Galois cohomology of elliptic curves

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Galois cohomology of elliptic curves

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Sokidaku mo Ogironaki kamo Kokibaku mo Yutakeki kamo Koko mireba Ube shi kamiyo yu Hajimekerashi mo



What a sweep of sea, The vastness of these waters! What abundant life In these teeming, boundless deeps! One has but to look – No wonder from the age of gods Began the building of this site.

From the Man’y¯oshu. ¯ See A Waka Anthology, Vol. 1: The Gem-Glistening Cup by E.A. Cranston, Stanford University Press, Stanford, 1993.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Basic Results from Galois Cohomology . . . . . . . . . . . . Preliminary results . . . . . . . . . . . . . . . . . . . . . . Poitou-Tate sequence . . . . . . . . . . . . . . . . . . . . . Cassels-Poitou-Tate sequence . . . . . . . . . . . . . . . . Cassels-Poitou-Tate sequence for elliptic curves . . . . . . 2 The Iwasawa Theory of the Selmer Group . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . The fundamental diagram . . . . . . . . . . . . . . . . . . Cyclotomic theory . . . . . . . . . . . . . . . . . . . . . . A theorem of Kato . . . . . . . . . . . . . . . . . . . . . . 3 The Euler Characteristic Formula . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclotomic theory . . . . . . . . . . . . . . . . . . . . . . The division field case . . . . . . . . . . . . . . . . . . . . 4 Numerical Examples over the Cyclotomic Zp -extension of Q Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Iwasawa theory over Q∞ for the curves of conductor 11 . Iwasawa theory over Q∞ for the curves of conductor 294 . 5 Numerical examples over Q(µp∞ ) . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . General strategy for the curves of conductor 11 over Q(µ5∞ ) The curve A0 . . . . . . . . . . . . . . . . . . . . . . . . . The curve A2 . . . . . . . . . . . . . . . . . . . . . . . . . Infinite descent on A1 over Q(µ5 ) . . . . . . . . . . . . . . The curves of conductor 294 over Q(µ7∞ ) . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure theory of Iwasawa modules . . . . . . . . . . . . Deeply ramified p-adic fields and cyclotomic extensions . . Tate parametrization of elliptic curves . . . . . . . . . . . Multiplicative Kummer generators . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

ii 1 3 3 5 7 9 16 16 18 19 26 29 29 30 44 48 48 51 57 61 61 62 65 72 72 81 84 84 88 94 95 97

Preface The genesis of these notes was a series of four lectures given by the first author at the Tata Institute of Fundamental Research. It evolved into a joint project and contains many improvements and extensions on the material covered in the original lectures. Let F be a finite extension of Q, and E an elliptic curve defined over F . The fundamental idea of the Iwasawa theory of elliptic curves, which grew out of Iwasawa’s basic work on the ideal class groups of cyclotomic fields, is to study deep arithmetic questions about E over F , via the study of coarser questions about the arithmetic of E over various infinite extensions of F . At present, we only know how to formulate this Iwasawa theory when the infinite extension is a p-adic Lie extension for a fixed prime number p. In fact, these notes will mainly discuss the simplest non-trivial example of the theory when our extension is the cyclotomic Zp -extension of F . However, we also make some comments about the theory when the extension is the field obtained by adjoining all p-power division points on E to F . When E does not admit complex multiplication, the Galois group of the extension of F obtained in this manner is an open subgroup of GL2 (Zp ), and it seems appropriate to refer to this situation as the GL2 Iwasawa theory of E. We only discuss algebraic aspects of the Iwasawa theory of elliptic curves, both because of our desire to keep the notes short, and also because the analytic theory is in a much greater state of flux, in view of the recent deep contributions to it by K. Kato [K]. However, we have discussed in detail a number of numerical examples, which illustrate the general theory beautifully. Also, we suspect that the study of numerical examples will be vital in formulating some aspects of the general GL2 Iwasawa theory of elliptic curves (for example, in studying the structure of the Selmer group as a module over the Iwasawa algebra). In addition, we have outlined some of the basic results in Galois cohomology which are used repeatedly in the study of the relevant Iwasawa modules. In the appendix, we briefly discuss some aspects of the cohomology of elliptic curves which seem to have been overlooked earlier and which turn out to be very useful in Iwasawa theory [C-S]. Finally, we refer the interested reader to ii

Preface

iii

Greenberg’s lecture notes [G2], which cover much the same ground as in these notes and more, but with a somewhat different view point. In conclusion, the first author would like to thank the Tata Institute of Fundamental Research, Bombay, and the second author the Isaac Newton Institute of Mathematical Sciences, Cambridge, in particular the organisers of the Arithmetic Geometry Programme there in 1998, for generous support while these notes were being written. Preface to the Second edition: The only changes we have made to the original notes have been to take modest account of the considerable progress which has been made in non-commutative Iwasawa theory in the intervening years. We have also included a short section on the deep theorems of Kato [K] on the cyclotomic Iwasawa theory of elliptic curves.

J. Coates R. Sujatha

Notation We summarize here the notation used frequently in these notes. We write p for a prime number which will always be odd unless the contrary is stated. If L is a field, L will denote the separable closure of L. For each integer m ≥ 1, µm will denote the group of m-th roots of unity in L, viewed as a Galois module. If K is a Galois extension of L, we write G(K/L) for the Galois group of K over L. Let µp∞ denote the group of all p-power roots of unity. We write F for a finite extension of Q, which will nearly always be the base field over which we are working. If v is any place of F , Fv will denote the completion of F at v. We let K∞ denote the cyclotomic Zp -extension of F , and put Γ = G(K∞ /F ). Throughout, S will denote a finite set of non-archimedean primes of F , which will always be assumed to contain all primes of F above p. Let FS denote the maximal extension of F unramified outside S and the archimedean primes of F . Put GS = G(FS /F ). Since S contains all primes of F dividing p, we have K∞ ⊂ FS , and we write GS,∞ = G(FS /K∞ ). Throughout, E will denote an elliptic curve, which will usually be taken to be defined over the number field F . If L is an extension field of F , we write E(L) for the group of L-rational points of F . If M is an abelian group, and m is an integer ≥ 1, we write Mm for the subgroup of elements of M which are annihilated by m. Put M (p) = ∪ Mpn , Tp (M ) = lim Mpn . ←

n≥1

Let Mtors denote the torsion subgroup of M . We write M ∗ = lim M/pn M ←

for the p-adic completion of M . If M is p-primary (i.e., M = M (p)), we write Mdiv for the maximal p-divisible subgroup of M . When M is a discrete p-primary abelian group or a compact pro-p-abelian group, we ˆ = Hom(M, Qp /Zp ) (here Hom denotes the group of continuous let M homomorphisms) for the Pontryagin dual of M . If φ : M1 → M2 is ˆ2 → M ˆ 1 for the dual a continuous homomorphism, we write φˆ : M homomorphism. Let G be a profinite group, and M a G-module. As 1

2

Notation

usual, M G will be the subgroup of elements fixed by G, and MG will be the largest quotient of M on which G acts trivially. If M is finite and p-primary, and µp∞ is also a G-module, we define M D = Hom(M, µp∞ ). We endow M D with the usual G-action, namely (σf )(x) = σf (σ −1 x) for σ ∈ G and x ∈ M . Similarly, if M is any p-primary G-module, we endow ˆ with the G-action (σf )(x) = f (σ −1 x) for σ ∈ G and x ∈ M . If M M is a discrete G-module, we write H i (G, M ) for the cohomology groups of G formed with continuous cochains. If L is a field, and G = G(L/L), we write as usual H i (L, M ) rather than H i (G, M ).

Chapter 1 Basic Results from Galois Cohomology In this chapter, we gather, without proofs, all the results from Galois cohomology that will be used in the subsequent chapters. In particular, we discuss the classical long exact sequence due to Poitou-Tate, which lies at the heart of many calculations in Galois cohomology. We also derive a canonical modification of it, due originally to Cassels, who first introduced it in the special case of elliptic curves. An excellent account of the proofs of many of these results can be found in the book [NSW].

Preliminary results 1.1. This paragraph recalls various pairings and duality results that will be used later. Weil pairing: If L is any field of characteristic zero, and E/L is an elliptic curve, there is a non-degenerate, alternate, bilinear pairing [Ta] wm : Em × Em → µm where Em = E(L)m and µm is the multiplicative group of m-th roots of unity. In fact, this is a particular case of such a pairing between Am and Atm , where A/L is an arbitrary abelian variety and At its dual abelian variety. Tate duality: Assume now that p is any prime, and L is a finite extension of Qp . Let E be an elliptic curve defined over L and let E(L) be the group of L-rational points of E. The Galois cohomology group H 1 (L, E) = H 1 (L, E(L)) classifies principal homogeneous spaces of E over L. There exists a non-degenerate bilinear pairing due to Tate (see [Si]) Φ : E(L) × H 1 (L, E) → Q/Z 3

4

Chapter 1

such that Φ induces a duality of locally compact groups, when E(L) is give the v-adic topology and H 1 (L, E) is endowed with the discrete topology. Let Lnr denote the maximal unramified extension of L, and E0 (L) be the subgroup of E(L) consisting of elements with non-singular reduction [Si, p.173]. We shall need the well-known fact (cf. [Mc]) that under this dual pairing, the subgroup Wv = H 1 (G(Lnr /L), E(Lnr )), is the exact orthogonal complement of E0 (L). Tate local duality for finite Galois modules: Recall that H i (L, M ) = 0 except when i = 0, 1, 2, where M is any discrete G(L/L) module. If M is a finite p-primary group which is also a G(L/L)-module, then for 0 ≤ i ≤ 2, there is a canonical dual pairing [Se] H i (L, M ) × H 2−i (L, M D ) → Qp /Zp where we have M D = Hom(M, µp∞ ), or equivalently, there is a canonical isomorphism H i\ (L, M ) ' H 2−i (L, M D ). We shall often tacitly treat this canonical isomorphism as an identification. Local Euler-Poincar´e characteristic: Put G = G(L/L). Given a finite G-module M , the groups H r (L, M ) are finite for all r and trivial for r ≥ 2, since the cohomological dimension of G is 2 [Se]. The Euler characteristic of M is defined as χ(G, M ) = # H 0 (L, M ).# H 2 (L, M )/# H 1 (L, M ). A basic result of Tate [Mi, p. 38] states that χ(G, M ) = 1/[R : mR], where R is the ring of integers in L, and m = # M . Global Euler-Poincar´e characteristic: Let F be a finite extension of Q and v an archimedean place of F . Put ( ordinary absolute value if v is real, | |v = square of ordinary absolute value if v is complex.

Basic Results from Galois Cohomology

5

If M is a finite p-primary GS -module with p 6= 2, then the cohomology groups H i (GS , M ) are finite for all i and trivial for i ≥ 3 [Ha, p. 21]. The Euler characteristic of M is defined as χ(GS , M ) = # H 0 (GS , M ).# H 2 (GS , M )/# H 1 (GS , M ). A well-known theorem of Tate [Ha, p. 30], [Mi, p. 82] gives the equality Y

χ(GS , M ) =

(# H 0 (Fv , M )/ | #M |v ).

v|∞

Kummer sequence: Let G be a profinite group, and let A be a discrete G-module which is divisible by p. The short exact sequence / Apn

0

n /A p /A

/0

induces the following short exact sequence in Galois cohomology, 0

/ A/pn A

/ H 1 (G, Apn )

/ H 1 (G, A)pn

/ 0,

and this exact sequence is usually referred to as the Kummer sequence associated to A. The results listed in the above paragraph will often be tacitly used in the subsequent sections. In Chapters 4 and 5, the duality results and the various pairings will be used explicitly in computations.

Poitou-Tate sequence 1.2. The next sequence we want to discuss is the celebrated Poitou-Tate sequence. As before, let F be a number field and M a finite p-primary GS -module. We assume throughout this section that p is odd, to avoid a discussion of the role of the infinite primes of F . For each v ∈ S, there are restriction maps γv,i : H i (GS , M ) → H i (Fv , M ) for i ≥ 0 which are induced by the composition of maps G(Fv /Fv )

/ / G(FS,w /Fv )  

/ GS

6

Chapter 1

where w denotes some prime of FS above v, and FS,w is the union of the completions at w of the finite extensions of F contained in FS . These restriction maps give canonical homomorphisms αM : H 0 (GS , M ) → ⊕ H 0 (Fv , M ), v∈S

1

βM : H (GS , M ) → ⊕ H 1 (Fv , M ), v∈S

2

γM : H (GS , M ) → ⊕ H 2 (Fv , M ), v∈S

where clearly αM is injective. Recall that the groups H i (GS , M ) are finite, for all i ≥ 0, with H i (GS , M ) = 0 for all i ≥ 3. Using Tate duality, we also have the canonical dual homomorphisms 2 D 0\ α d M : ⊕ H (Fv , M ) → H (GS , M ), v∈S

1 D 1\ βc M : ⊕ H (Fv , M ) → H (GS , M ), v∈S

0 D 2\ γc M : ⊕ H (Fv , M ) → H (GS , M ), v∈S

with α d M surjective. We thus have three sequences 0

γd / H 0 (GS , M ) αM / ⊕ H 0 (Fv , M ) M D / H 2 (G \ D S , M ), v∈S

H 1 (GS , M ) H 2 (GS , M )

γM

βM

/ ⊕ H 1 (Fv , M )

β[ MD

v∈S

/ ⊕ H 2 (Fv , M )

α \ MD

v∈S

/ H 1 (G \ , M D ), S

/ H 0 (G \ , M D) S

/ 0.

The homomorphisms H r (GS , M ) → ⊕ H r (Fv , M ) v∈S

have the property that Ker βM and Ker α d M are finite. Further, there is a canonical non-degenerate pairing Ker βM × Ker α d M → Qp /Zp . Even the definition of this pairing is rather involved. That it is nondegenerate follows using deep results of class-field theory (cf. [Ha, §1.4]). Granting these results, they can be used to construct a connecting homomorphism 1 \ D δM : H 2 (G S , M ) → H (GS , M ).

Basic Results from Galois Cohomology

7

The Poitou-Tate theorem connects the three sequences above using the connecting homomorphism δM . We refer to [Ha], [P] for more details and simply state the theorem. 1.3. Theorem. (Poitou-Tate) There is a long exact sequence 0

/ H 0 (GS , M ) αM

/ ⊕ H 0 (Fv , M )

γd MD

v∈S

/ H 2 (G \ D S, M ) δM

\ D o H 1 (G S, M ) 

β[ MD

⊕ H 1 (Fv , M ) o



βM

H 1 (GS , M )

v∈S

δd MD γM

H 2 (GS , M )

/ ⊕ H 2 (Fv , M ) v∈S

/ H 0 (G \ D S, M )

/ 0.

Cassels-Poitou-Tate sequence 1.4. We next discuss a modification of the Poitou-Tate sequence, originally due to Cassels [Ca]. Cassels considered the case of M = Epn , the pn -torsion points of E(F ), where E is an elliptic curve defined over F . We treat a more general situation than that considered by Cassels. From the Poitou-Tate sequence, we have Im δM = Ker βM . We suppose that for all v ∈ S, we are given a subgroup Wv (M ) of H 1 (Fv , M ). We define Wv (M D ) to be the orthogonal complement of Wv (M ) in the Tate pairing H 1 (Fv , M ) × H 1 (Fv , M D ) → Qp /Zp . We can thus canonically identify the dual of H 1 (Fv , M )/Wv (M ) with Wv (M D ) and vice-versa, which we do in what follows. Let φM be the composite φM : H 1 (GS , M )

βM

/ ⊕ H 1 (Fv , M ) v∈S

/ ⊕ H 1 (Fv , M )/Wv (M ) v∈S

where the second map is the natural projection. We also define maps ρM : Ker φM → ⊕ Wv (M ), v∈S

with ρM being the restriction of βM , and \ D ηM : H 2 (G S , M ) → Ker φM .

8

Chapter 1

The map ηM exists because Im δM = Ker βM ⊆ Ker φM . Observe that the sequence \ D H 2 (G S, M )

ηM

/ Ker φM ρM / ⊕ Wv (M ) v∈S

is exact. 1.5. Theorem. (Cassels-Poitou-Tate) There is a canonical exact sequence / H 1 (GS , M ) φM / ⊕ H 1 (Fv , M )/Wv (M ) v∈S

/ Ker φM

0

⊕ H 2 (Fv , M ) o

γM

v∈S





ηd MD

H 2 (GS , M ) o

ρd MD

\ Ker φM D

α \ MD

/ 0.

\ D H 0 (G S, M )

Proof. We only have to show that the sequence is exact at the term ⊕ H 1 (Fv , M )/Wv (M ), or equivalently that the dual sequence

v∈S

Ker φM D

ρM D

d / ⊕ Wv (M D ) φM / H 1 \ (GS , M ) v∈S

is exact at the term in the middle. The commutative diagram H 1 (GS , M D )

TTTT TTTTφM D TTTT βM D TTT)  1 D ⊕ H (Fv , M ) / ⊕ H 1 (Fv , M D )/Wv (M D )

v∈S

v∈S

yields the exact sequence 0

λM D

/ Ker β D M

/ Ker φ D M

/ Coker(β D ) M

/ Coker(φ D ) M

ρM D

/ ⊕ Wv (M D ) v∈S

/ 0.

Hence Im ρM D = Ker λM D . On the other hand, the original PoitouTate sequence yields a canonical inclusion 1\ βc M : Coker(βM D ) ,→ H (GS , M ).

d Moreover, the map φ M is the composite Wv (M D )

λM D

/ Coker(β

MD )

d β M

/ H 1\ (G , M ) S

d d and therefore Ker λM D = Ker φ M . Hence Im ρM D = Ker φM , as required. 

Basic Results from Galois Cohomology

9

1.6. We comment briefly on the various possible choices of the subgroups Wv (M ). (i): v 6= p. There are two natural choices which occur in literature. One is the case Wv (M ) = 0 and this is the case that is exploited by Wiles [Wi]. Note that with this choice, Wv (M D ) = H 1 (Fv , M D ). Another case is the subgroup Wv (M ) = H 1 (Gv /Iv , M Iv ), where Iv is the inertial subgroup of Gv = G(Fv /Fv ). The subgroup Wv (M ) is the group of unramified cocycles. A basic fact that we mention in this context is that in this case, Wv (M D ) = H 1 (Gv /Iv , (M D )Iv ). We refer to [Se, II 5.5] for a proof, remarking that the assumption there of M being unramified is unnecessary. (ii): In this case, we choose M = Epn , the pn -torsion points over F of an elliptic curve E defined over F , (more generally, one can consider an arbitrary abelian variety). Then the Kummer sequence gives a homomorphism E(Fv )/pn E(Fv ) 

 κE,pn/

H 1 (Fv , Epn )

and one considers the subgroup Wv (Epn ) = Im(κE,pn ). If one considers an arbitrary abelian variety A, then there is the non-degenerate Weil pairing Apn × Atpn → µpn where At is the dual abelain variety. Hence if M = Apn , then M D = Atpn . A basic fact (due to Tate) is that in this situation, Wv (M D ) = Wv (Atpn ) = Im(κAt ,pn ) where κAt ,pn is the corresponding map arising from the Kummer sequence for At . A generalization of these for arbitrary M has been exploited by Bloch-Kato [B-K], in their conjectures about the Tamagawa numbers of motives.

Cassels-Poitou-Tate sequence for elliptic curves 1.7. We now discuss how the Cassels-Poitou-Tate sequence can be refined further when one considers elliptic curves over F . In this case it is well-known that Epn is a GS -module [Si], where GS = G(FS /F ), and S is any finite set of primes containing all primes dividing p and all primes where E has bad reduction. We let M = Epn and note that M D = Epn since E is self-dual as an abelian variety. For the subgroup Wv (M ) ⊆ H 1 (Fv , M ) = H 1 (Fv , Epn ), we choose Wv (Epn ) = Im(κE,pn ). By the

10

Chapter 1

remarks in the previous paragraph, we have Wv (EpDn ) = Im(κE,pn ). Further, the group H 1 (GS , Epn ) ⊆ H 1 (F, Epn ) and the kernel of the natural map λS : H 1 (GS , Epn ) → ⊕ H 1 (Fv , E)pn ' ⊕ H 1 (Fv , Epn )/ Im(κE,pn ) v∈S

v∈S

is in fact the classical Selmer group of E relative to pn , which we denote by S(E/F, pn ) or simply S(E, pn ). The latter isomorphism above follows from the Kummer sequence applied to E over Fv for v ∈ S. We recall [Si, Chapter X, §4] that S(E/F, pn ) is defined by the exactness of Y H 1 (Fv , E) / / H 1 (F, Epn ) / S(E/F, pn ) 0 v

where v now ranges over all non-archimedean places of F , and that we have the exact sequence 0

X

/ E(F )/pn

/ S(E/F, pn )

X X

/

X(E/F )

pn

/0

(1)

X

where (E/F )pn or simply (E)pn denotes the pn -torsion subgroup of the Tate-Shafarevich group (E) of E. We recall that (E/F ) is defined by the exact sequence 0→

X(E/F ) → H (F, E) → ⊕H (F , E), 1

1

w

w

where w ranges over all archimedean and non-archimedean places of F . In particular, we see that with this choice of the subgroup Wv (Epn ), the group Ker λS is in fact independent of the choice of S. The CasselsPoitou-Tate sequence takes the form 0

/ S(E, pn )

/ H 1 (GS , Epn )

/ ⊕ H 1 (Fv , E)pn

/ S(E, \ pn )

/ H 2 (GS , Epn )

/ ⊕ H 2 (Fv , Epn )

/ E\ pn (F )

/ 0.

v∈S

v∈S

(2)

This is the sequence that was discovered by Cassels [Ca]. We can now vary n and pass to the limit as n → ∞. This can be done in two different ways. First, we can pass to the inductive limit using the inclusion

Basic Results from Galois Cohomology

11

Epn ,→ Epn+1 . Letting Ep∞ = lim Epn , the above equation then becomes →

0

/ S(E/F )

/ H 1 (GS , Ep∞ )

/ ⊕ H 1 (Fv , E)(p)

/ S(E/F \)

/ H 2 (GS , Ep∞ )

/ ⊕ H 2 (Fv , Ep∞ )

/ T\ p E(F )

/ 0.

v∈S

v∈S

(3)

Here S(E/F ) = lim S(E, pn ) ⊆ lim H 1 (GS , Epn ), →



while lim H 1 (GS , Epn ) = H 1 (GS , Ep∞ ) ⊆ H 1 (F, Ep∞ ) →

and S(E/F ) = lim S(E, pn ) ⊆ H 1 (F, Tp E). n←

Further, the Selmer group S(E/F ) is a discrete subgroup of H 1 (F, Ep∞ ) while S(E/F ) is a compact subgroup of H 1 (F, Tp E). By Tate duality, the group H 2 (Fv , Ep∞ ) = lim H 2 (Fv , Epn ) is dual to lim H 0 (Fv , Epn ), →n n← where the inverse limit is with respect to “multiplication by p” map from Epn+1 → Epn . But Epn (Fv ) is a finite group and hence the inverse limit above is trivial. Therefore we in fact have an exact sequence / S(E/F )

0

/ H 1 (GS , Ep∞ )

/ ⊕ H 1 (Fv , E)(p) v∈S

(4) / S(E/F \)

/ H 2 (GS , Ep∞ )

/ 0.

