreducibly on k 2 . (ii) H is not contained in the normalizer N = N (C) of a Cartan subgroup C ⊂ GL2 (k). (iii) P H (= the image of H in P GL2 (k)) is not isomorphic to A4 , S4 , A5 . (i) Note that c ∈ H and that the only c-stable lines in k 2 are X± = Ker(c ∓ I). If H acts reducibly on k 2 , then there exists ε = +, − such that Xε is stable by all g −1 cg (g ∈ G). This implies that, for each g ∈ G, the line g(Xε ) is c-stable, hence equal to one of the X± . As a result, G is contained in the normalizer of the split Cartan subgroup C = {g ∈ GL2 (k) | g(X± ) = X± }, which contradits our assumptions; thus H acts irreducibly on k 2 . (ii) Assume that H ⊂ N = N (C). Case (1): c ∈ C. As the eigenvalues of c are contained in k, C is split and there is a basis of k 2 in which ! ( !) ( !) 1 0 ∗ 0 0 ∗ c= , C= , N =C∪ . 0 −1 0 ∗ ∗ 0 As G is large enough, there exists a unipotent element g ∈ G − {I}. As p - |N |, g 6∈ N , hence gcg −1 ∈ N − C. 2 −2 A straightforward matrix calculation shows that the conditions gcg −1 ∈ N − C and ∈ N imply ! ! g cg 1 t 1 t that there exists x, t ∈ k ∗ such that g = x (resp., g = x , i ∈ k, i2 = −1) if −1 −1 ±t ∓1 ±it ∓i g 2 cg −2 ∈ C (resp., if g 2 cg −2 ∈ N − C). However, none of these matrices is unipotent. 0 02 Case (2): H ⊂ N , c 6∈ C. After extending ! scalars to a quadratic extension k /k, there is a basis of k in 0 1 which C, N are as in (1) and c = . Again, a straightforward calculation shows that, for g ∈ G − N , 1 0 the conditions gcg −1 , g −1 cg ∈ N impose such severe conditions on the matrix elements of g that g cannot be unipotent, which again contradicts the assumption that G is large enough. ∼ (iii) As A4 is not generated by its elements of order 2, we cannot have P H −→ A4 . In order to rule out −1 ∗ the remaining two cases, note that the group {g ∈ GL2 (k) | gcg ∈ k c} coincides with the normalizer of the split Cartan subgroup C = {g ∈ GL2 (k) | gcg −1 = c} of GL2 (k). As G ⊃ hSL2 (Fp )h−1 for some h ∈ GL2 (k), it follows that the number of elements of order 2 in P H is equal at least to
Im {gcg −1 | g ∈ G} −→ P GL2 (k) ≥

|SL2 (Fp )| |SL2 (Fp )| p(p + 1) ≥ = −1 2 |SL2 (Fp ) ∩ h Ch| 2(p − 1) 2

(where we have used the fact that SL2 (Fp ) ∩ h−1 Ch is contained in a split Cartan subgroup of SL2 (Fp ), ∼ hence its order divides p − 1). If P H −→ S4 or A5 , then p - |P H| ([Dic], §260). On the other hand, the inequalities (∀p - |S4 |) (∀p - |A5 |) imply that P H 6= S4 , A5 .

p(p + 1) 2 p(p + 1) |{elements of order 2 in A5 }| = 15 < 2

|{elements of order 2 in S4 }| = 9
, M ι . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.4.2 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.4.1 m(g) = m(g, K; P1 , . . . , Ps ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.6.4.9 N∞ (universal norms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.6, 11.3.4 normalized 2-cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 n(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 nSt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.3.1 νn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.9.2 427