The discrete Selmer group S(E/F ) and the compact Selmer group S(E/F ) are related as follows. 1.8. Lemma. There is an exact sequence 0

/ Ep∞ (F )

/ S(E/F )

/ Tp S(E/F )

/ 0.

12

Chapter 1

Proof. Choose n >> 0 so that pn kills Ep∞ (F ). Consider the exact sequence of G(F /F )-modules 0

n / Ep∞ (F ) p / Ep∞ (F )

/ Epn (F )

/ 0.

This induces the long exact sequence / Ep∞ (F )

0

/ H 1 (F, Ep∞ )

/ H 1 (F, Ep∞ )pn

/0

and also an exact sequence / Ep∞ (F )

0

/ S(E/F, pn )

/ S(E/F )pn

/ 0.

Taking inverse limits and observing that the groups S(E/F )pn are finite [Si], we get an exact sequence 0

/ Ep∞ (F )

/ S(E/F )

/ Tp S(E/F )

/0

and the lemma is proved.  We now record some important consequences of the Cassels-PoitouTate sequence. 1.9. Proposition. Assume that both E(F ) and Then

X(E/F )(p) are finite.

(i) H 2 (G(FS /F ), Ep∞ ) = 0. (ii) If λF is the natural map λF : H 1 (G(FS /F ), Ep∞ ) → ⊕ H 1 (Fv , E)(p), v∈S

then Coker λF is finite and #(Coker λF ) = #(E(F )(p)). Proof. The exact sequence 0

/ E(F )/pn

/

/ S(E, pn )

X(E)

pn

gives an exact sequence (on taking lim and noting that n←

0

/ E(F )∗

/ S(E/F )

/ Tp

X(E/F )

/0

X(E)

pn

/ 0.

is finite)

Basic Results from Galois Cohomology

13

X

X

Since (E)(p) is finite by assumption, we have Tp ( (E)) = 0 and hence E(F )∗ = S(E/F ). The exact sequence (4) therefore gives H 1 (G(FS /F ), Ep∞ )

λF

/ ⊕ H 1 (Fv , E)(p)

θF

v∈S

/ H 2 (G(FS /F ), Ep∞ )

/ E(F \ )∗

/ 0.

But E(F ) finite implies that E(F )∗ = E(F )(p) and θc F is the map (use Tate duality) ∗ θc F : E(F )(p) → ⊕ E(Fv ) v∈S

given by the natural inclusion. Since θc F is injective, it is plain that θF is surjective and the proposition follows.  1.10. Remark. As a special case, consider a modular elliptic curve E/Q such that the L-value L(E, 1) 6= 0. Then, by the deep theorem of Kolyvagin [K], the hypotheses of Proposition 1.9 are satisfied for all primes p. Hence we conclude that for all p 6= 2, H 2 (G(QS /Q), Ep∞ ) = 0 and Coker λQ is finite of order # E(Q)(p). We have H 2 (G(QS /Q), Ep∞ ) = 0 and Coker φQ is finite of order # E(Q)(p). We next establish a formula which is reminiscent of Tamagawa measures of E(Fv ) for v a finite place of F . 1.11. Lemma. Let E/F be an elliptic curve and v be a prime of F that does not lie above p. Then # H 1 (Fv , E)(p) ∼ cv /Lv (E, 1) where u ∼ v signifies that u/v is a p-adic unit. Proof. Let E0 (Fv ) ⊆ E(Fv ) be the subgroup which consists of points ˜ns (kv ). Here kv is whose reduction modulo v lies in the subgroup E ˜ ˜ns denotes the the residue field and E is the reduced curve while E subset consisting of the non-singular points of the reduced curve. Let Bv = E(Fv )/E0 (Fv ) and cv = #Bv . Since Bv is finite [Si, Chapter VII, §6], on taking p-adic completions of the exact sequence 0

/ E0 (Fv )

/ E(Fv )

/ Bv

/ 0,

14

Chapter 1

we obtain the exact sequence 0

/ E0 (Fv )∗

/ E(Fv )∗

/ B∗ v

/ 0.

Consider the exact sequence 0

/ E1 (Fv )

/ E0 (Fv )

/E ˜ns (kv )

/ 0.

The subgroup E1 (Fv ) can be identified with the points on the formal group of E at v. Now v does not divide p implies that p is an automor˜ns (kv )∗ . phism of E1 (Fv ) and hence E1 (Fv )∗ = 0. Therefore E0 (Fv )∗ ' E ˜ But it is well-known [Si] that # Ens (kv ) = N (v)/Lv (E, 1) and therefore # E(Fv )∗ ∼ cv /Lv (E, 1). The lemma now follows from Tate duality (1.1).  We will need the following useful lemma, part of which has already been referred to in (1.7). 1.12. Lemma. Let L be a non-archimedean local field and A/L an abelian variety. Then H 2 (L, A) = 0. Proof. Since the cohomological dimension of a local field is 2, the Kummer sequence (1.1) gives a surjection H 2 (L, Ap∞ ) → H 2 (L, A)(p) for any prime p. We prove that H 2 (L, Ap∞ ) = 0. By Tate duality (1.1), the groups H 2 (L, Apn ) are dual to A(L)pn for all n. Thus, on taking limits, we see that H 2 (L, Ap∞ ) is dual to the Tate module Tp (A(L)). But this latter group is zero as the p-primary subgroup of A(L) is finite. Since H 2 (L, A) is a torsion group with H 2 (L, A)(p) = 0 for any prime p, the lemma is proved.  The following is left as an exercise to the reader.

X

1.13. Exercise. Suppose E/F is an elliptic curve such that E(F ) and (E/F ) are both finite. Let T be a finite non-empty set of primes of F not containing any prime lying above p and define Y 1 H 1 (Fv , E)). T (E/F ) = Ker(H (F, E) →

X

v ∈T /

Basic Results from Galois Cohomology

15

Show that the index

X (E/F )(p) : X(E/F )(p)] ∼ # (E(F )(p))

[

−1

T

.

Y

cv .Lv (E, 1)−1 .

v∈T

(Hint: Use Lemma 1.11 and the Cassels-Poitou-Tate sequence). We do not know the answer to the following question.

X

1.14. Question. Is the above formula for the index valid if we do not assume that E(F ) and (E/F ) are finite?

Chapter 2 The Iwasawa Theory of the Selmer Group Introduction 2.1. Throughout this chapter, F will denote a finite extension of Q, and E will denote an elliptic curve defined over F . The main goal of the arithmetic of elliptic curves is to provide a p-adic approach to the conjecture of Birch and Swinnerton-Dyer for E over the base field F . We begin by describing a rather general setting in which the ideas of Iwasawa theory can be fruitfully applied. Let H∞ denote an infinite Galois extension of F whose Galois group Ω = G(H∞ /F ) is a p-adic Lie group of positive dimension. In these notes, we shall mainly be concerned with the case in which H∞ is the cyclotomic Zp -extension of F . We recall that the cyclotomic Zp -extension of F is the fixed field of the torsion subgroup of the Galois group of F (µp∞ ) over F , where µp∞ denotes the group of all p-power roots of unity. It is the unique subextension of F (µp∞ ) over F whose Galois group is topologically isomorphic to the additive group of the ring Zp of p-adic integers. However, we shall also discuss from time to time the case in which H∞ = F (Ep∞ ), where Ep∞ denotes the p-power torsion points of E(F ). Many other choices of the field H∞ are also possible. It is no exaggeration to say that one reason why the Iwasawa theory of elliptic curves is so rich in problems is because we are free to make many choices of the field H∞ . By direct analogy with the situation over F discussed in Chapter 1, we define the Selmer group S(E/H∞ ) of E over H∞ by Y S(E/H∞ ) = Ker(H 1 (H∞ , Ep∞ ) → H 1 (H∞,w , E)). w

Here w runs over all finite primes of H∞ , and, as usual for infinite extensions, H∞,w denotes the union of the completions at w of all finite 16

The Iwasawa Theory of the Selmer Group

17

extensions of F contained in H∞ . It is also easy to see that S(E/H∞ ) = lim S(E/L), →

where L runs over all finite extensions of F contained in H∞ , S(E/L) denotes the Selmer group of E over L as defined in Chapter 1, and the inductive limit is taken with respect to the restriction maps. Again, it follows immediately from Kummer theory for E over H∞ that we have the exact sequence 0

/ E(H∞ ) ⊗ Qp /Zp

/ S(E/H∞ )

/

X

X(E/H

∞ )(p)

/ 0,

where the Tate-Shafarevich group (E/H∞ ) of E over H∞ is defined by Y (E/H∞ ) = Ker(H 1 (H∞ , E) → H 1 (H∞,w , E)),

X

w

and w again runs over all finite primes of H∞ . The central idea of Iwasawa theory is to oberve that the Galois group Ω of H∞ over F has a natural left action on S(E/H∞ ), and to use this Ωmodule structure to study S(E/F ), and, in particular, to relate S(E/F ) to L-functions in the spirit of the conjecture of Birch and SwinnertonDyer. The action of Ω on S(E/H∞ ) is the obvious one arising from the action of Ω on H 1 (H∞ , Ep∞ ). This action makes S(E/H∞ ) into a discrete p-primary Ω-module. It will often be more convenient to \∞ ) = Hom(S(E/H∞ ), Qp /Zp ), which is study its compact dual S(E/H endowed with the left action of Ω given by (σf )(x) = f (σ −1 x) for f in \∞ ) and σ in Ω. S(E/H \∞ ) are continuous modules over the Clearly S(E/H∞ ) and S(E/H ordinary group ring Zp [Ω] of Ω with coefficients in Zp . But, as Iwasawa was the first to exploit in the case of the cyclotomic theory, it is much more useful to view them as modules over a larger algebra, which we denote by Λ(Ω) and call the Iwasawa algebra of Ω, and which is defined by Λ(Ω) = lim Zp [Ω/W ], ←

where W runs over all open normal subgroups of Ω. The algebra Λ(Ω) has an interpretation as the ring of measures on Ω with values in Zp , a fact which is very convenient for studying analysis on Ω and p-adic L-functions. A second, more algebraic, reason for the importance of Λ(Ω) is that all discrete p-primary Ω-modules, or compact Zp -modules

18

Chapter 2

on which Ω acts continuously, can be endowed with a natural structure as a module over Λ(G). If A is any p-primary Ω-module and X = Hom(A, Qp /Zp ) is its Pontrjagin dual, then we have A = ∪ AW ,

X = lim XW , ←

W

where W again runs over all open normal subgroups of Ω, and XW denotes the largest quotient of X on which W acts trivially. It is then clear how to extend by continuity the natural action of Zp [Ω] on A and X to an action of the whole Iwasawa algebra Λ(Ω).

The fundamental diagram 2.2. The crucial ingredient in studying S(E/H∞ ) as a module over the Iwasawa algebra Λ(Ω) is the single natural commutative diagram given below. Because of its importance, we shall henceforth call it the fundamental diagram. As in Chapter 1, let S be any finite set of primes of F which contains all the primes dividing p, and all primes where E has bad reduction. Let FS denote the maximal extension of F which is unramified outside of S and all the archimedean primes of F , and let GS be the corresponding Galois group. We assume that S also has the property that H∞ ⊂ FS (this is automatically true for the two choices of H∞ made in these notes). We set GS,∞ = G(FS /H∞ ). We thus have the following tower of extensions: FS  GS,∞

H∞  Ω

F For any finite extension L of F contained in H∞ , we write Jv (L) = ⊕ H 1 (Lw , E)(p)

(5)

w|v

where the direct sum is taken over all primes w of L lying above the prime v of F . We then define Jv (H∞ ) = lim Jv (L) →

(6)

where the inductive limit is taken with respect to the restriction maps,

The Iwasawa Theory of the Selmer Group

19

and L runs over all finite extensions of F contained in H∞ . Recall that Ep∞ denotes the GS -module of all torsion points in E(F ) whose order is a power of p. Then, as is explained in Chapter 1, we have the exact sequence / H 1 (GS , Ep∞ ) λ

/ S(E/F )

0

/ ⊕ Jv (F ), v∈S

(7)

where λ is the obvious localization map. There is an analogous exact sequence 0

/ S(E/H∞ )

/ H 1 (GS,∞ , Ep∞ ) λ∞ / ⊕ Jv (H∞ ), v∈S

(8)

which is obtained by taking the direct limit of the exact sequences (7) over all finite extensions of F contained in H∞ . We include the case p = 2, noting that in this case the corresponding Selmer groups are bigger than the classical ones because we have not imposed local conditions at the archimedean primes. Of course, the sequence (7) is a sequence of Ω-modules and on taking Ω-invariants we obtain the fundamental diagram 0

/ S(E/H∞ )Ω O

/ H 1 (GS,∞ , Ep∞ )Ω φ∞ O

α

0

/ S(E/F )

/ ⊕ Jv (H∞ )Ω v∈S O γ

β

/ H 1 (GS , Ep∞ )

λ

(9)

/ ⊕ Jv (F ), v∈S

where the rows are exact, and the vertical arrows are the obvious restriction maps with γ = ⊕ γv . We emphasize that all of our subsev∈S

quent arguments revolve around analyzing this diagram. In particular, the analysis of Ker γ and Coker γ is a purely local question whose answer at the primes v dividing p will depend on the nature of the reduction of E at v.

Cyclotomic theory 2.3. We now turn to the case with which we will be mainly concerned in these notes, namely when H∞ is the cyclotomic Zp -extension of the base field F . We write K∞ instead of H∞ in this case, and we also write Γ instead of Ω for the Galois group of K∞ over F . Thus Γ is topologically isomorphic to Zp , and it is well-known (cf. [Se1],[W], see

20

Chapter 2

Appendix A.1) that the Iwasawa algebra Λ(Γ) is isomorphic to the ring Zp [[T ]] of formal power series in an indeterminate T with coefficients in Zp . In fact, there is a unique isomorphism of topological Zp -algebras which maps any given topological generator of Γ to 1 + T . Also, the structure theory of finitely generated modules over Λ(Γ) is well-known, but we shall avoid using this as much as possible and for much of this chapter we will only need the fact that Λ(Γ) has no zero divisors. \∞ ) is 2.4. Lemma. For every prime p, the dual Selmer group S(E/K finitely generated over Λ(Γ). Proof. By a well-known variant of Nakayama’s lemma ([Se1],[W]), it \∞ ) is finitely generated over Zp , or equivsuffices to show that S(E/K Γ alently by duality, that S(E/K∞ )Γ has finite Zp -corank. But this latter statement is an easy consequence of the fundamental diagram (9) for the cyclotomic Zp -extension. Indeed, the fact that S(E/F ) has finite Zp corank follows easily from the well-known result that H 1 (GS , Ep∞ ) has finite Zp -rank (this can be easily deduced for example, from Tate’s global Euler characteristic formula, cf. (1.1)). Also Ker γ has finite Zp -corank because it is dual, via Tate duality (1.1), to a quotient of ⊕ E(Fv )∗ . v∈S

That ⊕ E(Fv )∗ is a finitely generated Zp -module follows from the wellv∈S

known structure of E(Fv ) [Si, Chapter VII] involving the formal group. Further, in the fundamental diagram, the map β is surjective because Γ has cohomological dimension 1, and from the Hochschild-Serre spectral sequence, Coker β injects into H 2 (Γ, Ep∞ (K∞ )) = 0. Thus the map α has cokernel of finite Zp -rank by the snake lemma, and so it follows that S(E/K∞ )Γ has finite Zp -corank, as required.  Let Q(Γ) denote the quotient field of Λ(Γ). If X is a finitely generated Λ(Γ)-module, we recall that the Λ(Γ)-rank of X is defined to be the Q(Γ)-dimension of X ⊗Λ(Γ) Q(Γ). It is natural to ask what the Λ(Γ)\∞ ) is. The conjectural rank of the finitely generated Λ(Γ)-module S(E/K answer for this depends on the nature of the reduction of E at the places v of F dividing p. We recall that E is said to have potential supersingular reduction at a prime v if there exists a finite extension L of Fv such that E/L has good supersingular reduction. For example, the elliptic curve y 2 = x3 − x over Q has bad but potentially supersingular reduction, at the prime 2. Let Pp (E/F ) be the set of places v of F dividing p such that E has potential supersingular reduction at v. We say that E/F is

The Iwasawa Theory of the Selmer Group

of supersingular type at p if Pp (E/F ) is non-empty. We define  X [Fv : Qp ] if E/F is of supersingular    type at p, v∈Pp (E/F ) rp (E/F ) =  if E/F is not of super  0 singular type at p.

21

(10)

The following conjecture is folklore: \∞ ) is 2.5. Conjecture. For every prime p, the Λ(Γ)-rank of S(E/K equal to rp (E/F ). The following partial result in the direction of this conjecture is wellknown, and we shall sketch its proof later in this chapter. \∞ ) is greater 2.6. Theorem. For all primes p, the Λ(Γ)-rank of S(E/K than or equal to rp (E/F ). 2.7. Examples. We shall illustrate our conjectures and results for the three elliptic curves of conductor 11 defined over Q, namely A0 : y 2 + y = x3 − x2 − 10x − 20 A1 : y 2 + y = x3 − x2 A2 : y 2 + y = x3 − x2 − 7820x − 263580. In fact, all three curves are isogenous over Q, and there are no other elliptic curves of conductor 11 defined over Q (cf. [Cr] and [Wi]). The curves have split multiplicative reduction at 11, good ordinary reduction at a set of primes of density 1, commencing with 3, 5, 7, 13, 17, . . ., and good supersingular reduction at an infinite set of primes starting with 2, 19, . . .. In fact, Conjecture 2.5 is true for all primes p for all the three curves Ai over Q, (i = 0, 1, 2), by the following arguments. It is well-known [Cr] that the complex L-function L(Ai /Q, s) of all three curves Ai does not vanish at s = 1. Hence, by [C-M], the dual of H 1 (GS,∞ , Ai,p∞ ) has Λ(Γ)-rank equal to 1 for all primes p. The case p = 2 is not dealt with in [C-M], but similar arguments work in this \∞ ) has Λ(Γ)-rank at most 1 case too. By (8), it follows that S(E/K for all primes p. Conjecture 2.5 is then clear for all primes where the Ai have good supersingular reduction from Theorem 2.6. The proof of Conjecture 2.5 for primes of good ordinary reduction follows from Theorem 2.8 below, since the deep theorem of Kolyvagin shows that in this case S(Ai /Q) is finite for all primes p because L(Ai /Q, 1) 6= 0, for

22

Chapter 2

i = 0, 1, 2. A similar argument also works for p = 11, since a simple numerical argument proves that the 11-adic logarithm of the 11-adic Tate period of the curves Ai over Q11 is not zero. These curves will be treated more fully in Chapters 4 and 5. It is one of the miracles of cyclotomic theory that there is an easy proof of an important case of Conjecture 2.5. 2.8. Theorem. Assume that the prime p satisfies the following: (i) E has potential good ordinary reduction at all primes v of F dividing p, (ii) S(E/F ) is finite (or equivalently both E(F ) and finite).

X(E/F )(p) are

\∞ ) is Λ(Γ)-torsion. Then S(E/K An important corollary of Theorem 2.8 is the following. 2.9. Corollary. Let E be a modular elliptic curve over Q with the property that its complex L-function L(E/Q, s) does not vanish at s = 1. \∞ ) is Λ(Γ)-torsion for all primes p where E Take F = Q. Then S(E/K has potential ordinary reduction. Proof. By a deep theorem of Kolyvagin [Ko], under the above hypotheses, the group S(E/Q) is finite for every prime p. The corollary now follows from Theorem 2.8.  The following classical lemma from the theory of Zp -extensions is needed in the proof of Theorem 2.8. 2.10. Lemma. Let X be a finitely generated Λ(Γ)-module. If XΓ is finite, then X is a torsion Λ(Γ)-module. We stress that the proof of this lemma (cf. [W], see Appendix, Corollary A.1.6), depends crucially on the structure theory of finitely generated Λ(Γ)-modules. It is the only place in this chapter where we use the structure theory of Λ(Γ)-modules. If we take H∞ = F (Ep∞ ), where E is assumed to have no complex multiplication, then the Galois group Ω = G(H∞ /F ) is an open subgroup of GL2 (Zp ). It is shown in [B-H] that the lemma is false in this case for the corresponding Iwasawa algebra. As Susan Howson has spointed out, the correct analogue of Lemma 2.10 in this case involves the Zp -coranks of all the higher cohomology groups pf Ω acting on the Pontryagin dual of X (not just the H 0 , as in Lemma 2.10).

The Iwasawa Theory of the Selmer Group

23

2.11. Proof of Theorem 2.8. We apply Lemma 2.10 with X = \∞ ), so that XΓ is dual to B = S(E/K∞ )Γ . Hence we must S(E/K show that the conditions imposed on p in Theorem 2.8 imply that B is finite. To do this, we appeal to the fundamental diagram (9). We recall that the map β in (9) is surjective. Thus, by the snake lemma and the finiteness of S(E/F ), B will certainly be finite if we can show that Ker γ is finite. If v ∈ S and v does not divide p, then Ker γv is clearly finite by Lemma 1.11. If v divides p, we will now explain how the finiteness of Ker γv follows from the results of [C-G] - and of course, it is here that we will use our hypothesis that E has potential ordinary reduction at all primes v dividing p. Fix a prime v of F dividing p, and let Γv denote the decomposition group of v in Γ. We also write v for some fixed prime of K∞ that extends the prime v of F . We have a tower of extensions: Fv

K∞,v  Γv

Fv By the inflation-restriction sequence, Ker γv = H 1 (Γv , E(K∞,v )). As is explained in [C-G, p.130] (see Appendix A.2), Tate duality implies that the group Ker γv is dual to E(Fv )/EU (K∞,v ), where EU (K∞,v ) is the group of universal norms in E(Fv ) for the Zp -extension K∞,v /Fv , i.e., EU (K∞,v ) = ∩0 NK 0 /Fv (E(K 0 )) K

where K 0 runs over all finite extensions of Fv contained in K∞,v and NK 0 /Fv denotes the norm map from K 0 to Fv on E. We now show, using the results of [C-G] (see Appendix A.2), that EU (K∞,v ) is of finite index in E(Fv ). By hypothesis, there exists a finite extension Lv of Fv such that E/Lv has good ordinary reduction. Then, in the terminology of [C-G], the composite extension L∞,v = Lv K∞,v is a deeply ramified p-adic field because it is a ramified Zp -extension of Fv (we recall that each prime of F dividing p is ramified in the cyclotomic

24

Chapter 2

Zp -extension K∞ , cf. [W]). Hence by Propositions 5.5 and 5.6 of [C-G] (see Appendix A.2.5), the group EU (L∞,v ) of universal norms in E(Lv ) for the Zp -extension L∞,v /Lv has finite index. We have: L∞,v

ww ww w w ww

K∞,v

Fv

v vv vv v v vv

Lv

But clearly, NLv /Fv (EU (L∞,v )) ⊆ EU (K∞,v ) ⊆ E(Fv ) as the norm map sends any open subgroup of E(Lv ) onto an open subgroup of E(Fv ), it follows that EU (K∞,v ) certainly has finite index in E(Fv ). Hence Ker γv is finite for all v dividing p, and the proof of Theorem 2.8 is complete.  2.12. Remark. There are various ways of deducing that EU (L∞,v ) has finite index in E(Lv ), or equivalently that H 1 (G(L∞,v /Lv ), E(L∞,v )) is finite when E has good ordinary reduction over L by using the main results of [C-G]. A second argument to do this is given in the proof of Proposition 3.5 in Chapter 3. 2.13. Proof of Theorem 2.6. By the result of Greenberg [G, Proposition 3], the dual of H 1 (GS,∞ , Ep∞ ) has Λ(Γ)-rank ≥ [F : Q]. On the other hand, we make the following claim: Λ(Γ)-rank of ⊕ Jv\ (K∞ ) = [F : Q] − rp (E/F ).

(11)

v∈S

We remark that the proof of Theorem 2.6 is immediate from these estimates and the exact sequence (8).

The Iwasawa Theory of the Selmer Group

25

We proceed to prove the claim above using the local arguments in [C-G]. Let v be any place in S, and let Γv ⊆ Γ be the decomposition group of v in Γ = G(K∞ /F ). Since Jv (K∞ ) = ⊕ H 1 (K∞,w , E)(p), w|v

and the number of summands in the direct sum on the right is [Γ : Γv ], it is clear that the Λ(Γ)-rank of Jv (K∞ ) is equal to tw , where \ tw = Λ(Γv )-rank of H 1 (K ∞,w , E)(p), and w is some fixed prime of K∞ above v. Also, Kummer theory (take direct limits in the Kummer sequence , cf. (1.1)) gives the exact sequence 0 −→ E(K∞,w )⊗Qp /Zp −→ H 1 (K∞,w , Ep∞ ) −→ H 1 (K∞,w , E)(p) −→ 0. Hence we have tw ≤ Λ(Γv ) − rank of H 1 (K\ ∞,w , Ep∞ ). If v does not divide p, then [G, Proposition 2] shows that H 1 (K\ ∞,w , Ep∞ ) is Λ(Γv )-torsion and therefore tw = 0. On the other hand, if v does divide p, we need the finer bounds for tw obtained by using [C-G, Proposition 4.9]. In this case, the integer g occuring in loc. cit. is 1, while the integer h occuring there has the value 1 or 2 accordingly as v 6∈ Pp (E/F ) or v ∈ Pp (E/F ) (cf. [C-G, p.150]). This is because v ∈ Pp (E/F ) if and only if the Galois submodule C of Ep∞ defined in [C-G, p.150] is equal to the whole of Ep∞ . By [C-G, Proposition 4.9], we therefore have ( [Fv : Qp ] if v 6∈ Pp (E/F ) tw = 0 if v ∈ Pp (E/F ). This proves (11) since [F : Q] =

X

[Fv : Qp ] and the proof of Theo-

v|p

rem 2.6 is now complete.  Theorem 2.8 has the following simple generalization, giving further examples where Conjecture 2.5 can be proven.

26

Chapter 2

2.14. Theorem. Assume that p satisfies the following: (i) E has potential good reduction at all primes v of F dividing p. (ii) S(E/F ) is finite. \∞ ) has Λ(Γ)-rank equal to rp (E/F ). Then S(E/K Proof. The crucial point of the proof is to show that the Zp -corank of S(E/K∞ )Γ is exactly equal to rp (E/F ). Indeed, granted the \ assertion, we let t be the Λ(Γ)-rank of S(E/K∞ ). Then it is clear from \∞ ) the structure theory of finitely generated Λ(Γ)-modules that S(E/K Γ \∞ ) is dual to S(E/K∞ )Γ , and also has Zp -rank at least t. But S(E/K Γ t ≥ rp (E/F ) by Theorem 2.6. Hence the above assertion shows that necessarily t = rp (E/F ), as required. We now proceed to show that S(E/K∞ )Γ has Zp -corank equal to rp (E/F ) by analysing the fundamental diagram (9). We recall that β is surjective since Γ has p-cohomological dimension equal to 1. Also, Ker β is finite by Imai’s theorem [I] (see also the Appendix). Our hypothesis that S(E/F ) is finite shows that Im α is finite. Also, by Proposition 1.9, Coker(λS (F )) is finite. Hence we see on applying the snake lemma to (9) that our assertion on the Zp -corank of S(E/K∞ )Γ is equivalent to the statement that Ker γ has Zp -corank equal to rp (E/F ). But γ is the direct sum of the local maps γv for v in S. By Lemma 1.11, Ker γv is finite for v not dividing p. As above, put tv = 0 or [Fv : Qp ], according as E has potential P ordinary or potential supersingular reduction at v. Thus rp (E/F ) = v|p tv . We claim that Zp -corank of Ker γv is equal to tv . If E has potential ordinary reduction at v, this is simply the assertion that Ker γv is finite, which is established in the proof of Theorem 2.8. If E has potential supersingular reduction at v, then the results of [C-G] show that γv is the zero map (cf. [C-G, Proposition 4.9]), and hence Ker γv = H 1 (Fv , E)(p). Then, by Tate duality, H 1 (Fv , E)(p) is dual to E(Fv )∗ , and this latter group has Zp -rank tv . This completes the proof of Theorem 2.14. 

A theorem of Kato 2.15. Since these notes were first written, Kato’s fundamental work on Euler systems [K] attached to elliptic modular forms has now appeared. In this last section, we briefly state some of the main results of this

The Iwasawa Theory of the Selmer Group

27

paper concerning elliptic curves E defined over Q. However, the proofs of these results rely heavily on the theory of modular forms and modular curves, as well as analytic arguments, and are beyond the scope of these notes. Assume for the rest of this section that E is an elliptic curve defined over Q, and let NE denote the conductor of E. For each prime q with (q, NE ) = 1, the integer aq is defined by ˜q (Fq )) = q + 1 − aq , #(E ˜q /Fq is the reduction of E modulo q. We recall the complex where E L-function of E over Q is defined in the half plane Re(s) > 3/2 by the Euler product Y Y (1 − εq q −s )−1 ˙ (1 − aq q −s + q 1−2s )−1 , L(E, s) = q|NE

(q,NE )=1

where εq = 0, 1, or −1 according as E has additive, split multiplicative, or non-split multiplicative reduction at the prime q dividing NE . Write L(E, s) =

∞ X an n=1

n

, fE (τ ) = s

∞ X

an q n ,

n=1

where τ in C satisfies Im(τ ) > 0 and q = e2πiτ . The following fundamental theorem is due to Wiles et al [Wi], [BCDT]:2.16. Theorem. Let E be an elliptic curve defined over Q of conductor NE . Then E is modular in the sense that fE (τ ) is a newform of weight 2 and trivial character for the subgroup Γ0 (NE ) of SL2 (Z consisting of all matrices   a b c d with a, b, c, d in Z, ad − bc = 1, and c ≡ 0 mod NE . A first major consequence of this theorem is that L(E, s) has an analytic continuation over the whole complex plane, and satisfies the functional equation ξ(E, s) = wE ξ(E, 2 − s), wE = ±1, where s/2

ξ(E, s) = NE (2π)−s Γ(s)L(E, s).

28

Chapter 2

A second major consequence of the above theorem is that it enables us to apply Kato’s arguments for Euler systems attached to modular forms to the elliptic curve E. If p is a prime number, recall that µp∞ denotes the group of all p-power roots of unity. 2.17. Theorem. ([K, Theorem 14.4]) Let E be an elliptic curve defined over Q, and let F be any finite abelian extension of Q. For each prime p, the group E(F (µp∞ ) is a finitely generated abelian group, Further, the work of [K] establishes Conjecture 2.5 in a very important case. For any prime p, take H∞ = Q(µp∞ ), F = Q(µ2p ), and define Ω = G(H∞ /Q), Γ = G(H∞ /F ), ∆ = G(F/Q). Then Ω ' Γ × ∆. We say that a finitely generated Λ(Γ)-module M is Λ(Ω)-torsion, if it is torsion as a Λ(Γ)-module (or equivalently if it has an annihilator in Λ(Γ) which is not a zero divisor). 2.18. Theorem. Assume that E is defined over Q, and put H∞ = Q(µp∞ ), where p is a prime of good ordinary reduction for E. Then \∞ ) is a torsion Λ(Γ)-module. S(E/H Thus this deep theorem establishes Conjecture 2.5 whenever E has good ordinary reduction at p, and F = Q(µ2p ), or any subfield of Q(µ2p ).

Chapter 3 The Euler Characteristic Formula Introduction 3.1. Again E will be an elliptic curve defined over a finite extension F of Q, and p will be an arbitrary prime number. Moreover, as in Chapter 2, H∞ will denote an infinite Galois extension of F , whose Galois group Ω = G(H∞ /F ) is a p-adic Lie group of positive dimension. Let S(E/H∞ ) be the Selmer group of E over H∞ . If this Selmer group is to be useful in studying the arithmetic of E over the base field F , we must be able to recover the basic arithmetic invariants of E over F from some exact formula attached to S(E/H∞ ). The simplest means of obtaining such an exact formula is to calculate the Ω-Euler characteristic of S(E/H∞ ). This is the question we shall study in this chapter, concentrating mainly on the case when H∞ is the cyclotomic Zp -extension of F . In all cases in which we have been able to calculate the Ω-Euler characteristic of S(E/H∞ ), it turns out to be very closely related to the conjectural exact formula of Birch and Swinnerton-Dyer for the leading term in the development of the complex L-function of E over F at the point s = 1 in the complex plane. Although we shall not discuss the aspects of analytic Iwasawa theory further in these notes, this provides a method for attacking the conjecture of Birch and SwinnertonDyer whenever we can prove a “main conjecture” for the Λ(Ω)-module S(E/H∞ ). To calculate Euler characteristics, we must study all of the cohomology groups H i (Ω, S(E/H∞ )) (i = 0, 1, . . .) and it is useful to make the following general remarks. Let d be the dimension of Ω as a p-adic Lie group. Then by Serre’s refinement [Se3] of Lazard’s theorem, Ω will have p-cohomological dimension equal to d provided Ω has no non-trivial ptorsion. Moreover, the fundamental diagram (9) relates the finiteness of H 0 (Ω, S(E/H∞ )) to the following issues. Let us assume that both Ker β 29

30

Chapter 3

and Coker β are finite (this is true when H∞ is either the cyclotomic Zp -extension of F or the field F (Ep∞ )). Then it follows from (9) that H 0 (Ω, S(E/H∞ )) is finite if and only if both S(E/F ) and Ker γ ∩ Im λ are finite. In practice, we shall assume that S(E/F ) is finite and use the local methods of [C-G] to show that Ker γ is finite under suitable conditions on the extension H∞ and the nature of the reduction of E at the primes v of F dividing p.

Cyclotomic theory 3.2. We now take H∞ to be the cyclotomic Zp -extension K∞ of F , and as in Chapter 2, we write Γ = G(K∞ /F ). Since Γ is topologically isomorphic to Zp , it has p-cohomological dimension equal to 1. Our aim is to calculate the Γ-Euler characteristic of S(E/K∞ ) under appropriate conditions on E and p. In general, if A is a discrete p-primary Γ-module, we shall say that A has finite Γ-Euler characteristic if both H 0 (Γ, A) and H 1 (Γ, A) are finite, and we then define its Euler characteristic by χ(Γ, A) = # H 0 (Γ, A)/# H 1 (Γ, A).

(12)

For each finite place v of F , let cv = [E(Fv ) : E0 (Fv )], where, as in Chapter 1, E0 (Fv ) denotes the subgroup of points with non-singular ˜v over kv for the reduction. We write kv for the residue field at v, and E reduction of E modulo v. Assuming S(E/F ) is finite, we define Y Y 2 (d(p) (13) ρp (E/F ) = # (E/F )(p)/#(E(F )(p))2 × c(p) v × v ) ,

X

v

v|p

˜v (kv )), and for any positive integer n, n(p) denotes the where dv = #(E largest power of p dividing n. The following is the main result proven in this chapter. 3.3. Theorem. (Main Theorem) Assume that (i) p > 2, (ii) E has good ordinary reduction at all places v of F dividing p, (iii) S(E/F ) is finite. Then S(E/K∞ ) has finite Γ-Euler characteristic given by χ(Γ, S(E/K∞ )) = ρp (E/F ).

The Euler Characteristic Formula

31

The proof of the theorem is rather involved. We immediately give the proof of this result and postpone the discussion of numerical examples until the next chapters. The essence of the proof is to make a more detailed study of the fundamental diagram (9). We start by analyzing the map γ = ⊕ γv , which is a purely local calculation that does not v∈S

need the hypothesis that S(E/F ) is finite. Recall that for any v ∈ S, Jv (K∞ ) = ⊕ H 1 (K∞,w , E)(p). w|v

Let Γv denote the decomposition group of v in Γ. By Shapiro’s lemma, we have H i (Γ, Jv (K∞ )) ' H i (Γv , H 1 (K∞,w , E)(p)),

∀i≥0

for some fixed prime w of K∞ above v. Hence by the Hochschild-Serre spectral sequence, and the fact that H 2 (Fv , E) = 0 [Lemma 1.12], we obtain Ker γv = H 1 (Γv , E(K∞,w )), Coker γv = H 2 (Γv , E(K∞,w )). (Note that these groups are p-primary since Γv is a pro-p group). 3.4. Lemma. Assume that v does not divide p. Then γv is surjective, (p) and Ker γv is finite of order cv . Proof. Let Fvnr be the maximal unramified extension of Fv , and put Wv = H 1 (G(Fvnr /Fv ), E(Fvnr )). It is well-known (1.1) that Wv is the exact orthogonal complement of E0 (Fv ) under the dual Tate pairing of H 1 (Fv , E) and E(Fv ). Hence Wv is dual to E(Fv )/E0 (Fv ) and # Wv (p) = c(p) v . Now assume that v does not divide p. Then K∞,w is the unique unramified Zp -extension of Fv , whence Ker γv = Wv (p), and so the proof of the second part of the lemma is complete. As for the first part, consider the following commutative diagram, where the vertical maps are the restriction maps and the horizontal ones are the natural maps arising from Kummer theory: H 1 (Fv , Ep∞ ) γv0



H 1 (K∞,w , Ep∞ )Γv

/ H 1 (Fv , E)(p) γv

 / H 1 (K∞,w , E)(p)Γv .

(14)

32

Chapter 3

We claim that the horizontal maps are in fact isomorphisms. Indeed, Kummer theory (1.1) shows that the maps are surjective, and they are injective because E(Fv ) ⊗ Qp /Zp = 0

and

E(K∞,w ) ⊗ Qp /Zp = 0,

as v does not divide p. The restriction map γv0 is surjective since the cokernel maps to H 2 (Γv , Ep∞ ) and Γv has p-cohomological dimension 1. The surjectivity of γv now follows from (14).  3.5. Proposition. Assume that v divides p and that E has good ordinary reduction at v. Then γv is surjective, and Ker γv is finite of order (p) ˜v (kv )). (dv )2 , where dv = #(E Proof. Let M∞,w be the maximal ideal of the ring of integers of K∞,w , ˆv be the formal group over the and let k∞,w be the residue field. Let E ring of integers of Fv which gives the kernel of reduction modulo v on E. Then we have the exact sequence 0

/E ˆv (M∞,w )

/ E(K∞,w )

/ E˜ (k v ∞,w )

/ 0,

(15)

ˆv (M∞,w ) where E˜v (k∞,w ) is a finite group since k∞,w is a finite field, and E ˆv is a formal group [Si, Chapter IV]. Note that is a Zp -module since E ˜ the map E(Fv ) → Ev (kv ) is surjective because E has good reduction at v [Si, Chapter VII]. Thus, taking Γv -cohomology of the exact sequence (15), we obtain the long exact sequence 0

/ H 1 (Γ , E ˆv (M∞,w )) v

/ H 1 (Γv , E(K∞,w ))

/ H 1 (Γ , E˜ (k v v ∞,w ))

/ H 2 (Γ , E ˆv (M∞,w )) v

/ H 2 (Γv , E(K∞,w ))

/ 0.

The last zero occurs in this sequence because H 2 (Γv , E˜v (k∞,w )) = H 2 (Γv , E˜v (k∞,w )(p)) = 0, as Γv has p-cohomological dimension 1.

(16)

The Euler Characteristic Formula

33

We now use the main result of [C-G] (see Appendix A.2) for the field K∞,w and the formal group Eˆv . Here we note that K∞,w is indeed deeply ramified in the sense of [C-G] (see Appendix A.2.2), since it contains the cyclotomic Zp -extension of Qp . Hence by [C-G, Corollary 3.2] (see Appendix, Theorem A.2.3) ˆv (M)) = 0 for i ≥ 1, H i (K∞,w , E

(17)

where M denotes the maximal ideal of the ring of integers of an algebraic closure of Fv . Fv

K∞,w  Γv

Fv By the inflation-restriction sequence [Se], it follows that we have the exact sequence ˆv (M))Γv ˆv (M∞,w )) −→ H i (Fv , E ˆv (M)) −→ H i (K∞,w , E 0 −→ H i (Γv , E for all i ≥ 1. Hence by (17), we obtain isomorphisms ˆv (M∞,w )) ' H i (Fv , E ˆv (M)) for i ≥ 1. H i (Γv , E

(18)

We now show that Proposition 3.5 follows from the lemma below, whose proof will be given shortly. 3.6. Lemma. Under the hypotheses of Proposition 3.5, we have ˆv (M)) is finite of order d(p) ˜ (i) H 1 (Fv , E v = # Ev (kv )(p), ˆv (M)) = 0 for all i ≥ 2. (ii) H i (Fv , E ˆv (M∞,w )) = 0, Indeed, by Lemma 3.6 (ii) and (18), we see that H 2 (Γv , E and so by (16) it follows that Coker γv = H 2 (Γv , E(K∞,w )) = 0. Moreover (16) then reduces to the short exact sequence ˆv (M∞,w )) −→ Ker γv −→ H 1 (Γv , E˜v (k∞,w )) −→ 0. 0 −→ H 1 (Γv , E (19)

34

Chapter 3

(p)

By Lemma 3.6 (i), the group on the left side of (19) has order dv . On the other hand, to compute the order of the group on the right side of (19), we may replace E˜v (k∞,w ) by its p-primary subgroup since Γv is pro-p. Further, as Γv is isomorphic to Zp and E˜v (k∞,w ) is finite, we note that # H 1 (Γv , E˜v (k∞,w )(p)) = # H 0 (Γv , E˜v (k∞,w )(p)) = d(p) v . (p)

Hence (19) shows that Ker γv has order (dv )2 , as required. Thus the proof of Proposition 3.5 is complete once we prove Lemma 3.6. 3.7. Proof of Lemma 3.6. Consider the Weil pairing Ev,pn × Ev,pn → µpn . ˆv,pn is its own orthogonal It is well-known that under the Weil pairing, E complement and therefore ˜v,pn = Hom(E ˆv,pn , µpn ) for n ≥ 1. E

(20)

ˆv,pn (Fv ) = pn , by Tate’s local Euler characteristic theorem Since # E (cf. (1.1)), we have ˆv,pn ) = pdn+a+b , # H 1 (Fv , E where ˆv,pn ), pb = # H 2 (Fv , E ˆv,pn ). d = [Fv : Qp ], pa = # H 0 (Fv , E Assume for the rest of the proof that n >> 0. Now, by Tate local duality (1.1) and (20), we have ˆv,pn ) = # H 0 (Fv , E ˜v,pn ) = #(E˜v (kv )pn ) = d(p) pb = # H 2 (Fv , E v . (21) Consider the exact sequence 0

/E ˆ

v,pn

/E ˆ

pn v,p∞

/E ˆ

v,p∞

/0

of G(Fv /Fv )-modules. By our assumption on n, the associated long exact Galois cohomology sequence gives 0

/E ˆ

v,p∞ (F )

/ H 1 (F , E ˆv,pn ) v

/ H 1 (F , E ˆv,p∞ )pn v

/ 0.

The Euler Characteristic Formula

35

ˆv,p∞ (F ) = pa , we have Since # E ˆv,p∞ )pn = pdn+b . # H 1 (Fv , E

(22)

ˆv,p∞ ) ' (Qp /Zp )d ⊕ B H 1 (Fv , E

(23)

Therefore (p)

where B is a finite group with # B = pb = dv by (21). We also have the exact sequence 0

/E ˆ

v,pn

pn /E /E ˆv (M) ˆv (M)

/ 0.

(24)

Taking G(Fv /Fv )-cohomology and then taking the direct limit as n → ∞, we obtain the exact sequence ˆv (M) ⊗ Qp /Zp −→ H 1 (Fv , E ˆv,p∞ ) −→ H 1 (Fv , E ˆv (M)) −→ 0. 0 −→ E Since the group on the left of this sequence is divisible of Zp -corank d, ˆv,p∞ ) by (23), and it must be the maximal divisible subgroup of H 1 (F, E so it follows from (23) that ˆv (M)) = # B = d(p) H 1 (Fv , E v . This proves (i) in the lemma. To prove (ii), we take the G(Fv /Fv )-cohomology of (24) and obtain the exact sequence ˆv (M)) H 1 (Fv , E

pn

/ H 1 (F , E ˆv (M)) v

α

/ (H 2 (F , E ˆv (M)))pn v

/ H 2 (F , E ˆv,pn ) v

/ 0.

ˆv (M)) (recall that we have chosen n >> 0), By (i), pn annihilates H 1 (Fv , E and so α is injective. But then α is surjective by (i) and (21). Hence ˆv (M)) = 0. This completes the proof of Lemma 3.6. H 2 (Fv , E  The following lemma, first established by Imai [I], can be proven by local or global means (see the Appendix Theorem A.2.8 for an outline of one proof). 3.8. Lemma. The group H 0 (K∞ , Ep∞ ) is finite.