O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1, 10.6 o(πv ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.9 ω, ωR (dualizing complex) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5, 2.7, 2.10.3 (P ) (absence of real primes if p = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.5 0 P 0 , P, P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.7 e P e 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.7.11 P, Pk,ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2 Pontrjagin duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 p-ordinary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.6.1-2 pseudo-null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Φ−d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.11 Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 π(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.5 π(f )v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.6 π(µ, µ0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.6 $P , $P 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.12 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 R = R[[Γ]] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 RΓ{m} (−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 RΓcont (−, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.6 dc,cont (GK,S , −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.7.2 RΓ gf (−) = RΓ gf (GK,S , −; ∆(−)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 RΓ RΓur (Gv , −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.13 RΓIw (G, H; −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.3.4 RΓIw (K∞ /K, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.5.1 RΓc,Iw (K∞ /K, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 RΓ(KS /K∞ , −), RΓc (KS /K∞ , −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 gf,Iw (K∞ /K, −), RΓ gf (KS /K∞ , −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.5 RΓ g g gf,Iw,Σ (L/K, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.8 RΓf,Σ (L, −), RΓf,Σ (KS /L, −), RΓ (R Mod)f t ,(R Mod)cof t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 ad

Mod (admissible R[G]-modules) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1  ind−ad R[G] Mod (ind-admissible R[G]-modules) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5

R[G]

ran (−, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 ran (K, −, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2.3 recv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5 resv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 resc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3.4 res+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2.3 resf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4.3 ring class character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1.5 S, Sf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Sbad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.8 Sex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Ssp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1.5 str SX (−), SX (−) (Greenberg Selmer groups) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Sel(B/K, pn), Sp (B/K) (classical Selmer groups for abelian varieties) . . . . . . . . . . . . . . . . . 9.6.7.1, 9.6.7.4 428

Sel(−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Sk (n, ϕ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 St, St(µ) = St ⊗ µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.6 s12 , s23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 sn , s0n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.15 sh (map in Shapiro’s Lemma) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 shc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3.2 sh+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2.1 shf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4.1 Σ, Σ0 = Sf − Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 σ≤i , σ≥i (na¨ıve truncations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.11 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5.4 Tv+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 T (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 T (P), T (P 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.10 e T (P e 0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.11 T (P), Tamv (T, p) (local Tamagawa factors) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.10.1 t = tv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.15 (−)tors , (−)R−tors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.13 Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5, 8.1.6.1-2 τ≤i , τ≥i (truncations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.11 θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 θ(β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 (U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Uv+ (−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1.1 Uv− (−), US± (−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 U (M • ), U ± (M • ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 (−)U = Z[G/U ] ⊗ (−), U (−) = Hom(Z[G/U ], −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 u±,π,hv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 V (f ), Vp (f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.9 (−)v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1 − (−)+ v , (−)v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1, 7.8.2 wj (−) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.10 X∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5.1 0 X∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.2.2 ∪ (cup product for group cohomology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 ∪r,h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 ∪ij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.9 c ∪, ∪c (cup product of cohomology and cohomology with compact support) . . . . . . . . . . . 5.3.3.2, 5.3.3.3 b, ∪ b c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 c∪ • • ∪ = ∪π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 ∪±,π,hv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 ∪π,r,h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 ∪π,hS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 ∪π,hS ,i,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 429

⊥π,hv , ⊥π,hS (orthogonality of local conditions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1.3-4 ⊥⊥π,hv , ⊥⊥π,hS (local conditions are orthogonal complements of each other) . . . . . . . . . . . . . . . . . . . 6.2.7 ⊥π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 ⊥⊥π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.7.4 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5