36

Chapter 3

We now return to the proof of Theorem 3.3. We split the fundamental diagram (9) into two commutative diagrams with exact rows, namely / S(E/K∞ )Γ O

0

α

β

/ S(E/F )

0

/ Im φ∞ O

/ H 1 (GS,∞ , Ep∞ )Γ O

/0

(25)

δ

/ H 1 (GS , Ep∞ )

/0

/ Im λ

and / Im φ∞ O

0

/ ⊕ Jv (K∞ )Γ v∈S O γ

δ

/ Im λ

0

/ Coker φ∞ O

/0

(26)



/ ⊕ Jv (F )

/ Coker λ

v∈S

/ 0.

where δ and  are the obvious induced maps. Note that Ker β, and therefore Ker α, are finite by Lemma 3.8 and the Hochschild-Serre spectral sequence. Also, β is surjective because Γ has p-cohomological dimension 1. Moreover, Ker δ is finite because it is contained in Ker γ, which is finite by Lemma 3.4 and Proposition 3.5. Recall that we are assuming that S(E/F ) is finite, whence it has the same order as (E/F )(p) (see Chapter 1). Applying the snake lemma to (25), we obtain the exact sequence

X

0

/ Ker α

/ Ker β

whence # S(E/K∞ )Γ /#

/ Ker δ

/ Coker α

/ 0,

X(E/F )(p) = # Coker α/# Ker α = # Ker δ/# Ker β.

(27)

But, as Ep∞ (K∞ ) is finite, we have # Ker β = # H 1 (Γ, Ep∞ (K∞ )) = # Ep∞ (K∞ )Γ = # E(F )(p). (28) To compute the order of Ker δ, we apply the snake lemma to (26). We recall that γ is surjective by Lemma 3.4 and Proposition 3.5 (whence also  is surjective), and that δ is surjective because β is surjective. Hence we obtain from (26) the exact sequence 0

/ Ker δ

/ Ker γ

/ Ker 

/ 0.

(29)

The Euler Characteristic Formula

37

But Lemma 3.4 and Proposition 3.5 yield Y Y 2 # Ker γ = c(p) × (d(p) v v ) . v

v|p

By Proposition 1.9 and our hypothesis that S(E/F ) is finite, we have Coker λ is finite of order # Coker λ = # E(F )(p). But Coker φ∞ is also then finite, since  is surjective, and # Ker  = # E(F )(p)/# Coker φ∞ . Thus by (29), we have # Ker δ = (# Coker φ∞ /# E(F )(p)) ×

Y

Y 2 (d(p) v ) .

c(p) v ×

v

v|p

Combining this formula with (27) and (28), we finally obtain # S(E/K∞ )Γ /# Coker φ∞ = ρp (E/F ), where ρp (E/F ) is given by (13). Thus, to complete the proof of Theorem 3.3, it suffices to show that H 1 (Γ, S(E/K∞ )) is finite of order # H 1 (Γ, S(E/K∞ )) = # Coker φ∞ .

(30)

To show this, we use the following result, whose proof will be given a little later. Note that the non-trivial part of the exactness of (31) is the surjectivity of λ∞ . 3.9. Proposition. Under the hypotheses of Theorem 3.3, we have the exact sequence 0

/ H 1 (GS,∞ , Ep∞ ) λ∞ / ⊕ Jv (K∞ )

/ S(E/K∞ )

v∈S

/ 0.

(31)

Assuming the exactness of (31), we take Γ-invariants and obtain the exact sequence H 1 (GS,∞ , Ep∞ )Γ

φ∞

/ ⊕ Jv (K∞ )Γ v∈S

/ H 1 (Γ, S(E/K∞ )) 

H 1 (Γ, H 1 (GS,∞ , Ep∞ )). (32)

38

Chapter 3

We claim that the group on the right hand end of this sequence is zero. Indeed, by Proposition 1.9 (i), we have H 2 (GS , Ep∞ ) = 0 since S(E/F ) is finite. As Γ has p-cohomological dimension 1, the Hochschild-Serre spectral sequence H p (Γ, H q (GS,∞ , Ep∞ )) =⇒ H n (GS , Ep∞ ) implies easily that H 1 (Γ, H 1 (GS,∞ , Ep∞ )) = 0,

(33)

as asserted. But (30) is now clear from (32), and so the proof of Theorem 3.3 is now complete, granted Proposition 3.9. We note that there is a well-known proof of Proposition 3.9 (see [P-R]), which works more gen\∞ ) is torsion over Λ(Γ). Howerally under the hypothesis that S(E/K ever, we shall give a different and simpler proof, which crucially uses the hypothesis that S(E/F ) is finite. 3.10. Proof of Proposition 3.9 Let S 0 = S \ {v | p}. We define S 0 (E/F ) and S 0 (E/K∞ ) as λ0

S 0 (E/F ) = Ker(H 1 (GS , Ep∞ ) −→ ⊕ Jv (F )) v∈S 0

λ0∞

S 0 (E/K∞ ) = Ker(H 1 (GS , Ep∞ ) −→ ⊕ Jv (K∞ )), v∈S 0

where λ0 and λ0∞ are the obvious localization maps. The commutative triangle λ / ⊕ Jv (F ) H 1 (GS , Ep∞ ) v∈S NNN NNN 0 NλNN π NNN & 

⊕ Jv (F )),

v∈S 0

where π is the projection, gives the exact sequence 0

θ

/ S(E/F )

/ S 0 (E/F )

/ ⊕ Jv (F )

/ Coker λ

/ Coker λ0

/ 0.

v|p

(34)

The Euler Characteristic Formula

39

Similarly, we have the anlagous exact sequence 0

θ∞

/ S(E/K∞ )

/ S 0 (E/K∞ )

/ Coker λ∞

/ Coker λ0 ∞

/ ⊕ Jv (K∞ )

(35)

v|p

/ 0.

Our first step will be to show that Coker λ0∞ = 0. To do this, we first note that Coker λ0 = 0. Indeed, by Proposition 1.9 (ii), Coker λ is dual to E(F )(p) because S(E/F ) is assumed finite. But the dual map θb is the natural map θb : E(F )(p) → ⊕ E(Fv )∗ , v|p

and thus θb is clearly injective, whence θ is surjective, as required. Let φ0∞ : H 1 (GS,∞ , Ep∞ )Γ −→ ⊕ Jv (K∞ )Γ v∈S 0

be the map induced by λ0∞ . Since λ0 is surjective, and γv is surjective for all v ∈ S 0 by Lemma 3.4, the obvious commutative diagram H 1 (GS , Ep∞ )

λ0

/ ⊕ Jv (F ) v∈S 0

γ 0 = ⊕ γv

β



H 1 (GS,∞ , Ep∞ )Γ

φ0∞



v∈S 0

/ ⊕ Jv (F∞ )Γ v∈S 0

shows that φ0∞ is surjective. Taking Γ-cohomology of the short exact sequence 0

/ S 0 (E/K∞ )

/ H 1 (GS,∞ , Ep∞ )

/ Im λ0



/ 0,

and noting that (33) remains valid (the argument used to prove it only uses the fact that S(E/F ) is finite), we obtain the exact sequence −→ H 1 (GS,∞ , Ep∞ )Γ −→ (Im λ0∞ )Γ −→ H 1 (Γ, S 0 (E/K∞ )) −→ 0. (36) 1 0 We also obtain that H (Γ, Im λ∞ ) = 0 because Γ has p-cohomological dimension 1. Using H 1 (Γ, Im λ0∞ ) = 0 and noting that φ0∞ is surjective, it follows easily that (Coker λ0∞ )Γ = 0

and

H 1 (Γ, S 0 (E/K∞ )) = 0.

(37)

40

Chapter 3

Hence Coker λ0∞ = 0 because Coker λ0∞ is a discrete p-primary Γmodule. We also get the second assertion of (37) for free from this argument. \φ∞ is Λ(Γ)-torsion. Since Our next step will be to show that Coker \λ∞ )Γ is dual to (Coker λ∞ )Γ , it suffices by Lemma 2.10 to show (Coker that (Coker λ∞ )Γ is finite. Taking Γ-invariants of the exact sequence 0

/ S(E/K∞ )

/ H 1 (GS,∞ , Ep∞ )

/ Im λ∞

/ 0,

and using (33) again, we obtain the exact sequence / (Im λ∞ )Γ

/ H 1 (GS,∞ , Ep∞ )Γ

/ 0,

/ H 1 (Γ, S(E/K∞ ))

and also that H 1 (Γ, Im λ∞ ) = 0. Because of this latter fact, we also have the exact sequence 0

/ (Im λ∞ )Γ

/ ( ⊕ Jv (K∞ ))Γ v∈S

/ (Coker λ∞ )Γ

/ 0.

Hence we obtain the exact sequence 0

/ H 1 (Γ, S(E/K∞ ))

/ Coker φ∞

/ (Coker λ∞ )Γ

/ 0.

But, as noted above, Coker φ∞ is finite, and so (Coker λ∞ )Γ is indeed finite. To complete the proof of Proposition 3.9, we make the following Claim. The Λ(Γ)-torsion submodule of ⊕ Jv\ (K∞ ) is zero. v|p

Indeed, dualizing (35) and recalling that Coker λ0∞ = 0, it follows \λ∞ injects into ⊕ Jv\ that the torsion module Coker (K∞ ), and so must v|p

be zero, granted the claim. Hence it remains to prove the claim and we proceed to do so below. Proof. (of claim) By definition, Jv (K∞ ) = ⊕ H 1 (K∞,w , E)(p). w|v

Pick some fixed prime w of K∞ lying above a given prime v of F dividing p, and let Γv denote the decomposition group of w in Γ. It clearly suffices to show that the Λ(Γv )-module H 1 (K\ ∞,w , E)(p) has no Λ(Γv )-torsion. Here we appeal to the theory of deeply ramified fields in [C-G] (see Appendix A.2), noting that K∞,w is deeply ramified because v divides

The Euler Characteristic Formula

41

p and it is a ramified Zp -extension of Fv . Thus, by [C-G, Proposition 4.8] (see Appendix A.2.4), we have ˜v,p∞ ), H 1 (K∞,w , E)(p) ' H 1 (K∞,w , E

(38)

˜v,p∞ denotes the Galois module of all p-power torsion points on where E the reduction E˜v of E modulo v. Now we apply a result of Greenberg ([G, Corollary 1], see Appendix) to the right hand side of (38). Note that ˆv,p∞ ' Hom(Tp (E ˜v,p∞ ), µp∞ ), E and by a basic result of Imai ([I], see Appendix, Theorem A.2.8), we ˆv,p∞ (K∞ ) is indeed finite. Thus [G, Corollary 1] shows that have that E the dual of the right hand side of (38) has no Λ(Γv )-torsion, and the claim is proved. Note that the proof of Proposition 3.9 is at last complete. This also finally completes the proof of Theorem 3.3.  We now discuss an interesting complement to Theorem 3.3, which is a variant of a result of Greenberg [G1], and which is very useful in the study of numerical examples. Our method of proof is different from that used by Greenberg; we state the most general result which emerges from our arguments. 3.11. Theorem. Assume that (i) p > 2, (ii) S(E/F ) is finite, (iii) For each P 6= 0 in E(F )(p), there exists a place v of F dividing p such that E has good reduction modulo v and the reduction of P modulo v is not zero. Then H 1 (Γ, S(E/K∞ )) = 0. 3.12. Remarks. (i) Let A be any discrete p-primary Γ-module and let X = Aˆ be its Pontryagin dual. Assume that X is a finitely generated Λ(Γ)-module. Then H 1 (Γ, A) = 0 is equivalent to the assertion that X has no finite non-zero Γ-submodules and that the characteristic power series (see Appendix A.1.5) of the Λ(Γ)-torsion submodule of X does not vanish at T = 0.

42

Chapter 3

(ii) Note that condition (iii) of the theorem above is automatically true if E(F )(p) = 0. (iii) Condition (iii) is also automatically valid if there exists a place v of F dividing p such that E has good reduction at v and the absolute ramification index ev of v satisfies ev < p − 1. For, it is well-known [Si, Chapter VII §3] that the formal group Eˆv of E at v will have a non-zero point of order p defined over Fv only if ev ≥ p − 1. (iv) As we shall see in Chapter 5, some of the most interesting applications of Theorem 3.11 occur when F = Q(µp ), and so the ramification index of p in F is equal to p − 1. (v) The results of this chapter can be used to give a simple proof that S(E/K∞ ) = 0 under the following conditions. We assume that the hypotheses of Theorem 3.3 are valid for E/F and p. Further, we assume that the quantity ρp (E/F ) defined by (13) is equal to 1. Hence, by Theorem 3.3, we have χ(Γ, S(E/K∞ )) = 1. Finally, we assume that condition (iii) of Theorem 3.11 is also valid for E/F and p. Thus Theorem 3.11 implies that H 1 (Γ, S(E/K∞ )) = 0, whence H 0 (Γ, S(E/K∞ )) = 0 because χ(Γ, S(E/K∞ )) = 1. But, as S(E/K∞ ) is a discrete Γ-module, H 0 (Γ, S(E/K∞ )) = 0 gives S(E/K∞ ) = 0. This rather curious argument works very well for certain numerical examples, including A1 = X1 (11) and p = 5, as we shall see in Chapter 5. 3.13. Proof of Theorem 3.11. We consider the following commutative diagram with exact rows H 1 (GS,∞ , Ep∞ )Γ

φ∞

O

(39)

γ

β

H 1 (GS , Ep∞ )

/ ⊕ Jv (K∞ )Γ v∈S O

λ

/ ⊕ Jv (F ) v∈S

θ

/ E(F \ )(p)

/ 0.

Here the commutative square is simply the right hand part of the fundamental diagram (9), and the exactness of the bottom row follows from our hypotheses (i) and (ii) and Proposition 1.9. We claim that \ θ(Ker γ) = E(F )(p).

(40)

Assuming (40), and using the fact that γ is surjective, a simple diagram chase shows that γ ◦ λ is then surjective. Hence the map φ∞ must certainly be surjective because φ∞ ◦ β = γ ◦ λ. We now prove (40).

The Euler Characteristic Formula

43

For each v ∈ S, let EU (K∞,v ) be the subgroup of universal norms in E(Fv ) as defined in (2.11). Then we claim that (40) is equivalent to the assertion that the natural map τ : E(F )(p)



⊕ E(Fv )/EU (K∞,v )

v∈S

is injective. This follows on noting that the dual of θ is, via the Tate pairing, the natural inclusion E(F )(p)

,→

⊕ E(Fv )∗ ,

v∈S

and that the exact orthogonal complement of Ker γ under this pairing is ⊕ EU (K∞,v )∗ (note that E(Fv )/EU (K∞,v ) is automatically pro-p v∈S

because it is dual to Ker γv ). We now proceed to prove the injectivity of τ . Suppose, on the contrary, that there exists P 6= 0 in E(F )(p) such that τ (P ) = 0. It follows that P ∈ EU (K∞,v ) for all places v of F dividing p. For each n ≥ 0, let Kn,v be the unique sub-extension of K∞,v of degree pn over Fv , and let kn,v be the residue field of Kn,v . Since v is assumed to divide p, it is a basic property of the cyclotomic Zp -extension K∞ over F that there exists an integer n0 such that K∞,v /Kn,v is totally ramified for all n ≥ n0 (cf. Appendix). Hence kn,v = kn0 ,v for all n ≥ n0 . As P ∈ EU (K∞,v ), we know that P = NKn,v /Fv (Pn ) for some Pn ∈ E(Kn,v ). We now use condition (iii) of the theorem, and choose v dividing p such that E has good reduction modulo v and such that the reduction P˜ of P modulo v is non-zero in E˜v (kv ). We have the commutative diagram E(Kn,v ) NKn,v /Fv



E(Fv )

/ E˜ (k ) v n,v 

(41)

NK^ n,v /Fv

/ E˜ (k ), v v

where the left vertical map is the norm map, and the right vertical map is its reduction modulo v, and the horizontal maps are reduction modulo v. Now choose n so large such that pn−n0 annihilates the pprimary subgroup of E˜v (kn,v ). Since the norm map from Kn,v to Kn0 ,v is multiplication by pn−n0 on E˜v (kn,v ), it follows that the reduction

44

Chapter 3

modulo v of Pn is sent to an element R of E˜v (kv ) of order prime to p by the right vertical map of (41). But, by the commutativity of (41), R = P˜ . But P˜ is non-zero of p-power order, this is a contradiction, and the proof of Theorem 3.11 is complete. 

The division field case 3.14. In the rest of the chapter, we outline, without proofs, how far we can go towards computing the Euler characteristic of the Selmer group over the field F∞ = F (Ep∞ ). We do not give proofs, and refer the interested reader to [C], [C-H] and [H] for these. For the remainder of this chapter, we shall assume that E does not admit complex multiplication. Indeed, the case when E admits complex multiplication is less interesting and essentially well-known. We again write Σ for the Galois group of F∞ over F , so that Σ can be identified with a subgroup of GL2 (Zp ), which is open by Serre’s theorem. We assume that p ≥ 5, which guarantees that Σ has no non-trivial p-torsion. Hence, by [Se3], Σ has p-cohomological dimension equal to 4. If A is a discrete p-primary Σ-module, we shall say that A has finite Σ-Euler characteristic if H i (Σ, A) is finite for i = 0, . . . , 4, and we then define its Euler characteristic by χ(Σ, A) =

4 Y i (# H i (Σ, A))(−1) .

(42)

i=0

Our aim is to compute χ(Σ, S(E/F∞ )). Unfortunately, no unconditional analogue of Theorem 3.3 has been proved to date, and so we begin by stating a conjecture (see [C-H], [H]). Let v be any place of F . As before, (cf. (3.2)), let cv = [E(Fv ) : E0 (Fv )],

dv = # E˜v (kv )

where kv is the residue field of v. Let B = B(E) be the (possibly empty) set of places v of F where ordv (jE ) < 0; here jE denotes the j-invariant of E. For each v ∈ B, let Lv (E, s) denote the Euler factor at v of the complex L-function of E over F . Hence Lv (E, s) is 1, (1/(1 − N v)s ) or (1/(1+N v)s ) according as E has additive reduction at v, split multiplicative reduction at v, or non-split multiplicative reduction at v. Assuming

The Euler Characteristic Formula

45

S(E/F ) is finite, we define ξp (E/F ) =

X

# (E/F )(p) Y (p) 2 Y × (dv ) × (cv /Lv (E, 1))(p) . # (E(F )(p))2

(43)

v∈B

v|p

(p)

Note that, since p ≥ 5, cv = 1 for all v 6∈ B [Si], and so ξp (E/F ) is related to ρp (E/F ) (cf. (13)) by the formula Y ξp (E/F ) = ρp (E/F ) × (1/Lv (E, 1))(p) . (44) v∈B

3.15. Conjecture. Assume that (i) p ≥ 5, (ii) E has good ordinary reduction at all places v of F dividing p, (iii) S(E/F ) is finite. Then S(E/F∞ ) has finite Σ-Euler characteristic given by χ(Σ, S(E/F∞ )) = ξp (E/F ). We now discuss what can be proven in the direction of this conjecture. Let M be a finitely generated left Λ(Σ)-module. We say that M is Λ(Σ)-torsion if every element of M can be annihilated by an element of Λ(Σ) which is not a zero divisor. 3.16. Theorem. [C-H] Assume the hypotheses of Conjecture 3.15. If \∞ ) is Λ(Σ)-torsion, then S(E/F∞ ) has finite Euler characteristic S(E/F given by χ(Σ, S(E/F∞ )) = ξp (E/F ). Moreover, H i (Σ, S(E/F∞ )) = 0 for i = 2, 3, 4. \∞ ), We now explain a general conjecture about the Λ(Ω)-module S(E/F which is made in [CFKSV], and which is considerably stronger than the \∞ ) is Λ(Ω)-torsion. For each finite extension L of assertion that S(E/F cyc Q, let L denote the cyclotomic Zp -extension of L. Also, we define Φ = G(F∞ /F cyc ), so that Φ is a closed normal subgroup of Ω such that Ω/Φ ' Zp .

46

Chapter 3

Following [CFKSV], we then define MΦ (Ω) to be the category of all finitely generated left Λ(Ω)-modules M such that M/M (p) is finitely generated over the sub-algebra Λ(Φ) of Λ(Ω); here M (p) denotes the p-primary sub-module of M . As is explained in §2 of [CFKSV], every module in the category MΦ (Ω) is S ∗ -torsion where S ∗ is a certain canonical Ore set of non-zero divisors in Λ(Ω); in particular, every module in MΦ (Ω) is Λ(Ω)-torsion. Then the following conjecture is made in [CFKSV, Conjecture 5.1]:3.17. Conjecture. Assume that E has good ordinary reduction at all \∞ ) always belongs to the category places v of F dividing p. Then S(E/F MΦ (Ω). As is explained in §5 of [CFKSV], there are many reasons why it is hoped that this conjecture is always true. in particular, as is explained there, it enables one to formulate a “main conjecture” for the Λ(Ω)-module \∞ ). The best evidence to date in support of this conjecture is S(E/F given by the following theorem proven in [CFKSV] (see Prop. 5.6). 3.18. Theorem. Assume that p ≥ 5, and that E has good ordinary reduction at all places v of F dividing p. Suppose further that there exists a finite extension L of F contained in F∞ , and an elliptic curve E 0 defined over L, satisfying (i) G(F∞ /L) is pro-p, (ii) E 0 is isogenous 0 /Lcyc ) is a finitely generated Z -module. \ to E over L, and (iii) S(E p \ Then S(E/F∞ ) belongs to the category MΦ (Ω). For example, this theorem applies to any of three elliptic curves A0 , A1 , A2 defined over Q of conductor 11, which are defined at the beginning of the next chapter, and the prime p = 5. Theorem 5.4 of Chapter 5 shows that we can take E 0 = A1 and L = Q(µ5 ). Finally we mention the following result, which is used in the proof of Theorem 3.16, which we shall prove in the Appendix (see A.2.9). Let E/F be any elliptic curve and p be a prime such that the Galois group Σ has no element of order p (this is automatically true if p ≥ 5). Let d be the dimension of Σ as a p-adic Lie group. Thus d = 4 if E has no complex multiplication and by the theory of complex multiplication, d = 2 when E admits complex multiplication. We define d Y i χ(Σ, Ep∞ ) = (# H i (Σ, Ep∞ ))(−1) . i=0

The Euler Characteristic Formula

47

3.19. Theorem. We have χ(Σ, Ep∞ ) = 1 and H i (Σ, Ep∞ ) = 0 for all i ≥ d. Note that the case p = 2 is allowed in Theorem 3.19. This result was originally proven by Serre [Se5] when E/F has no complex multiplication and p ≥ 5. In the Appendix, we give a different and simpler proof to Serre, which easily generalizes to arbitrary abelian varieties defined over a number field (see [C-S]).