430

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436

SELMER COMPLEXES – ERRATA

(9.5.6.1) Replace E∞ on the second line by E∞ ⊗ Qp /Zp . (11.4.10) The remark is incorrect, as the assumption 11.4.1.5 is not satisfied in this case. (12.4.4.3) In the supercuspidal case it is the group Gv , but not necessarily Iv , which acts absolutely irreducibly on V (f )v . In the monomial case, the quadratic extension Ew /Fv can be unramified. (12.6.4.9) The sentence “If v splits in K/F . . . ” in the proof of (iii) is incorrect, in general. As a result, one (P ···P ) has to add the following assumption to 12.6.4.9(iii): no prime v | p that splits in K/F divides n(g)St 1 s . This assumption also has to be added to (12.6.4.12), but not to (12.6.4.11). (12.6.4.12) In the proof of 12.6.4.12(iii), the inequality m ≥ m0 should be replaced by m ≤ m0 . (12.7.14.2)(iii) The statement and the proof are correct only in the case when Iv acts absolutely irreducibly ∗ ∼ Gv 0 ∼ 0 0 ∗ v on Vv . If this is not the case, then Vv −→ IndG Gw (µ) and Vv −→ IndGw (µ ), where µ, µ : Ew −→ Lp with µ0 /µ unramified. (12.7.14.3)(iii), (12.7.14.5)(i) By the previous remark, the proofs (which refer to (12.7.14.2)(iii)) are incomplete in the supercuspidal case. In fact, for p > 2, both statements follow from [De2, Thm. 6.5] and (12.7.14.2.2). In general, these are special cases of (the second half of the proof of) Prop. 2.2.4 in [Ne5, Doc. Math. 12 (2007), 243–274]. Both references use Galois-theoretical ε-factors, which coincide with the automorphic ones. 0 (12.9.6) In fact, if 2 - [F : Q], then there is no exceptional KQ /F , either. If it existed, put η 0 = Q extension 0 0 ηK 0 /F . As K /F is unramified outside ∞P , we have 1 = v ηv (−1) = v|∞P ηv0 (−1) = (−1)[F :Q] ηP0 (−1), hence ηP0 (−1) = −1. In particular, P is ramified in K 0 /F . As f is p-ordinary, πP (g) is not supercuspidal, which implies (by 12.6.1.2.3) that π(g)P = π(µ, µηP0 ) for some µ : FP∗ −→ C∗ . The central character of π(g)P is trivial; thus µ2 ηP0 = 1 and ηP0 (−1) = µ(−1)−2 = 1, contradiction. As a result, we can omit the assumption “Assume that g does not have CM by . . . ” in (12.9.5), (12.9.8), (12.9.9), (12.9.11) and (12.9.12). Similarly, in (12.9.7), we can omit the assumptions (i), (ii) in the case 2 - [F : Q]. (P )

(P )

(12.9.8), (12.9.9) Add the following assumption: no prime v | p that splits in K/F divides n(g)St n(g 0 )St . (12.9.13) In the proof, we choose P 0 such that g 0 also satisfies (n(g 0 )St , (p)) = (1). (12.11.12) In the proof of (iii), SL2 (Fp ) ∩ h−1 Ch is contained in a not necessarily split Cartan subgroup, which implies that its order is less than or equal to p + 1, hence the number of elements of order 2 in P H is equal at least to
Im {gcg −1 | g ∈ G} −→ P GL2 (k) ≥

|SL2 (Fp )| |SL2 (Fp )| p(p − 1) ≥ = . −1 2 |SL2 (Fp ) ∩ h Ch| 2(p + 1) 2

∼

If P H −→ S4 or A5 , then (assuming that H is not large enough) p - |P H| ([Dic], §260). On the other hand, the inequalities (∀p - |S4 |) (∀p - |A5 |)

p(p − 1) 2 p(p − 1) |{elements of order 2 in A5 }| = 15 < 2

|{elements of order 2 in S4 }| = 9
3, then H is not solvable. Indeed, non-solvability of G implies that, either (1) G contains hSL2 (Fp )h−1 (which is not solvable, as ∼ ∼ p > 3), hence so does H, by Lemma; or (2) P G −→ A5 . In the latter case the image of c in P G −→ A5 is an element of order 2, and the subgroup of A5 generated by all conjugates of such an element is equal to A5 ; ∼ thus P H = P G −→ A5 , which implies that H is not solvable.

2