Chapter 4 Numerical Examples over the Cyclotomic Zp-extension of Q Introduction 4.1. In this chapter, we discuss some numerical examples of elliptic curves over Q, which illustrate the general theory developed in Chapters 2 and 3. We study the Iwasawa theory of these curves over the cyclotomic Zp -extension Q∞ of Q. As usual, we write Γ = G(Q∞ /Q), and let Λ(Γ) denote the Iwasawa algebra of Γ. We work with the single isogeny class of the curves of conductor 11, and one isogeny class of curves of conductor 294. We are grateful to T. Fisher [F] for pointing out to us the latter isogeny class, and providing us with data about a 7-descent on it. The isogeny class of curves of conductor 11 was already introduced in Chapter 2. We recall that these three curves are A0 : y 2 + y = x3 − x2 − 10x − 20 A1 : y 2 + y = x3 − x2 A2 : y 2 + y = x3 − x2 − 7820x − 263580. The curve A0 is the modular curve X0 (11), and A1 is the modular curve X1 (11). All three curves are isogenous over Q, the isogenies being of degree 5. When there is no need to distinguish between the three curves, we shall simply write A to denote any one of them. The complex Lfunction of A over Q is given by L(A, s) =

∞ X n=1

48

an /ns ,

Numerical Examples

49

where the integers an (n = 1, 2, . . .) are defined by the expansion q

∞ Y

(1 − q n )2

n=1

∞ Y

(1 − q 11n )2 =

n=1

∞ X

an q n .

n=1

This is the q-series expansion of a cusp form of weight two for the subgroup Γ0 (11) of SL2 (Z) consisting of all matrices   a b c d with a, b, c, d in Z, ad−bc = 1 and c ≡ 0 mod 11. It is easily verified by computation (cf. [Cr]) that L(A, 1) 6= 0. Hence, by Kolyvagin’s theorem, it follows that Ai (Q) and (Ai /Q) are both finite for 0 ≤ i ≤ 2.

X

4.2. The three curves A0 , A1 and A2 can be distinguished by the structure of their groups of 5-division points considered as Galois modules for the Galois group G = G(Q/Q). On A0 , the point (5, 5) is a non-trivial 5-division point and we have A0,5 = Z/5Z ⊕ µ5

(45)

as G-modules. On A1 , the point (0, 0) is a non-trivial 5-division point, and we have the exact sequence of G-modules 0 → Z/5Z → A1,5 → µ5 → 0

(46)

the quotient being µ5 by virtue of the Weil pairing. This exact sequence is non-split as a sequence of G(Q11 /Q11 )-modules because the discriminant ∆(A1 ) is −11. Indeed, A1 has split multiplicative reduction at 11, and hence A1 is a Tate curve over Q11 (cf. Appendix A.3). We have ∗

Z A1 (Q11 ) = Q11 /qA , 1

where qA1 is the Tate period of of A1 at 11 (cf. Appendix A.3). If the whole group of 5-division points were rational over a finite extension L of Q11 , then the absolute ramification index of L would have to be divisible by 5 because qA1 has order 1 at 11. In particular, this proves that (46) cannot split over Q11 (µ5 ) = Q11 . On A2 , there are no nontrivial rational points, and we have the exact sequence of G-modules 0

/ µ5

/ A2,5

/ Z/5Z

/ 0.

(47)

50

Chapter 4

Again this sequence is not split as a sequence of G(Q11 /Q11 )-modules as the discriminant ∆(A2 ) = −11. We note that A1 = A0 /µ5 and A2 = A0 /(Z/5Z). Also, A0 = A2 /µ5 , and so we see that A2,5∞ contains a cyclic subgroup of order 25 which is invariant under G. 4.3. The second isogeny class of elliptic curves is given by the following two curves of conductor 294: B1 : y 2 + xy = x3 − x − 1 B2 : y 2 + xy = x3 − 141x + 657. These are the curves B1 and B2 of conductor 294 in Cremona’s tables [Cr]. In making his tables, Cremona has verified that all isogeny classes listed are complete, and so there are no other elliptic curves over Q which are isogenous to B1 and B2 . The discriminants of the curves are given by ∆(B1 ) = −2 · 3 · 72 , ∆(B2 ) = −27 · 37 · 72 . Cremona’s tables show that neither B1 nor B2 possesses any rational point of order prime to 7. This can also be seen from the fact that the number of points modulo 5 or modulo 11 is 7. From Cremona [Cr], we see that B2 has a rational point of order 7, and that B1 = B2 /(Z/7Z), where Z/7Z denotes the subgroup generated by the point of order 7. Both curves have split multiplicative reduction at 2 and 3, and bad reduction at 7. In fact, both curves achieve good ordinary reduction over the field Q7 (µ7 ). This can be seen explicitly for B1 as follows (and then it follows automatically for the isogenous curve B2 ). If we make the change of variables x = x0 + 4, y = y 0 + 3x0 + 5, we obtain the new equation for B1 y 2 + 7xy + 14y = x3 + 14.

(48)

Making the change of variables, x = π 2 x0 , y = π 3 y 0 , where π is any local parameter of Q7 (µ7 ), we clearly obtain an equation for B1 over Q7 (µ7 ) which has good reduction because 7 has ramification index 6 in Q7 (µ7 ). One also verifies easily that, on taking π = 1 − ζ, where ζ is a primitive 7-th root of unity, the reduction of this last equation is the curve y 2 = x3 − 2

(49)

Numerical Examples

51

over the field F7 . The number of points on this last curve over F7 is equal to 7, and so it is ordinary. Let Φ denote the Galois group of the totally ramified extension Q7 (µ7 ) over Q7 . Now the theory of the N´eron model shows that Φ injects into the automorphism group of (49) over F7 (see [Se2, §5.6]), and so it must coincide with the whole automorphism group, since the latter has order 6. But the automorphism group of (49) is generated by (x, y) 7→ (ωx, −y) where ω is a primitive cube root of unity, and so it certainly does not act trivially on the F7 -rational points of (49). We conclude that the Galois group Φ also acts non-trivially on the F7 -rational points of (49), a fact which we shall use later. We thank B. Totaro for making this very useful remark. The structure of the groups of 7-division points on B1 and B2 are given by the exact sequences of G-modules 0

/ µ7

0

/ Z/7Z

/ B1,7 / B2,7

/ Z/7Z

/ 0,

(50)

/ µ7

/ 0.

(51)

Since B1 and B2 form a complete isogeny class over Q, and neither B1 nor B2 admit complex multiplication, we see that neither of these exact sequences can split as G-modules. Of course, one could also use a local argument at the primes 2 or 3 to prove this for B1 , but not for B2 because ∆(B2 ) = −27 · 37 · 72 . Finally, we note that B1 and B2 are both modular, and their complex L-function does not vanish at s = 1 (see [Cr]). Hence Kolyvagin’s theorem tells us that Bi (Q) and (Bi (Q)) are finite for i = 1, 2.

X

Iwasawa theory over Q∞ for the curves of conductor 11 4.4. The study of the Iwasawa theory of the curves of conductor 11 was begun by Mazur in his seminal paper [Ma]. The results we describe now are essentially folklore (see [G2]) and have been well-known to the experts for quite some time. In describing the Iwasawa theory for a given prime p, we shall assume that (Ai /Q)(p) = 0 (0 ≤ i ≤ 2). (52)

X

This is in accord with the conjecture of Birch and Swinnerton-Dyer, which predicts that the Tate-Shafarevich group of each Ai over Q is trivial (cf. [Cr]). It should be possible to prove (52) for all primes p by

52

Chapter 4

computing non-trivial Heegner points and using the formula of GrossZagier (cf. [Gr]), but the details do not appear in the literature. We let Γ = G(Q∞ /Q) and recall that Λ(Γ) denotes the Iwasawa algebra of Γ. We shall also let A denote any one of the three curves Ai , 0 ≤ i ≤ 2, and take the set S to be {p, 11}. 4.5. Theorem. Let p be an odd prime such that (52) holds. Assume that A has good supersingular reduction at p (e.g., p = 19, 29, . . .). Then \∞ ) ' Λ(Γ) S(A/Q as a Λ(Γ)-module. \∞ ). By Theorem 2.14, we know that Proof. Let X = S(A/Q Λ(Γ) - rank of X = 1. We claim that S(A/Q∞ )Γ ' Qp /Zp .

(53)

In order to prove the claim, we simply analyse the fundamental diagram (9) with F = Q and H∞ = Q∞ : 0

/ S(A/Q∞ )Γ O α

0

/ S(A/Q)

/ H 1 (GS,∞ , Ap∞ )Γ φ∞ / ⊕ Jv (Q∞ )Γ v∈S O O γ

β

/ H 1 (GS , Ap∞ )

λ

/ ⊕ Jv (Q). v∈S

Observe that A(Q)(p) must be zero because this group injects into ˜ 5 ) = Z/5Z under reduction modulo 5. By virtue of (52), and the A(F fact that A(Q)(p) = 0, we have S(A/Q) = 0. Thus it follows that the \ by Proposition 1.9. map λ above is surjective as Coker λ ' A(Q)(p) Note also that β is injective because Ker β = H 1 (Γ, Ap∞ (Q∞ )) and Ap∞ (Q∞ ) = 0 since H 0 (Γ, Ap∞ (Q∞ )) = 0. As the map β is always surjective, we deduce that it is an isomorphism in this case. In view of these remarks, we conclude that Ker γ ' S(A/Q∞ )Γ . As S = {p, 11}, and we have γ = γp ⊕ γ11 . Now ( 5 for i = 0 c11 (Ai ) = 1 for 1 ≤ i ≤ 2.

(54)

Numerical Examples

53

Hence it follows from Lemma 3.4 that γ11 is injective. We are thus left with analysing γp . We prove that γp is in fact the zero map. Indeed, if w denotes the unique prime of Q above p, we have H 1 (Q∞,w , A)(p) = 0, by Proposition 4.8 of [C-G]. Note that since A has supersingular reduction at p, the Galois submodule C of Ap∞ defined in [C-G, p. 150] is equal to the whole of Ap∞ . Thus Ker γp = H 1 (Qp , A)(p) which is dual to A(Qp )∗ by Tate local duality. Our claim (53) now follows on observing that A(Qp )∗ is a free Zp -module of rank 1. Indeed, any b but A b torsion element of A(Qp )∗ will have to lie on the formal group A, is torsion free as the ramification index of Qp is < p − 1 (cf. [Si, Chapter VII]). The final part of the proof consists of a simple argument with Λ(Γ)-modules, (cf. [Appendix A.1]) along with (53) and the fact that X has Λ(Γ)-rank equal to 1. Let Y denote the Λ(Γ)-torsion submodule of X, so that we have the exact sequence /Y

0

/X

/ X/Y

/0

of Λ(Γ)-modules. Now (X/Y )Γ = 0 because X/Y has no non-zero Λ(Γ)torsion, and thus we obtain the exact sequence / YΓ

0

/ XΓ

/ (X/Y )Γ

/ 0.

(55)

But XΓ = Zp because of (53) and the fact that XΓ is dual to S(A/Q∞ )Γ . Also X/Y has Λ(Γ)-rank equal to 1, and thus Zp − rank of (X/Y )Γ ≥ 1. We conclude from (54) that YΓ is finite. Hence YΓ = 0 because it injects into XΓ ' Zp . But Y is a compact Λ(Γ)-module, and so YΓ = 0 =⇒ Y = 0. The structure theory for finitely generated Λ(Γ)-modules gives (cf. Appendix A.1) an exact sequence 0

/X

/ Λ(Γ)

/B

/ 0,

54

Chapter 4

where B is a finite Λ(Γ)-module. Taking Γ-invariants of this last exact sequence, we obtain the exact sequence 0

/ BΓ

/ XΓ

/ Zp

/ BΓ

/ 0.

But XΓ is a free Zp -module of rank 1, and so B Γ = 0 because it is finite. This proves in turn that B = 0, and the proof of the theorem is now complete.  4.6. Theorem. Let p 6= 5 be an odd prime satisfying (52). Assume that A has good ordinary reduction at p (e.g., p = 3, 7, 13, . . .). Then S(A/Q∞ ) = 0. Proof. The proof involves analysing the fundamental diagram as before. Since A has good ordinary reduction at p, by Theorem 3.3, the Euler characteristic χ(Γ, S(A/Q∞ )) is finite, and χ(Γ, S(A/Q∞ )) = ρp (A/Q) where ρp (A/Q) is defined by (13). We now proceed to calculate ρp (A/Q). Let A˜p denote the reduced curve over Fp . Then we have # A˜5 (F5 ) = 5,

A˜p (Fp )(p) = 0 f or p 6= 5.

This latter assertion follows from Hasse’s bound on the order of A˜p (Fp ) and the fact that 5 must divide the order of A˜p (Fp ) for all p 6= 11. We also have c11 (A0 ) = 5, c11 (Ai ) = 1 f or 1 ≤ i ≤ 2. Since p 6= 5, we again have A(Q)(p) = 0. Hence assuming (52), we deduce from (13) that ρp (A/Q) = 1. This implies that χ(Γ, S(A/Q∞ )) = 1. On the other hand, since A(Q)(p) = 0, Theorem 3.11 shows that H 1 (Γ, S(A/Q∞ )) = 0. This in turn implies that H 0 (Γ, S(A/Q∞ )) = 0 and hence S(A/Q∞ ) = 0, as S(A/Q∞ ) is a discrete Γ-module. The theorem is therefore proved. 

Numerical Examples

55

4.7. Theorem. For p = 5, we have S(A1 /Q∞ ) = 0 \ S(A 0 /Q∞ ) = Λ(Γ)/(5) 2 \ S(A 2 /Q∞ ) = Λ(Γ)/(5 ). Proof. We will only sketch the proof of this theorem, as it is wellknown, and treated in detail in [G2]. In the case p = 5, we have # A0 (Q)(5) = 5, # A1 (Q)(5) = 5, # A2 (Q) = 0. Further, it can be shown by a 5-descent argument [G2] that

X(A (Q))(5) = 0 f or all i, 0 ≤ i ≤ 2. i

Hence we deduce from Theorem 3.3 that   5 χ(Γ, S(Ai /Q∞ )) = 1   2 5

for i = 0 for i = 1 for i = 2.

Moreover, since there is no 5-torsion on the formal group of Ai over Q, Theorem 3.11 shows that H 1 (Γ, S(Ai /Q∞ )) = 0 f or all i, 0 ≤ i ≤ 2. Therefore it follows that H 0 (Γ, S(A1 /Q∞ )) = 0 and so S(A1 /Q∞ ) = 0, as it is a discrete p-primary Γ-module. \ Let fi (T ) denote a characteristic power series for S(A i /Q∞ ) as a Λ(Γ)module (see Appendix); here we are identifying Λ(Γ) with the ring Zp [[T ]] of formal power series in T with coefficients in Zp . Hence we conclude from Propostion A.1.7 of the Appendix that f0 (0) = 5u0 , f2 (0) = 52 u2

(56)

where u0 and u2 are 5-adic units. There are now two ways to complete the proof of Theorem 4.7. The first is to invoke Proposition 5.7 of [G2], which is applicable because A0 and A2 have good ordinary reduction at 5, and Ai,5∞ contains a non-trivial subgroup Di which is stable

56

Chapter 4

under the action of G. In the case of A0 , D0 = µ5 , and so we deduce from Proposition 5.7 of [G2] that f0 (T ) is divisible by 5, whence it is clear from the first equation in (56), that we can take f0 (T ) = 5. In the case of A2 , D2 has order 25, and the action of complex conjugation on D2 is odd because µ5 ⊂ D2 . We conclude in this second case that f2 (T ) is divisible by 52 , and so by (56), we can take f2 (T ) = 52 . Alternatively, we could use the results of Perrin-Riou [P-R2] or Schneider [Sch], giving the change in the characteristic power series of the dual of Selmer under an isogeny. Since, we know that the characteristic power \ series of S(A 1 /Q∞ ) is 1, we deduce also in this way that we can take f0 (T ) = 5, f2 (T ) = 52 . \ As H 1 (Γ, S(A0 /Q∞ )) = 0, it follows that S(A 0 /Q∞ ) has no non-zero finite Λ(Γ)-submodule. Hence, since f0 (T ) = 5, the structure theory \ for Λ(Γ)-modules implies that there is an embedding of S(A 0 /Q∞ ) in Λ(Γ)/5 with finite cokernel. But Λ(Γ)/5 is the ring F5 [[T ]], and so is a \ discrete valuation ring. Thus the image of S(A 0 /Q∞ ) in Λ(Γ)/5 must be a principal ideal and so itself isomorphic to Λ(Γ)/5. This proves the assertion of Theorem 4.7 for A0 . We omit the detailed proof of Theorem 4.7 for A2 and refer the reader to [G2] for a full explanation.  4.8. To avoid possible confusion with our notation for the p-adic numbers, we write Q(n) for the unique subfield of Q∞ of degree pn over Q, and put Γn = G(Q∞ /Q(n) ). We are still totally ignorant about the behaviour of the p-primary subgroup of (A) in the tower Q∞ when A has good supersingular reduction at p. It does not appear to be known whether (A/Q(n) )(p) (n = 1, 2, . . .) are even finite. Moreover, assuming the finiteness, nothing seems to be known about the asymptotic behaviour of the order of (A/Q(n) )(p) as n → ∞. By contrast, it is interesting to note that one can obtain information about the arithmetic of the elliptic curves Ai over the finite layers of Q∞ /Q from Theorem 4.5 when p is an ordinary prime, but we restrict our discussion for simplicity, to p 6= 5. Again A denotes any of the three curves Ai .

X

X

X

X

4.9. Theorem. Let p be any ordinary prime 6= 5 such that (52) is valid. Then A(Q(n) ) is finite of order prime to p and (A/Q(n) )(p) = 0 for all n ≥ 0. Proof. By Theorem 4.6, we have S(A/Q∞ ) = 0. Applying the fundamental diagram (9) with base field F = Q(n) , and noting that we have

Numerical Examples

57

A(Q(n) )(p) = 0 because A(Q)(p) = 0, we conclude that the map αn : S(A/Q(n) ) → S(A/Q∞ )Γn = 0 must be injective, whence S(A/Q(n) ) = 0. The assertions of the theorem follow from this. 

Iwasawa theory over Q∞ for the curves of conductor 294 4.10. In describing the Iwasawa theory of B1 and B2 over Q∞ for a given prime p, we shall assume that

X(B /Q)(p) = 0 i

(i = 1, 2).

(57)

In fact, the conjecture of Birch and Swinnerton-Dyer predicts that B1 and B2 both have trivial Tate-Shafarevich group over Q (see [Cr]), and again this should be provable for all p using Heegner points (see [Gr]). Exactly analogous statements to Theorems 4.5, 4.6 and 4.9 hold for all primes p 6= 2, 3, 7 for both the curves B1 and B2 , assuming (57) holds. As the proofs are entirely similar, we do not repeat them. However we now discuss the prime p = 7 for B1 and B2 , where subtler arguments are required. In fact, (57) is known to be true for p = 7, thanks to calculations of T. Fisher [F]. We also remark that despite the fact that B1,7 has a subgroup isomorphic to µ7 as a G-module, Greenberg’s Proposition 5.7 in [G2] does not apply to B1 because B1 has bad additive reduction at 7 (if it did apply, we would have a positive µ-invariant for S(B1 /Q∞ )). In fact, the following result is true. 4.11. Theorem. We have S(B1 /Q∞ ) = 0. Proof. We can take S = {2, 3, 7}. As always, we use the fundamental diagram (9) with F = Q and H∞ = Q∞ . 0

/ S(B1 /Q∞ )Γ O α

0

/ S(B1 /Q)

/ H 1 (GS,∞ , B1,7∞ )Γ φ∞ / ⊕ Jv (Q∞ )Γ v∈S O O γ

β

/ H 1 (GS , B1,7∞ )

λ

/ ⊕ Jv (Q). v∈S

58

Chapter 4

Note that λ is surjective, because B1 (Q) has no 7-torsion. Also β is injective for the same reason, and is surjective as always. In addition, S(B1 /Q) = 0, by Fisher’s calculations. Hence we deduce that S(B1 /Q∞ )Γ ' Ker γ. We now show that γ is injective, and so it will follow that S(B1 /Q∞ ) = 0 as required. But γ2 and γ3 are injective because c2 = c3 = 1 for our curve B1 . Hence it remains to show that γ7 is also injective. To do this, we appeal once again to the theory of [C-G], where the canonical exact sequence is given by 0

/C

/ B1,7∞

/D

/0

(58)

where D is the group of all 7-power division points on the curve (49) over F7 . Here D is isomorphic to Q7 /Z7 as an abelian group, but the inertial subgroup of the Galois group G(Q7 /Q7 ) acts on D via the finite quotient ∆ = G(Q7 (µ7 )/Q7 ), as described earlier in 4.3. Note also that B1 (Q7 ) is a pro-7 group because we have the exact sequence 0

/B ˆ1 (Q7 )

/ B1 (Q7 )

/ F7

/ 0,

ˆ1 denotes the formal group of B1 at 7 (this is because c7 = 1 and where B the Kodaira symbol of the reduction at 7 is II). We shall show that every element of B1 (Q7 ) is a universal norm for Q∞,7 , by using Theorem 5.2 of [C-G]. Observe that the map πQ7 of [C-G] is surjective because B1 has potential good reduction at 7. Thus Theorem 5.2 of [C-G] shows that the universal norm subgroup of B1 (Q7 ) is dual to H 1 (Q7 , D)/ Ker ρ, where ρ : H 1 (Q7 , D) → H 1 (Q∞,7 , D) is the restriction map. Put L = Q7 (µ7 ), L∞ = Q7 (µ7∞ ). Now ρ is injective because its kernel is contained in D(L∞ )∆ , and this latter group is zero because ∆ acts non-trivially on D; here we have identified ∆ with G(L∞ /Q∞ ). As is explained in the proof of Theorem 5.2 in [C-G], we have a natural surjection H 1 (Q7 , B1 ) → H 1 (Q7 , D).

(59)

By Tate duality, the universal norm subgroup will be the whole of B1 (Q7 ) provided we can show that (59) is an isomorphism. As B1 has

Numerical Examples

59

good reduction over L, it is explained in [C-G, 5.31] that we have an exact sequence 0

/ E(F ˜ 7)

/ H 1 (L, B1 ) φ

/ H 1 (L, D)

/ 0,

(60)

˜ denotes the curve (49) over F7 . Recall that ∆ acts non-trivially where E ˜ on E(F7 ). We may identify (59) with the map induced by φ in (60) between the ∆-invariants, as ∆ is of order prime to 7. This shows that (59) is an isomorphism. This completes the proof of Theorem 4.11.  4.12. Theorem. We have S(B2 /Q∞ ) = 0. Proof. We again have S = {2, 3, 7} and the fundamental diagram / S(B2 /Q∞ )Γ O

0

/ H 1 (GS,∞ , B2,7∞ )Γ φ∞ / ⊕ Jv (Q∞ )Γ v∈S O O

α

/ S(B2 /Q)

0

γ

β

/ H 1 (GS , B2,7∞ )

λ

/ ⊕ Jv (Q). v∈S

By Proposition 1.9 of Chapter 1, Coker λ can be identified with the dual of B2 (Q)(7), and so has order 7. Also, Ker β has the same order as B2 (Q)(7), and so it has order 7. Again β is surjective, and S(B2 /Q) = 0 by Fisher’s calculations [F]. We conclude from the above diagram that we have the exact sequence 0

/ Ker β

/ Im λ ∩ Ker γ

/ S(B2 /Q)Γ

/ 0.

(61)

We claim that Ker γ has order 72 . Indeed, both Ker γ2 and Ker γ3 have order 7 because c2 = c3 = 7 for B2 . Exactly as in the proof of Theorem 4.11, but working with the curve B2 rather than B1 , one can show that γ7 is injective. We omit the details. We next show that #(Im λ ∩ Ker γ) = 7.

(62)

Note that, in view of (61), this will imply that S(B2 /Q∞ )Γ = 0, and so S(B2 /Q∞ ) = 0. To prove (62), it suffices to show Ker γ maps onto Coker λ = B2\ (Q)(7). To lighten the notation, put B = B2 for the rest of the proof. By Tate duality, this latter assertion is equivalent to showing that the map B(Q)(7) → ⊕ B(Qv )/BU (Q∞,v ) (63) v∈S

60

Chapter 4

is injective, where BU (Q∞,v ) denotes the group of universal norms in B(Qv ) from Q∞,v . We shall now show using the Tate parametrization of B over Q2 (see [Ta2] or [Si2] and the remark in Appendix A.3) that the universal norm subgroup BU (Q∞,2 ) contains no 7-torsion. Hence it cannot contain B(Q)(7) and so the map in (63) is indeed injective, as required. Let q denote the 2-adic Tate period of B over Q2 . Then we have an analytic isomorphism ξ : Q∗2 /q Z → B(Q2 ) and it is well-known (see [Si2], p.431) that ξ(Z∗2 ) = B0 (Q2 ), the subgroup of points of B(Q2 ) with non-singular reduction. We claim that ξ(Z∗2 ) = BU (Q∞,2 ). Indeed, as c2 = 7, it follows that ξ(Z∗2 ) has index 7 in B(Q2 ). But we also know that BU (Q∞,2 ) has index 7 in B(Q2 ) (cf. [Lemma 3.4]). Moreover, as Q∞,2 is the unramified Z7 -extension of Q2 , every element of Z∗2 is the norm of a unit in every finite extension of Q2 contained in Q∞,2 . Thus, by the Tate parametrisation, ξ(Z∗2 ) is contained in BU (Q∞,2 ) and therefore these two groups must be equal. This completes the proof of Theorem 4.12. 

Chapter 5 Numerical examples over Q(µp∞ ) Introduction 5.1. There is considerable interest in studying the Iwasawa theory of elliptic curves over the field Q(µp∞ ), for at least two different reasons. Firstly, as R. Greenberg pointed out to the first author seven or eight years ago, one gets a whole variety of phenomena occurring for the Selmer groups over Q(µp∞ ), which simply do not occur over the cyclotomic Zp -extension of Q (for example, the existence of finite nonzero Γ-submodules in the dual of the Selmer group over Q(µp∞ )). Secondly, as was already mentioned in 2.18, there exist elliptic curves E over Q without complex multiplication for which Q(Ep∞ ) is a pro-p extension of Q(µp ). For such curves E, we can often exploit knowledge of the Iwasawa theory of E over the field Q(µp∞ ) to prove deep statements, such as Conjectures 2.16 and 3.15, for the GL2 -Iwasawa theory of E over Q(Ep∞ ) (see [C], [C-H]). We believe that much further interesting work can be done in this direction, both algebraically and analytically, and this is why we discuss some numerical examples in detail in this chapter. Most of the chapter is devoted to studying the isogeny class of elliptic curves of conductor 11 over the field Q(µ5∞ ). The results we establish were communicated to the first author orally by R. Greenberg, and they are stated without proof in [G1], and partly proven in [G2]. We give here somewhat different proofs. We also discuss the Iwasawa theory of the isogeny class consisting of the two elliptic curves B1 and B2 of conductor 294 (see 4.3) over the field Q(µ7∞ ), using descent data about B1 over Q(µ7 ), which has been proved by T. Fisher [F]. 61

62

Chapter 5

General strategy for the curves of conductor 11 over Q(µ5∞ ) 5.2. Again, we let A denote any of the three curves A0 , A1 , A2 of conductor 11. We first outline the general strategy that we will use to study the curves of conductor 11 over Q(µ5∞ ). Put F = Q(µ5 ), K∞ = Q(µ5∞ ), Γ = G(K∞ /F ). We shall show later by arguments of classical descent theory, that we have S(A1 /F ) = 0. This, in turn, implies that both S(A0 /F ) and S(A2 /F ) are finite. In fact, we shall show later that S(A0 /F ) = 0 and S(A2 /F ) has order 54 . We assume throughout that the set S consists of the primes of F that lie above 5 and 11. Now A/F satisifes the hypotheses of Theorem 3.3 and so from Proposition 3.9, we have an exact sequence 0

/ H 1 (GS,∞ , Ap∞ ) λ∞ / ⊕ Jv (K∞ )

/ S(A/K∞ )

v∈S

/ 0.

(64)

Taking Γ-invariants along this sequence, we obtain the exact sequence H 1 (GS,∞ , Ap∞ )Γ

φ∞

/ ⊕ Jv (K∞ )Γ v∈S

/ H 1 (Γ, S(A/K∞ ))

/0

(65) since (cf. (33)) H 1 (Γ, H 1 (GS,∞ , Ap∞ )) = 0. Thus it is clear that in order to compute H 1 (Γ, S(A/K∞ )), we need to analyse Coker φ∞ . We do this as before, by considering the following commutative square that is a part of the fundamental diagram (9): H 1 (GS,∞ , Ap∞ )Γ

φ∞

O

/ ⊕ Jv (K∞ )Γ v∈S O γ

β

H 1 (GS , Ap∞ )

(66)

λ

/ ⊕ Jv (F ). v∈S

We recall that the maps β and γ are surjective and since S(A/F ) is finite, we know by Proposition 1.9 that \ Coker λ ' A(F )(p).

Numerical examples over Q(µp∞ )

63

Clearly, Coker φ∞ = Coker(φ∞ ◦ β) = Coker(γ ◦ λ). Let τ : A(F )(p) →

⊕ A(Fv )/AU (K∞,v )

v∈S

be the natural composite homomorphism A(F )(p) ,→

⊕ A(Fv ) 

v∈S

⊕ A(Fv )/AU (K∞,v )

v∈S

where AU (K∞,v ) is the subgroup of universal norms, i.e., AU (K∞,v ) = ∩0 NK 0 /Fv (E(K 0 )), K

as K 0 varies over all finite extensions of Fv contained in K∞,v and NK 0 /Fv denotes the norm map. Clearly, from (66), we have an exact sequence Ker γ

τb

/ Coker λ

/ Coker(γ ◦ λ)

/ 0,

and we claim that the arrow on the left is the dual of τ . In particular, we get an isomorphism Coker τb ' Coker φ∞ .

(67)

We now justify the above claim. Let Γv := G(K∞,v /Fv ) for v ∈ S, where K∞,v is as before. We have (cf. 3.13) Ker γ ' ⊕ H 1 (Γv , A)(p). v∈S

The group H 1 (Γv , A)(p) is dual to ⊕ A(Fv )/AU (K∞,v ), (cf. 3.13). v∈S

Using the above remarks on the dual groups, it is clear that the claim is justified. Observe therefore that (67) can be re-written as \τ ' Coker φ∞ . Ker

(68)

Thus our strategy in computing Coker φ∞ will be to analyse the homomorphism τ .

64

Chapter 5

5.3. To help the reader through the computations that follow, we present in this paragraph a preview of the results and the methods used. Because of (68), we focus on the study of the homomorphism τ . We have A0 (F )5 ' Z/5 ⊕ µ5 , A1 (F )5 ' Z/5, A2 (F )5 ' µ5 . The above assertions hold for A1 and A2 because the exact sequences (46) and (47) do not split as G(Q/Q)-modules. It is very easy to see (cf. Lemma 5.6) that the subgroup of points corresponding to Z/5 of A1 (F )5 and A0 (F )5 do not lie in the universal norm subgroup for the unique prime in F above 5, and therefore do not lie in Ker τ . For the points µ5 of A0 (F )5 and A2 (F )5 , the situation is more subtle, and we prove that in both cases µ5 lies in the universal norm subgroup at the unique prime above 5. However, in the case of A0 , we show that µ5 does not belong to the universal norm subgroup for each of the four primes v dividing 11. Hence µ5 too does not lie in Ker τ and therefore Ker τ is trivial. For the curve A2 (F ), and the four primes v dividing 11, the quotient group A2 (Fv )/A2,U (Fv ) is itself trivial, and hence in this case Ker τ is of order 5. These remarks determine Ker τ for all three curves, and in view of (68) and (65), we get   if i = 0 0 1 H (Γ, S(Ai /K∞ )) = 0 if i = 1   Z/5 if i = 2. In fact, more is true for the curve A1 , as the following theorem shows. 5.4. Theorem. We have S(A1 /K∞ ) = 0 where K∞ = Q(µ5∞ ). Proof. We begin by giving a direct proof of the fact that H 1 (Γ, S(A1 /K∞ )) = 0 using Theorem 3.11. Indeed, all the hypotheses of this theorem hold with F = Q(µ5 ) and p = 5, since A1 (F )(5) ' Z/5 is not contained in the kernel of reduction modulo 5. Further, by Theorem 3.3, the Γ-Euler characteristic of S(A1 /K∞ ) is finite and given by χ(Γ, S(A1 /K∞ )) = ρ5 (A1 /F ).

Numerical examples over Q(µp∞ )

65

But ρ5 (A1 /F ) = 1 since cv (A1 ) = 1 for each prime v of F above 11, and A1 (F )(5) = 5, A˜1 (F5 ) = Z/5,

X(A /F )(5) = 0; 1

the last statement is true because S(A1 /F ) = 0 (cf. Theorem 5.18 below). Hence H 0 (Γ, S(A1 /K∞ )) = 0 which implies that S(A1 /K∞ ) = 0, thereby proving the theorem.  We remark that this example shows that Theorem 3.11 is sometimes applicable in cases where Proposition 4.15 of [G2] cannot be used. However, Theorem 3.11 cannot be used for the curves A0 and A2 over F , and this is why we must make a more detailed study of the universal norms in these cases.

The curve A0 5.5. We now start on the computations for the curve A0 and the cyclotomic Z5 -extension K∞ = Q(µ5∞ ) over F = Q(µ5 ). In order to simplify notation, we shall denote the curve A0 by X for the whole of this section. All other notation is as in the previous sections. Let K∞,5 = Q5 (µ5∞ ) and let XU (K∞,5 ) ⊆ X(F5 ) be the universal norm subgroup where F5 denotes the completion of F at the unique prime above 5. We shall study the homomorphism τ (cf. 5.2) componentwise. Let τ5 be the homomorphism τ5 : X(F )5 → X(F5 )/XU (K∞,5 ) which is the component of τ for the unique prime above 5. 5.6. Lemma. The map τ5 restricts to an injective map on the subgroup Z/5 ⊆ X(F )5 . Proof. We use essentially the same arguments as in the proof of Theorem 3.11. Regarding Z/5 as lying in X(F5 ), it suffices to show that every non-zero point P in Z/5 does not belong to XU (K∞,5 ). Take any non-zero point P in Z/5. We will show that it is not a norm even from F1,5 := Q5 (µ52 ). Consider the commutative diagram X(F1,5 ) NF1,5 /F5



X(F5 )

/ X(F ˜ 5) 5

 / X(F ˜ 5)

(69)

66

Chapter 5

where the left vertical arrow is the norm map and the right one is its reduction modulo 5 and the horizontal arrows are the reduction maps. If a non-zero point P of Z/5 is a norm from F1,5 , it is clear from (69) that the reduction of P modulo 5 must be zero. But P cannot be in the kernel of reduction modulo 5, since it is defined over Q5 . Hence the lemma is proved.  5.7. We now study the image of the subgroup µ5 ⊂ X(F5 )5 under τ5 . We show that this subgroup in fact does lie in the universal norm subgroup, or equivalently that τ5 (µ5 ) = 0. The proof of this proceeds as follows. We first prove that a result about the divisible subgroup H 1 (F5 , A˜5∞ )div of H 1 (F5 , A˜5∞ ) is equivalent to the assertion τ5 (µ5 ) = 0. The first step is the following general proposition. Let p be any prime, L a finite extension of Qp and ML the maximal ideal of the ring of integers ˆ E ˜ for the formal of L. Let E be an elliptic curve over L, and write E, group and reduction of E respectively. 5.8. Proposition. Assume that E/L has good ordinary reduction and ˆ that E(M L )(p) is of order p. Let L∞ be the cyclotomic Zp -extension of L. Then the following statements are equivalent: ˜p∞ )div , where ρ is the restriction map (i) Ker ρ ⊆ H 1 (L, E ˜p∞ ) → H 1 (L∞ , E ˜p∞ ). ρ : H 1 (L, E ˆ (ii) Every element of E(M L )(p) is a universal norm from L∞ . Proof. Let EU∗ (L∞ ) be the maximal pro-p subgroup of EU (L∞ ) (the former group is the same as the p-adic completion of the latter group). Then assertion (ii) is equivalent to the following statement: EU∗ (L∞ ) has non-trivial torsion.

(70)

ˆ Clearly EU∗ (L∞ ) has non-trivial torsion if it contains E(M L )(p). Con∗ versely, suppose that EU (L∞ ) has non-trivial torsion. Then, a norm argument similar to that in (3.13) shows that any element of E(L)(p) ˆ which is in EU (L∞ ) is necessarily in E(M L )(p). Therefore we have to prove that (i) is equivalent to (70). Consider the Pontryagin dual ∗ (L ) of E ∗ (L ). By duality, (70) is equivalent to the statement E\ ∞ ∞ U

U

∗ that E\ U (L∞ ) is not divisible. We now appeal to one of the main results

Numerical examples over Q(µp∞ )

67

of [C-G] (cf. [C-G, Theorem 5.2]), which gives an explicit description of the dual of EU∗ (L∞ ). Under the Tate pairing, the dual of EU∗ (L∞ ) is the ˜p∞ )/ Ker ρ (cf. Appendix A.2.5). It is immediate now group H 1 (L, E that (70) is equivalent to the assertion ˜p∞ )/ Ker ρ is not divisible. H 1 (L, E

(71)

ˆ Since E(M L )(p) is assumed to be of order p, it is easily seen that (71) ˜p∞ )div . This is equivalent to the assertion (i) that Ker ρ ⊆ H 1 (L, E completes the proof of the proposition.  5.9. In this paragraph, we formulate the equivalent statements of Proposition 5.8 in yet another way for the curve X = A0 over F5 = Q5 (µ5 ). Let ˆ M5 be the maximal ideal of the ring of integers of F5 . Then X(M 5 )(5) 2 ˆ is of exact order 5 because it cannot contain a 5 -division point of X. 2 ˆ Indeed, a point of order 5 on X can be rational only over an extension of Q5 of ramification index 20 (cf. [Si], Chapter IV, §6). Hence by Proposition 5.8, in order to prove that τ5 (µ5 ) = 0, it suffices to show ˜ 5∞ )div where that Ker ρ ⊆ H 1 (F5 , X ˜ 5∞ ) → H 1 (K∞,5 , X ˜ 5∞ ) ρ : H 1 (F5 , X is the restriction map as before. Now Tate duality gives a dual pairing ˜ 5∞ ) → Q5 /Z5 . ˆ 5∞ )) × H 1 (F5 , X H 1 (F5 , T5 (X

(72)

Indeed, since the Weil pairing restricts to a non-degenerate pairing ˆ 5n × X ˜ 5n → µ5n , X the usual cup-product ˜ 5n ) → H 2 (F5 , µ5n ) ' Z/5n ˆ 5n ) × H 1 (F5 , X H 1 (F5 , X

(73)

is a non-degenerate pairing. The pairing in (72) in then obtained from the above pairing on taking limits. The exact orthogonal complement ˜ 5∞ )div in (72) is the torsion subgroup H 1 (F5 , T5 (X ˆ 5∞ ))tors . of H 1 (F5 , X ˆ This group is easily computed using Kummer theory on X. There is an exact sequence 0

/ X(M ˆ 5)

/ H 1 (F , T (X ˆ 5∞ )) 5 5

/ T (H 1 (F , X)) ˆ 5 5

/ 0.

68

Chapter 5

ˆ is zero as H 1 (F5 , X) ˆ is a finite group (by The group T5 (H 1 (F5 , X)) Lemma 3.6) and hence ˆ ˆ 5∞ ))tors ' X(M H 1 (F5 , T5 (X 5 )(5). It emerges from the above discussion that in order to show that τ5 (µ5 ) ˆ is zero, it suffices to show that the group X(M 5 )(5) is orthogonal to 1 ˜ Ker ρ ⊆ H (F5 , X5∞ )div in (72). But ˜ 5 ) = Hom(Γ5 , X ˜ 5 ) ' Z/5. Ker ρ ' H 1 (Γ5 , X We have therefore shown τ5 (µ5 ) = 0 ⇐⇒

1 ˆ ˜ µ5 = X(M 5 )(5) is orthogonal to H (Γ5 , A5 ) under the Tate local duality pairing.

(74) 5.10. We next discuss how the right hand side of (74) follows from ˆ 5 (M5 ). a cup-product calculation. Let α be a generator of µ5 ' X Consider the composite 1 ˆ 5 (M5 ) → X(M ˆ ˆ X 5 )/5 ,→ H (F5 , X5 )

where the first map is the canonical one and the second map is part of ˆ Denote by α ˆ5) the Kummer sequence for X. ˜ the image of α in H 1 (F5 , X under this composite map. Similarly, if β is a generator of ˜ 5 ) ,→ H 1 (F5 , X ˜ 5 ), Z/5 ' H 1 (Γ5 , X ˜ 5 ). Then the pairing (α, β) under the let β˜ denote its image in H 1 (F5 , X ˜ in (73), which Tate local duality pairing is just the cup-product (˜ α, β) in turn is the restriction of the cup-product H 1 (F5 , X5 ) × H 1 (F5 , X5 ) → H 2 (F5 , µ5 ) ' Z/5 induced by the Weil pairing. Since the group X5 is rational over F5 , we have H 1 (F5 , X5 ) = Hom(G(F 5 /F5 ), X5 ). We carry out this cupproduct calculation now, using well-known folklore relating the Weil pairing and multiplicative Kummer generators which is explained in general in Appendix A.4. Let F1,5 = Q5 (µ52 ). Since Γ5 has a unique quotient of order 5, which is G(F1,5 /F5 ), we have ˜ 5 ) = Hom(G(F1,5 /F5 ), X ˜ 5 ). H 1 (Γ5 , X

Numerical examples over Q(µp∞ )

69

ˆ 52 )/F5 and that determined by The extension determined by α ˜ is F5 (X ˜ β is F5 (µ52 )/F5 (cf. Corollary A.4.2). Since µ5 ⊂ F5 , let ξ be such that ˆ 52 ) = F5 (ξ 1/5 ). Let ζ be a primitive 5th -root of unity. Then by F5 (X ˜ = 0 if and only if the Lemma A.4.4, we see that the Tate pairing (α ˜ , β) 5-Hilbert norm residue symbol (ξ, ζ)5 = 1. Summarizing, we now have τ5 (µ5 ) = 0 ⇐⇒ (ξ, ζ)5 = 1. 5.11. Proposition. The 5-Hilbert norm-residue symbol (ξ, ζ)5 = 1. Proof. Recall that the Hilbert symbol (ξ, ζ)5 = 1 if ξ is a norm from F1,5 = F5 (ζ 1/5 ). Write (ξ, F1,5 /F5 ) for the local Artin symbol of ξ in G(F1,5 /F5 ). Hence we must show that (ξ, F1,5 /F5 ) = 1. But by the basic functoriality of the Artin symbol, we have (ξ, F1,5 /F5 ) = (η, F1,5 /Q5 ),

(75)

where η = NF5 /Q5 (ξ), and the right hand side of (75) denotes the Artin 5 symbol for the abelian extension F1,5 /Q5 . We now show that η ∈ Q∗5 , which will clearly prove the proposition. Write ∆ = G(F5 /Q5 ) and ω : ∆ → (Z/5Z)∗ be the character giving the action of ∆ on µ5 . The crucial point is that ˆ 52 ) is also abelian over Q5 . Hence ∆ = G(F5 /Q5 ) acts the extension F5 (X ˆ 52 )/F5 ) via inner automorphisms. But multiplicative trivially on G(F5 (X Kummer theory gives a ∆-isomorphism 5

ξF5∗ /F5∗

5

ˆ 52 )/F5 ), µ5 ). ' Hom(G(F5 (X 5

5

Thus, we must have that ∆ acts on ξF5∗ /F5∗ via ω, where ω is as above. 5 Hence NF5 /Q5 (ξ) ∈ Q∗5 because ω is a non-trivial character of ∆ and the proposition is proved.  We have thus finally established the following result. 5.12. Corollary. The subgroup µ5 ⊂ X(F )5 lies in the universal norm subgroup XU (K∞,5 ) ⊆ X(F5 ). 

70

Chapter 5

5.13. Let Kn,5 denote the field Q5 (µ5n+1 ). Corollary 5.12 shows that, given any R ∈ µ5 ⊂ X(F5 ), then for each n ≥ 1 there exists Rn ∈ X(Kn,5 ) such that NKn,5 /F5 (Rn ) = R. We remark that in fact we can choose Rn ˆ to lie on the formal group X(M n,5 ), where Mn,5 denotes the maximal ideal of the ring of integers of Kn,5 . This is because we can clearly replace Rn by any point of the form Rn + Wn , where Wn belongs to Z/5 ⊂ X(F5 ). Since the subgroup Z/5 ⊂ X(F5 ) maps surjectively onto ˜ the reduction X(Z/5) of X modulo 5, we simply choose Wn so that ˆ Rn + Wn reduces to zero modulo 5, and therefore lies in X(M n,5 ). 5.14. We turn our attention now to the primes v ∈ S that lie above 11. The prime 11 splits completely in F and it suffices to consider any one of the four primes that lie above 11. Fix one such prime v. Then Fv = Q11 and K∞,v = Q11 (µ5∞ ) is the unique unramified Z5 -extension of Q11 . Recall that we are interested in computing Ker τ . Also recall from Lemma 5.6 that τ is injective on the subgroup Z/5 of X(F )(5). By Corollary 5.12, the subgroup µ5 maps into the universal norm subgroup XU (K∞,5 ). However, we will now show that for v dividing 11, the vcomponent of τ τv : X(F )(5) → X(Q11 )/XU (K∞,v ) is injective. 5.15. Proposition. The subgroup µ5 ⊂ X(F )(5) does not lie in the universal norm subgroup XU (K∞,v ), i.e., τv (µ5 ) 6= 0, for v dividing 11. Proof. Let v be any prime of F dividing 11. Since X has split multiplicative reduction at v, it admits an 11-adic Tate parametrization over Q11 (see [Ta2] or [Si2], and the remarks in Appendix A.3). Let q0 denote the 11-adic Tate period of X. Then we have an analytic surjection ∗

ξ : Q11 → X(Q11 )

(76)

whose kernel is precisley q0Z . As K∞,v is the unique unramified Z5 extension of Q11 , the index of XU (K∞,v ) in X(Q11 ), which is equal to (5) the order of H 1 (Γv , X(K∞,v )), is equal to c11 = 5 (see Lemma 3.4). Let Z∗11 denote the group of units of Z11 . Since K∞,v is the unramified Z5 -extension of Q11 , every element of Z∗11 is a norm from every finite extension of Q11 which is contained in K∞,v . Thus, as ord11 (q0 ) = 5, we must have XU (K∞,v ) = ξ(Z∗11 ). (77)

Numerical examples over Q(µp∞ )

71

In order to avoid confusion between groups of points on X and subgroups of Q∗11 , let us, for the rest of the proof, write W for the subgroup µ5 of X(F ). In view of (77), the proposition will follow provided we can show ξ(µ5 ) 6= W.

(78)

But this is clear from the isogeny 0 → W → X = A0 → A1 → 0. Indeed, let q1 be the 11-adic Tate period of A1 . By virtue of the existence of this isogeny, we must have q15 = q0 (see [Ta2], p. 344), and the isogeny is then given by the commutative diagram 

1 0

/Z

0

 /Z

/ q0

5

1



/ Q∗ 11

/ A0 (Q11 )

/0

1

 / A1 (Q11 )

/ 0.

 / Q∗ 11 / q1

It is plain from this diagram that ξ(µ5 ) is not equal to the kernel of the right hand vertical map, proving (78). This completes the proof of the proposition.  Note that we have finally proved 5.16. Theorem. For the curve A0 , the prime p = 5 we have H 1 (Γ, S(A0 /Q(µ5∞ )) = 0. Proof. Recall that τ is a direct sum of τ5 and the τv ’s, where v runs over the primes dividing 11. Suppose τ (z) = 0, where z = x + y, with x ∈ Z/5 and y ∈ µ5 . But, by Corollary 5.12, τ5 (y) = 0 and so τ5 (x) = 0. On the other hand, by Lemma 5.6, τ5 is injective on Z/5 and so x = 0. Hence τ (y) = 0 and so τv (y) = 0, whence y = 0 by Proposition 5.15. Thus Ker τ = 0, but by (5.2), (65), and (68), \τ ' Coker φ∞ ' H 1 (Γ, S(A0 /K∞ )). Ker 

72

Chapter 5

The curve A2 We can straightaway reap the benefits of the work done in the previous section now to study the curve A2 . 5.17. Theorem. For the curve A2 and the prime p = 5, we have H 1 (Γ, S(A2 /Q(µ5∞ )) = Z/5. Proof. We have (cf. 5.3) A2 (F )(5) ' µ5 . Further, A2 = A0 /(Z/5), and the isogeny from A0 to A2 induces an isomorphism from Aˆ0 to Aˆ2 , where Aˆi denotes the formal group of Ai at the prime 5; this is clear because Z/5 does not lie on Aˆ0 . Now, as was remarked in (5.13), each element of µ5 ∈ Aˆ0 (M5 ) is a norm of a point in Aˆ0 (Mn,5 ) for all n ≥ 0, where Mn,5 is the maximal ideal of the ring of integers of Q5 (µ5n+1 ). Hence the same is true for µ5 when viewed as a subgroup of the points on the isomorphic formal group Aˆ2 . In particular, we see that the subgroup µ5 of A2 (F5 ) lies in the universal norm subgroup A2,U (K∞,5 ). However, for A2 , the points in µ5 are also universal norms from K∞,v , for each prime v of F dividing 11. This is because A2,U (K∞,v ) is the whole of A2 (Q11 ) by Lemma 3.4, since cv = 1 for each v dividing 11, as the discriminant of A2 is −11. Thus τ (µ5 ) = 0 \τ ' H 1 (Γ, S(A2 /K∞ )) as before, and Ker τ is of order 5. Since Ker the theorem is proved. 

Infinite descent on A1 over Q(µ5 ) The aim of this section is to use classical descent theory `a la Mordell-Weil to prove the following result: 5.18. Theorem. We have S(A1 /F ) = 0 when F = Q(µ5 ) and p = 5. 5.19. As has already been explained (see the proof of Theorem 5.4), Theorem 5.18 implies easily that S(A1 /Q(µ5∞ )) = 0. Our arguments to prove Theorem 5.18 have been inspired by those in §5 of Greenberg’s article [G2], where it is shown that S(A1 /Q) = 0 for p = 5. However, more delicate arguments are required to carry out the descent overQ(µ5 ), and we have therefore taken the liberty of including rather full details.

Numerical examples over Q(µp∞ )

73

We shall use the following notation throughout this section. Put F = Q(µ5 ). We recall that 5 is totally ramified and 11 splits completely in F , and we let S denote the set of primes of F above 5 or 11. As usual, FS will denote the maximal extension of F unramified outside S and the archimedean primes of F , and we put GS = G(FS /F ) as always. It will be more convenient for us to work with the 5-Selmer group S(A1 /F, 5) which is defined by the exactness of the sequence 0

/ S(A1 /F, 5)

/ H 1 (GS , A1,5 )

/ ⊕ (H 1 (Fv , A1 ))5 . v∈S

Then, as is explained in the proof of Lemma 1.8, we have the exact sequence 0

/Ω

/ S(A1 /F, 5)

/ (S(A1 /F ))5

/ 0,

(79)

where Ω = Z/5Z denotes the subgroup of A1,5 which is generated by the rational point (0, 0) of order 5. Here we have used the fact that A1,5∞ (F ) = Ω. Hence Theorem 5.18 is equivalent to showing that Ω = S(A1 /F, 5), or equivalently, that every element of S(A1 /F, 5) arises from dividing some multiple of (0, 0) by 5. We analyse S(A1 /F, 5) via the exact sequence (46) of GS -modules. To avoid possible confusion with subgroups of the multiplicative group, let us put Θ = A1,5 /Ω. Thus Θ is isomorphic to µ5 as a Galois module, sequence of Galois modules (which is just (46) notation) /Ω / A1,5 /Θ 0

and we have the exact re-written in our new / 0.

(80)

Now A1,5 (F ) = Ω because (80) does not split, which in turn is because the discriminant of A1 is −11. Hence, taking GS -cohomology of (80), we obtain the exact sequence 0



/ H 1 (GS , Ω)

g

/ H 1 (GS , A1,5 )

f

/ H 1 (GS , Θ)

(81)

where f and g denote the induced maps as indicated. We recall that S(A1 /F, 5) is a subgroup of H 1 (GS , A1,5 ). 5.20. Proposition. We have an inclusion S(A1 /F, 5) ⊂ Ker f .

74

Chapter 5

Proof. Observe that since µ5 is rational over F , we have H 1 (GS , Θ) = Hom(GS , µ5 ). The proof is immediate from the local Lemma 5.21 given below, and assertion (i) of Proposition 5.26, about the global arithmetic of F . Indeed, if α is any element of S(A1 /F, 5), let Lα be the fixed field of the kernel of f (α). By Lemma 5.21 below, Lα must be a cyclic extension of F , which is unramified outside 5, and in which every prime of F dividing 11 splits completely. Also, the degree of Lα over F must divide 5. By (i) of Proposition 5.26 below, there is no non-trivial extension of F with these properties. Hence Lα = F , and so f (α) = 0, completing the proof of the proposition.  5.21. Lemma. For α ∈ S(A1 /F, 5), let Lα denote the fixed field of f (α). Then Lα is a cyclic extension of F of degree dividing 5, which is unramified outside the unique prime of F dividing 5, and in which all the four primes of F dividing 11 split completely. Proof. All is clear, except the final assertion. Let v be any prime of F dividing 11, and fix an identification of Fv with Q11 . Let sv denote the restriction map from H 1 (GS , Θ) to H 1 (Q11 , Θ). We must show that sv (f (α)) = 0 for all α ∈ S(A1 /F, 5). As A1 is a Tate curve over Q11 , we have the canonical exact sequence of G(Q11 /Q11 )-modules / C11

0

/ D11

/ A1,5∞

/ 0,

(82)

where C11 is isomorphic to µ5∞ and D11 is isomorphic to Q5 /Z5 (with trivial Galois action) as G(Q11 /Q11 )-modules. We claim that the global subgroup Ω of A1 (Q11 ) can be identified with (C11 )5 in the exact sequence (82). To prove this, we note that we have an isogeny ϕ /Ω

0

/ A1

ϕ

/ A0

/ 0.

Let q0 be the 11-adic Tate period of A0 , and q1 the 11-adic Tate period of A1 . As was already remarked in the proof of Proposition 5.15 we must have q0 = q15 . Thus, at the level of Tate curves, the isogeny φ must be of the form (see [Ta2], p. 324) 

1 0 0

/Z 

/ Q∗ 11

1

/Z

1

/ q1



5



/ Q∗ 11 / q0

/ A1 (Q11 ) 

/0

ϕ

/ A0 (Q11 )

/ 0.

Numerical examples over Q(µp∞ )

75

It is clear from this diagram that Ω = Ker ϕ = (C11 )5 . It follows from (82) that we can identify Θ with (D11 )5 as G(Q11 /Q11 )-modules. Now let α denote any element of S(A1 /F, 5). Let tv be the composite tv : H 1 (GS , Θ)

sv

/ H 1 (Q11 , (D11 )5 ) uv / H 1 (Q11 , D11 ),

where uv is the map induced by the inclusion of (D11 )5 into D11 . Note that uv is clearly injective, since G(Q11 /Q11 ) acts trivially on D11 . We claim that we always have tv (f (α)) = 0. Indeed, we have the commutative diagram H 1 (GS , A1,5 ) f



H 1 (G

S , Θ)

h

/ H 1 (GS , A1,5∞ )

tv

rv

/ H 1 (Q11 , A1,5∞ )  / H 1 (Q11 , D11 )

where h is the induced map, and rv is the restriction map. But h(α) is in S(A1 /F ), and so rv (h(α)) must be zero because it lies inside the subgroup A1 (Q11 ) ⊗ Q5 /Z5 = 0 of H 1 (Q11 , A1,5∞ ). Thus it is clear from the diagram that tv (f (α)) = 0, whence sv (f (α)) = 0 because, as was remarked above, the map uv is injective. This completes the proof of the lemma.  The beauty of Proposition 5.20 is that it tells us that the image of g in the exact sequence (81) contains S(A1 /F, 5). We now proceed to study the properties of the extensions of F which arise from those β in Hom(GS , Ω) with g(β) in S(A1 /F, 5). The first lemma we need is already established in §5 of [G2] as an important step in the proof given there that S(A1 /Q) = 0 for p = 5. Let T = {5, 11}. Clearly, FS is also the maximal extension of Q unramified outside 5, 11 and ∞. Put GT = G(FS /Q). 5.22. Lemma. Let Q(µ11 )+ be the maximal real subfield of the field Q(µ11 ) of 11-th roots of unity. Then the kernel of the natural map η : H 1 (GT , Ω) → H 1 (GT , A1,5∞ ) is precisely Hom(G(Q(µ11 )+ /Q), Ω). Proof. Let W denote the kernel of η. Now it is clear that H 1 (GT , Ω) = Hom(G(R/Q), Ω)

76

Chapter 5

where R is the compositum of Q(µ11 )+ and the first layer of the cyclotomic Z5 -extension of Q. Hence W has order at most 52 . On the other hand, W is certainly non-trivial because it must contain the inverse image of the subgroup Ω of H 1 (GT , A1,5 ) under the natural map η by Proposition 5.20 and (79). Hence the assertion of the lemma will follow if we can show that every element of W must factor through an extension of Q which is unramified at 5. Let A˜1 denote the reduction of A1 modulo 5, and let A˜1,5∞ denote the group of all 5-power division points on A˜1 . Let I5 denote the inertial subgroup of some fixed prime of Q above 5. We clearly have the commutative diagram H 1 (GT , Ω) 

H 1 (I5 , Ω)

/ H 1 (GT , A1,5∞ )

/ H 1 (Q , A ˜1,5∞ ) S

(83)

 / H 1 (I , A ˜1,5∞ ). 5

But Ω maps injectively under reduction modulo 5, and the bottom horizontal arrow is then injective because I5 acts trivially on Ω and A˜1,5∞ . Thus every element in W must map to zero in the left vertical arrow, and therefore the extensions of Q corresponding to elements of W must be unramified at 5, as required. This completes the proof of the lemma.  We now analyse the local conditions which arise for those β in Hom(GS , Ω) such that g(β) ∈ S(A1 /F, 5). The next lemma covers the case of primes v of F dividing 11. Put G11 = G(Q11 /Q11 ). Taking G11 -invariants of the non-split exact sequence (80) of G11 -modules, we obtain the exact sequence 0



/ Hom(G11 , Ω) g11 / H 1 (Q11 , A1,5 ).

(84)

Put Y11 = A1 (Q11 )/5. Local Kummer theory on A1 gives, as usual, the canonical injection Y11 ,→ H 1 (Q11 , A1,5 ). If β ∈ Hom(G11 , Ω), we write Rβ for the fixed field of the kernel of β. As usual, we write Q11 (µ11 )+ = −1 Q11 (ζ11 + ζ11 ), where ζ11 is any primitive 11-th root of unity. 5.23. Lemma. Let x be any element of the subgroup Y11 of H 1 (Q11 ,A1,5 ). Let Mx denote the compositum of all fields Rβ for β running over all elements of Hom(G11 , Ω) with g11 (β) = x. If x 6= 0, then Mx is an unramified extension of Q11 (µ11 )+ . If x = 0, then Mx is contained in an unramified extension of Q11 (µ11 )+ .

Numerical examples over Q(µp∞ )

77



Proof. Let ξ : Q11 → A1 (Q11 ) denote the Tate parametrization. We have already shown in the proof of Lemma 5.21 that ξ(µ5 ) = Ω. It follows easily that the image of Θ is Hom(G11 , Ω) in (84) is generated 1/5 1/5 1/5 by the homomorphism σ 7→ ξ(σ(q1 ))/q1 , where q1 is any fixed 5-th root of the Tate period q1 of A1 . It is plain that the fixed field of the ker1/5 nel of this homomorphism is Q11 (q1 ). On the other hand, Lemma 5.22 shows that the natural map from Hom(GT , Ω) to H 1 (GT , A1,5 ) sends Hom(G(Q(µ11 )+ /Q), Ω) to the subgroup Ω of H 1 (GT , A1,5 ). Write H = Q11 (µ11 )+ . It follows immediately from this last fact that g11 (Hom(G(H/Q11 ), Ω)) ⊂ Y11 .

(85)

But # Y11 = 5 because c11 (A1 ) = 1, and thus we see that the inclusion in (85) must in fact be an equality because the group on the left of 1/5 1/5 (85) is not zero. Hence Mx = H(q1 ) if x 6= 0 and Mx = Q11 (q1 ) if 1/5 x = 0. But q1 is a local parameter of Q11 , and both H and Q11 (q1 ) are tamely ramified cyclic extensions of Q11 . Let Qnr 11 denote the maximal unramified extension of Q11 . By the theory of tame ramification (see nr 1/5 for example, [Se2]), we necessarily have HQnr 11 = Q11 (q1 ), whence the assertion of Lemma 5.23 is now clear.  We next discuss the unique prime of F above 5. Let w denote this unique prime and put Gw = G(F w /Fw ). We consider the natural map g5 : Hom(Gw , Ω) → H 1 (Fw , A1,5 ). Put Y5 = A1 (Fw )/5, which we view, as usual, as a subgroup of w , A1,5 ). Again if β ∈ Hom(Gw , Ω), we write Rβ for the fixed field of the kernel of β.

H 1 (F

5.24. Lemma. Let x be any element of the subgroup Y5 of H 1 (Fw , A1,5 ). Let Mx denote the compositum of all fields Rβ for β running over all elements of Hom(Gw , Ω) with g5 (β) = x. Then Mx is an unramified extension of Fw . Proof. Let A˜1 denote the reduction of A1 modulo 5, and let A˜1,5∞ be the group of all 5-power torsion points on A˜1 . Let δ denote the canonical composite map δ : H 1 (Fw , A1,5 ) → H 1 (Fw , A1,5∞ ) → H 1 (Fw , A˜1,5∞ ).

78

Chapter 5

The crucial point in the proof of the lemma is δ(Y5 ) = 0.

(86)

Indeed, (86) is a special case of a very general principle (see [C-G] or [G2]). It can be justified by remarking that δ(Y5 ) clearly maps to ˜ 5 ). But zero under the natural map from H 1 (Fw , A˜1,5∞ ) to H 1 (Fw , A˜1 (F ˜ the latter map is injective because A1 (F5 ) is a torsion group. Let Iw denote the inertial subgroup of Gw . Now, analogous to (83), we have the commutative diagram Hom(Gw , Ω) 

Hom(Iw , Ω)

/ H 1 (Fw , A1,5 )

δ

/ H 1 (F , A ˜1,5∞ ) w  / H 1 (I , A ˜1,5∞ ). w

Again the bottom horizontal arrow is injective, since Ω maps injectively under reduction modulo w and Iw acts trivially on Ω and A˜1,5∞ . Hence, if β is any element of Hom(Gw , Ω) with g5 (β) ∈ Y5 , it follows from (86) and the above commutative diagram that the restriction of β to Iw must be trivial, i.e. Rβ is an unramified extension of Fw . This completes the proof of the lemma.  5.25. Proof of Theorem 5.18. We can now complete the proof of Theorem 5.18. If β ∈ Hom(GS , Ω), we write Kβ for the fixed field of Ker β. Let r denote the dimension of S(A1 /F, 5) as a vector space over F5 . We define Λ to be the compositum of all the fields Kβ as β ranges over g −1 (S(A1 /F, 5)). Since Ker g has order 5, we see immediately that [Λ : F ] = 52r . Now Lemma 5.22 shows that if we define P to be the compositum of Q(µ11 )+ and F , P = Q(µ11 )+ F, (87) then we must have g(Hom(G(P/F ), Ω)) = Ω ⊂ S(A1 /F, 5). This means that one of the fields Kβ for β in g −1 (S(A1 /F, 5)) is the field P . Hence [Λ : P ] = 52r−1 . But, combining Lemmas 5.23 and 5.24, we see immediately that Λ must be an unramified extension of P . Note

Numerical examples over Q(µp∞ )

79

that, since r ≥ 1, this shows by class field theory that 5 must divide the class number of P . On the other hand, by (ii) of Proposition 5.26, 5 divides the class number of P to the first power only. Hence r = 1, and so S(A1 /P, 5) = Ω. In view of the exact sequence (79), this completes the proof of Theorem 5.18.  Finally, we are left to prove the following proposition about the global arithmetic of F , which played a central role in the above descent argument. 5.26. Proposition. (i) There is no non-trivial abelian 5-extension of F = Q(µ5 ), which is unramified outside the unique prime of F above 5, and in which every prime of F above 11 splits completely. (ii) The class number of the field Q(µ55 ) is 10. Proof. Assertion (ii) is very old, and is given, for example, in the table of class numbers at the end of [W]. Recall that Λ is the subfield of Q(µ55 ) which is given by (87), namely it is the compositum of Q(µ5 ) and Q(µ11 )+ . Since [Q(µ55 ) : Λ] = 2, it follows from (ii) that the class number of Λ must be of the form 5 · 2k for some integer k ≥ 0, and it is this latter fact which is needed in the proof of Theorem 5.18. We also remark in passing that the arguments in the proof of Theorem 5.18 show a priori that the class number of Λ, and therefore also the class number of Q(µ55 ), must be divisible by 5. We do not see any elegant proof of (i), and we just blast it out by studying congruences in the Kummer generator. Thus we suppose that, contrary to (i), there is a cyclic extension L of F of degree 5, which is unramified outside the unique prime of F above 5, and in which the primes of F above 11 split completely. By Kummer theory, we then have L = F (β 1/5 ), where β can only be divisible by the unique prime of F above 5. It follows from Hensel’s lemma that a prime v of F above 11 will split completely in L if and only if the congruence X 5 ≡ β mod v is soluble in the ring of integers of Fv . This latter congruence √ will be soluble if and only if we have β ≡ ±1 mod v. Let α2√= (1 + 5)/2 be the fundamental unit of the maximal real subfield Q( 5) of F . We define w to be one of the two primes of F above 11 such that α2 ≡ 4 mod w. Write ∆ for the Galois group G(F/Q). Thus we conclude that the congruence σ(β) ≡ ±1

mod w for all σ ∈ ∆

(88)

80

Chapter 5

must hold. Let ζ denote a primitive 5-th root of unity. Then the unique prime of F√ above 5 is generated by ζ − 1. Thus, putting α1 = ζ, α2 = (1 + 5)/2, α3 = ζ − 1, we see that we can write β = α1n1 α2n2 α3n3 , where n1 , n2 , n3 are integers, which we are only interested in modulo 5. We proceed to show by a brutal calculation that (88) yields four contradictory linear equations for n1 , n2 , n3 modulo 5, unless all of n1 , n2 , n3 are divisible by 5. Now 3 is of order 5 mod 11, and so we fix our choice of ζ by assuming that ζ ≡ 3 mod w. Write σi (i = 1, . . . , 4) for the element of ∆ which acts on µ5 by raising to the i-th power. The following table gives the values of the σ(αj ) modulo w:

α1 α2 α3

σ1 3 4 2

σ2 −2 −3 3

σ3 5 −3 4

σ4 4 4 3.

Taking the squares of the congruence (88), and noting that 4 ≡ 34 mod 11, and 52 ≡ 3 mod 11, we deduce immediately that the following four linear equations hold for n1 , n2 , n3 viewed modulo 5: 2n1 + 3n2 + 4n3 ≡ 0 4n1 + 2n2 + 2n3 ≡ 0 n1 + 2n2 + 3n3 ≡ 0 4n1 + 4n2 + n3 ≡ 0. But the only solutions of these equations are n1 ≡ n2 ≡ n3 ≡ 0 mod 5. This contradicts the hypothesis that L is an extension of F of degree 5, and so the proof of Proposition 5.26 is now complete. 

The Selmer group of A0 and A2 over Q(µ5∞ ) 5.27. Let fi (T ) denote the characteristic power series for the dual of S(Ai /Q(µ5∞ )). Theorem 5.4 shows that we can take f1 (T ) = 1. 5.28. Theorem. We have f0 (T ) = 52 and f2 (T ) = 54 . Proof. We apply the formula of Perrin-Riou [P-R2] and Schneider [Sch] on the change of the characteristic power series under isogeny to the two

Numerical examples over Q(µp∞ )

isogenies

81

0

/ µ5

/ A0

/ A1

/0

0

/ Z/5

/ A0

/ A2

/ 0.

Since µ5 lies on the formal group of A0 at the unique prime of F above 5, we conclude from the first exact sequence and the fact that f1 (T ) = 1, that we can take f0 (T ) = 52 . Similarly, since Z/5 does not lie on the formal group of A0 at the unique prime above 5, we conclude from the second exact sequence and the fact that f0 (T ) = 52 , that we have f2 (T ) = 54 .  5.29. Corollary. We have

X(A /F )(5) = 0, # X(A /F )(5) = 5 . 0

2

4

Proof. This follows immediately by combining Theorem 5.28 with the Euler characteristic formula (Theorem 3.3), recalling that (see Appendix, Proposition A.1.7) fi (0) = χ(Γ, S(Ai /K∞ ))

(0 ≤ i ≤ 2).



The curves of conductor 294 over Q(µ7∞ ) 5.30. Finally, we outline the arguments for studying the isogeny class of curves B1 and B2 (see 4.3) of conductor 294 over Q(µ7∞ ). We now put F = Q(µ7 ), K∞ = Q(µ7∞ ), and we recall that B1 and B2 have good ordinary reduction of F above 7. We shall assume without proof the following result of T. Fisher [F], which is parallel to Theorem 5.18 for the curve A1 /Q(µ5 ). 5.31. Theorem. We have S(B1 /F ) = 0 when F = Q(µ7 ) and p = 7. It is remarkable that B1 /F satisfies Theorem 5.31 and also has cv = 1 for all primes v of F dividing 2 and 3. As B1 has good ordinary reduction at the unique prime of F above 7, we can apply Theorem 3.3 to it, with

82

Chapter 5

p = 7. Recalling that the reduction (49) of B1 at the unique prime of F above 7 has 7 points in F7 , we conclude from Theorem 5.31 and Theorem 3.3 that χ(Γ, S(B1 /K∞ )) = 1. In particular, this shows that the characteristic power series of the dual of S(B1 /K∞ ) is a unit in Λ(Γ), and so S(B1 /K∞ ) must be finite by the structure theory for finitely generated torsion modules over Λ(Γ). We are very grateful to B. Totaro (unpublished) for providing us with the elegant proof given below of the following stronger result: 5.32. Theorem. We have S(B1 /K∞ ) = 0. Proof. We will use Theorem 3.11 to show that H 1 (Γ, S(B1 /K∞ )) = 0. Since χ(Γ, S(B1 /K∞ )) = 1, this will imply that H 0 (Γ, S(B1 /K∞ )) = 0 and hence the assertion of the theorem. By Theorem 3.11, it suffices to show that the subgroup µ7 of B1 (F )(7) does not lie on the formal group of B1 at the unique prime v of F above 7. If, on the contrary, µ7 lay on the formal group, we would have an exact sequence of Galois modules 0

/ µ7

/ B1,7

/ E(F ˜ 7)

/ 0,

(89)

˜ denotes the curve (49) over F7 . Now, as was explained in where E ˜ 7 ) in (89) is non-trivial. Hence the 4.3, the action of G(F/Q) on E(F existence of (89) would contradict the fact that, by the Weil pairing, the determinant of the Galois representation on B1,7 is the character giving the action of G(Q/Q) on µ7 . Hence µ7 cannot lie on the formal group of B1 at the unique prime of F above 7, proving Theorem 5.32.  5.33. Theorem. The characteristic power series of S(B2 /Q(µ7∞ )) is 73 . Proof. We apply the formula of Perrin-Riou [P-R2] and Schneider [Sch] to the isogeny / µ7 / B1 / B2 /0 0 and use the fact that µ7 does not lie on the formal group of B1 at the unique prime of F above 7.  5.34. Corollary. We have

X(B /F )(7) = 0. 2

Proof. This follows immediately from Theorems 5.33 and 3.3. 

Numerical examples over Q(µp∞ )

83

5.35. We make the following final remark. Theorem 5.4 shows that none of the curves A0 , A1 , A2 of conductor 11 has a point of infinite order in the field Q(µ5∞ ). Similarly, Theorem 5.32 shows that none of the curves B1 , B2 of conductor 294 has a point of infinite order in the field Q(µ7∞ ). However, it is unknown whether the first three curves have any point of infinite order in their common field of 5-power division points. Similarly, it is unknown whether either of the curves B1 , B2 has a point of infinite order in their common field of 7-power division points. Nevertheless, we are grateful to K. Matsuno for pointing out to us that the three curves A0 , a1 , A2 should have points of infinite order in the field ˙ 11 )= , L = Q(µ52 )Q(µ where Q(µ11 )+ denotes the maximal real subfield of Q(mu11 ). Here L is a subfield of the field of all 5-power division points on these curves, of degree 100 over Q. Matsuno calculated that the complex L-series of any of these curves over the field L has a zero at s = 1 of order 4, whence the conjecture of Birch and Swinnerton-Dyer predicts that the rank of their Mordell-Weil groups should also be 4. However, it is a reflection of our lack of knowledge and lack of computing power not to be able to prove at present that the rank is 4.

Appendix In this appendix, we expand on some results that have been used in the notes. It is, of course, beyond the scope of these notes to provide detailed proofs of the results. However, we felt that it would be of use to the reader, if we isolated them and elaborated a little more, giving additional references to a fuller treatment.

A.1. Structure theory of Iwasawa modules A.1.1. The main reference for this section is Washington’s book [W]. The results in this section have been used mainly in Chapters 2 and 3 of the notes. Let F be a number field. We say a Galois extension F∞ /F is a Zp -extension if Γ = G(F∞ /F ) is topologically isomorphic to Zp . Let Γn = pn Γ and write Fn for the fixed field of Γn , so that the degree of Fn over F is pn . F∞  Γn

Fn

F Recall that the Iwasawa algebra of Γ, denoted Λ(Γ) (we shall abbreviate the notation to Λ) is defined as Λ(Γ) := lim Zp [Γ/Γn ], ←

where the inverse limit is defined via the ring homomorphism induced by the natural maps G(Fn+1 /F ) → G(Fn /F ). There is an isomorphism Zp [[T ]] ' Λ under which T maps to (γ−1), where γ is any fixed topological generator of Γ. Thus Λ is a complete local noetherian ring. 84

Appendix

85

Given a compact or discrete Γ-module M , the action of Γ can be extended to an action of the whole of Λ on M , by linearity and continuity. The structure of Λ-modules is well-understood, at least up to the weaker notion of pseudo-isomorphism. Recall that two Λ-modules M and M 0 are pseudo-isomorphic if there is a Λ-homomorphism M → M 0 with finite kernel and cokernel. If M and M 0 are pseudo-isomorphic, we write M ∼ M 0 . The main structure theorem is the following, which was originally proved by Iwasawa, and later by Serre in a more direct fashion. A.1.2. Theorem. Let M be a finitely generated Λ-module. Then s

t

i=1

j=1

M ∼ Λr ⊕ ( ⊕ Λ/pni ) ⊕ ( ⊕ (Λ/fj (T )mj ) where r, s, t, ni , mj ∈ Z and fj (T ) ∈ Zp [T ] are irreducible monic distinguished polynomials. (A monic polynomial f (T ) ∈ Zp [T ] is distinguished if p divides all its coefficients except the leading one). The r that appears in such an expression is the rank of M as a Λ-module. The module M is Λ-torsion if and only if r = 0. Put Y Y g(T ) = ( pni )( fj (T )mj ). i

j

If r = 0, we define the characteristic power series of M to be any generator of the principal ideal (g(T )) in Λ. Note that a characteristic power series is a well-defined element Ps of Λ, up to mulitiplication by a unit. Assuming rP = 0, the integer i=1 ni is called the µ-invariant of M and the integer tj=1 mj deg(fj (T )) is called the λ-invariant of M . We now state an analogue of Nakayama’s lemma for Λ-modules. We refer the reader to [W] for a proof, or to [B-H] for a detailed proof in a much more general context. A.1.3. Lemma. Let M be a compact Λ-module and m the unique maximal ideal of Λ. If mM = M , then M = 0. A.1.4. Corollary. Let M, Λ, m be as above. If M/mM is finitely generated as a Λ/m(= Z/p)-module, then M is finitely generated as a Λ-module.

86

Appendix

Proof. The hypothesis implies that M/mM is finite as a Λ/m-module. Let m1 , . . . , mn ∈ M be elements of M which generate M/mM and let N =< m1 , . . . , mn >. Then N + mM = M and M/N is compact. But m(M/N ) = M/N , and so Lemma A.1.3 implies that M = N and hence M is finitely generated as a Λ-module.  A.1.5. Lemma. Let A be a discrete p-primary Γ-module and suppose b its Pontryagin dual, is a finitely generated Λ-module of rank r. Then A, the following are equivalent: (i) H 1 (Γ, A) = 0 bΓ=0 (ii) (A) b Γ has Zp -rank r and A b has no non-zero finite submodule. (iii) (A) Proof. (i) ⇐⇒ (ii) : This follows from a standard duality result in group cohomology. (i) ⇐⇒ (iii) : We first observe the following general reuslt which follows from the structure theory of Λ-modules and the identification of Zp [[T ]] with Λ under which T 7→ γ − 1, γ a topological generator of Γ. Suppose M is a finitely generated Λ-module such that M ∼ Λk ⊕ (⊕(Λ/(fi (T ))) i

where fi (T ) ∈ Zp [T ] is a monic distinguished polynomial or a constant of the form pm . Clearly, (Λ)Γ ' Zp and (Λ/fi (T ))Γ ' Zp ⇐⇒ T divides fi (T ). Thus, Zp − rank of MΓ = k + δ where δ = #{i | fi (0) = 0}.

(A1)

Now, since T maps to γ − 1, if x ∈ M Γ , then γ(x) = x and hence (γ − 1)x = T x = 0. It is therefore clear that Z ' Λ/(T ) =⇒ Z Γ ' Zp , and Z ' (Λ/pm ) =⇒ Z Γ = 0.

Appendix

87

If Z is of the form Λ/fi (T ) where fi (T ) is a monic distinguished polynomial, then one sees that Zp [T ]/(fi (T )) ' Zp [[T ]]/(fi (T )) and therefore 0 6= x ∈ Z Γ ⇐⇒ T | fi (T ). Noting that pseudo-isomorphism does not affect the Zp -rank, it follows that Zp − rank of M Γ = δ where δ is as in (A1) above. The implications (i) ⇐⇒ (iii) is now an ˆ easy exercise, on taking M = A.  The arguments in the above proof give the following result. A.1.6. Corollary.

Let M, Λ, Γ be as before. Then MΓ f inite =⇒ M is Λ − torsion. 

We mention below how the characteristic power series of a torsion Λ(Γ)-module relates to the Γ-Euler characteristic of its Pontryagin dual. Let M be a finitely generated Λ-torsion module, and let fM (T ) be a charˆ = Hom(M, Qp /Zp ) is a discrete acteristic power series of M . Then M Γ-module. A.1.7. Proposition. With notation as above, the following assertions hold: ˆ ) or H 1 (Γ, M ˆ ) is finite. (i) fM (0) 6= 0 if and only if either H 0 (Γ, M (ii) If fM (0) 6= 0, we have 0 1 ˆ ˆ | fM (0) |−1 p = # H (Γ, M )/# H (Γ, M ).

The proof is an easy consequence of the structure theory of Λmodules (see [G2, §4] for a detailed proof). In the notes, we use this \∞ ), where E is an elliptic curve over a numproposition for M = S(E/K ber field F having good ordinary reduction at all primes of F dividing p, and such that the selmer group S(E/F ) is finite, with K∞ the cyclotomic Zp -extension of F .

88

Appendix

A.2. Deeply ramified p-adic fields and cyclotomic extensions A.2.1. One of the chief inputs in the local cohomology calculations in Chapter 3 is the theory of deeply ramified extensions. The main reference for this is [C-G]. We shall state the principal results from this theory that are used in these notes. Let M denote the maximal ideal of the ring of integers of Fv . Suppose K/Fv is an algebraic extension. Then the Galois group G(Fv /K) acts on M, and we have the cohomology groups H i (K, M) = H i (G(Fv /K), M). A.2.2. Definition. An algebraic extension K of Qp is said to be deeply ramified if H 1 (K, M) = 0. There are various equivalent conditions that define a deeply ramified extension (cf. [C-G, p. 143]), but for the purposes of these notes, the cohomological condition above is the most useful. Let L be any finite extension of Qp . A basic example of a deeply ramified field K is a ramified Zp -extension of L. More generally, a large class of examples occuring from a theorem of Sen are Galois extensions K of L such that G(K/L) is a p-adic Lie group with infinite inertial subgroup [C-G, Theorem 2.13]. Let E/L be an elliptic curve. Recall ˆv which is the that associated to E are two natural objects, namely E ˜ formal group over the ring of integers of L and Ev which is the reduced curve, defined over the residue field kv (cf. [Si, Chapters IV and VII]). The formal group is in fact the kernel of the reduction modulo v. The principal result of [C-G] which is used in these notes is the following (cf. [C-G, Theorem 3.1 and Corollary 3.2]). A.2.3 Theorem. Let E/L be an elliptic curve. for each deeply ramified extension K of L, we have ˆv (M)) = 0 ∀ i ≥ 1. H i (K, E

A.2.4. One of the main advantages of working with deeply ramified extensions is that it vastly simplifies the computations of local cohomology groups whose coefficient modules are not torsion groups. This is amply borne out in the local computations of Chapter 3 for a prime v dividing p such that E has good ordinary reduction at v. The deeply

Appendix

89

ramified extension of Fv that one works with here is K∞,v , the cyclotomic Zp -extension of Fv . Suppose E/Fv and K∞,v are as above. Then we have (cf. [C-G, Proposition 4.8]) ˜p∞ ). H 1 (K∞,v , E(Fv ))(p) ' H 1 (K∞,v , E

(A2)

The main ingredient in proving (A2) is the explicit description of the image of the Kummer homomorphism for deeply ramified extensions (cf. [C-G, Proposition 4.3]). A.2.5. A second striking application of the theory of deeply ramified extensions is in the study of universal norm subgroups of elliptic curves [C-G, §5]. As before, Fv is a finite extension of Qp and E/Fv an elliptic curve. For any profinite abelian group X, let X ∗ denote its maximal pro-p subgroup. Suppose K/Fv is an algebraic extension. Recall that the group of universal norms of E from K is defined as the subgroup EU (K) = NK/Fv (E) = ∩ NL/Fv (E(L)) L

where L runs over all finite extensions of Fv in K, and NL /Fv denotes the norm map. We have the non-degenerate Tate pairing H 1 (Fv , E)(p) × E ∗ (Fv ) → Qp /Zp , and the restriction map r : H 1 (Fv , E)(p) → H 1 (K, E)(p). It is very well-known that EU∗ (K) is the exact orthogonal complement of Ker r in the Tate pairing. However, it is not a priori easy to determine either of these groups without the help of the theory of deeply ramified fields. We illustrate what happens when we assume that E has good ordinary reduction over Fv and K is any deeply ramified extension of Fv . Indeed, under the above hyotheses, in the notation of [C-G], we then have ˆp∞ , ˜p∞ , C=E D=E ˆ (resp. E) ˜ is the associated formal group (resp. reduced curve). where E Further, the homomorphism ˜p∞ ) πFv : H 1 (Fv , Ep∞ ) → H 1 (Fv , E

90

Appendix

is surjective (cf. [C-G], Proposition 5.3]). Let ˜p∞ ) ˜p∞ ) → H 1 (K, E ρ : H 1 (F, E denote the restriction map. As K/Fv is a deeply ramified extension, we see from [C-G, Theorem 5.2] that the group ˜p∞ )/ Ker ρ Ω(E, K/Fv ) := H 1 (Fv , E is canonically isomorphic to the image of the restriction map r : H 1 (Fv , E)(p) → H 1 (K, E)(p). From this, one deduces that the subgroup EU∗ (K) of E(Fv ) is canonically dual to Ω(E, K/Fv ). This gives the desired description of EU∗ (K) in ˜p∞ . terms of the cohomology of the p-primary Galois module E A.2.6. In this paragraph, we discuss briefly the cyclotomic Zp -extension of a number field F . The main reference for this is [W], see also [L]. Let F (µp∞ ) be the Galois extension of F obtained by adjoining the p-power roots of unity. Recall that the cyclotomic Zp -extension of F , denoted K∞ , is by definition, the fixed field of the torsion subgroup of G(F (µp∞ )/F ). The extension K∞ /F is unramified at all primes v of F which do not lie above p. On the other hand, every prime lying above p ramifies in K∞ . Further, there exists n ≥ 0 such that every prime which ramifies in K∞ /Kn is totally ramified. We note there are only finitely many primes that lie above p. More generally, it is also easy to see that there are only finitely many primes of K∞ above every rational prime. If w is any prime of K∞ , we write K∞,w for the union of the completions of all the finite extensions of F contained in K∞ . It is plain now that if w is a prime of K∞ that does not lie above p, then the extension K∞,w is the unique unramified Zp -extension of the completion of F at w. If w divides p, then K∞,w is one of the basic examples of a deeply ramified field. A.2.7. In this paragraph, we discuss Imai’s theorem and give a nonstandard proof for elliptic curves using Lie algebra cohomology. Our main motivation for this is that it leads to a simple proof of Serre’s theorem (see Theorem 3.19) on the Euler characteristics of elliptic curves. Let us also recall that we used Imai’s theorem in Chapter 3, to prove the finiteness of Ker β in the fundamental diagram.

Appendix

91

Let A be an abelian variety defined over a finite extension L of Qp , and let L∞ be the cyclotomic Zp -extension of L. Then, Imai proved that the p-primary subgroup of A(L∞ ) is finite, provided that A has potential good reduction. We will only need the following global statement for an elliptic curve E defined over a finite extension F of Q. A.2.8. Theorem. The group Ep∞ (K∞ ) is finite, where K∞ is the cyclotomic Zp -extension of F . Proof. Consider the tower of extensions F∞  ∆

K  ∞ Γ

F where we recall that F∞ = F (Ep∞ ). Let Σ = G(F∞ /F ) and ∆ = G(F∞ /K∞ ). As usual, we define Tp (E) = lim Epn , n←

and Vp (E) = Tp E ⊗ Qp . The module Tp E is a free Zp -module of rank two and the action of Σ on it gives an injective homomorphism Σ ,→ GL2 (Zp ) via which we can identify Σ as a compact subgroup of GL2 (Qp ). Hence Σ is a p-adic Lie group. To prove the assertion of the theorem, we can clearly assume that µp ⊂ F if p > 2 and µ4 ⊂ F if p = 2, so that K∞ = F (µp∞ ). By the Weil pairing, ∆ = G(F∞ /K∞ ) is given by the intersection of SL2 (Zp ) with Σ. Let d denote the Lie algebra of ∆. We recall that H 0 (d, Vp (E)) is the Qp -subspace of all vectors in Vp (E) which are annihilated by all elements in the Lie algebra d. The starting point of our proof is to show that H 0 (d, Vp (E)) = 0.

(A3)

92

Appendix

To do this, it clearly suffices to prove that there exist invertible elements in d. If E does not have complex multiplication, we appeal to Serre’s deep theorem that Σ is open in GL2 (Zp ), whence ∆ is open in SL2 (Zp ). Thus, in this case, d is the Lie algebra of SL2 (Zp ) which is well-known to consist of all matrices in M2 (Qp ) with trace zero. Therefore d clearly has invertible elements. Assume now that E has complex multiplication. We use the local results given in the appendix of Serre [Se6] to show the existence of an invertible element in d. Pick any place v of F lying above p. Since E automatically has potential good reduction at v (because E has complex multiplication), the Lie algebra of the decomposition group of v is given by Theorem A.2.2 of [Se6] when E has potential supersingular reduction at v, and by Corollary 2 of A.2.4. in [Se6] when E has potential ordinary reduction at v. It is clear from the explicit description of these Lie algebras that they always contain invertible elements of trace zero. But the Lie algebra of the decomposition group is a subalgebra of d. This completes the proof of (A3). We now appeal to results from the cohomology of Lie algebras to conclude from (A3) that H i (d, Vp (E)) = 0 ∀ i ≥ 0.

(A4)

By a well-known result in the cohomology of Lie algebras [H-S, Theorem 10], it suffices to show that d is a reductive Lie algebra and that Vp (E) is a semisimple d-module. This is clear when E does not have complex multiplication, as d is then equal to the Lie algebra of SL2 (Zp ) which is well-known to be semisimple. When E has complex multiplication, Vp (E) is a faithful semisimple representation of the group Σ and hence also for the Lie algebra g of Σ. This shows that g is indeed a reductive Lie algebra, whence d is also, because it is an ideal in g. Also, Vp (E) is a semisimple d-module. Hence (A4) follows from (A3). By a deep theorem of Lazard [La, V.2.4.10], the group H i (∆, Vp (E)) is a Qp -subspace of H i (d, Vp (E)) for all i ≥ 0. Hence H i (∆, Vp (E)) = 0 ∀ i ≥ 0.

(A5)

Consider the short exact sequence 0

/ Tp (E)

/ Vp (E)

/ Ep∞

/0

of Galois modules. Now since H i (∆, Vp (E)) = 0, we see that the group H i (∆, Tp (E)) is finite for all i. Indeed H i (∆, Tp (E)) is a Zp -module of

Appendix

93

finite type [La, V.3.2.7], isomorphic to H i−1 (∆, Ep∞ ). But this latter group is torsion and therefore H i (∆, Tp (E)) is necessarily finite for all i, and (A6) H i (∆, Ep∞ ) is finite ∀ i ≥ 0. The case i = 0 is Theorem A.2.8. The finiteness of the higher cohomology groups does not seem to have been pointed out before (see [C-S]).  A.2.9. In fact, the above argument leads to a new and simple proof of Theorem 3.19, which has the merit of generalizing almost immediately to all abelian varieties over a number field (see [C-S]). Again, let K∞ denote the cyclotomic Zp -extension of F and Γ = G(K∞ /F ). Then Γ has p-cohomological dimension equal to 1, and the Hochschild-Serre spectral sequence yields the short exact sequence, for all i ≥ 1, 0

/ H 1 (Γ, H i−1 (∆, Ep∞ ))

/ H i (Σ, Ep∞ )

.0 o

H 0 (Γ, H i (∆, Ep∞ ))



It was shown in (A6) that H i (∆, Ep∞ ) is finite for all i ≥ 0 and all primes p. Moreover, if Y is any finite group on which Γ acts, the groups H 0 (Γ, Y ) and H 1 (Γ, Y ) have the same cardinality. Hence, if we let hi denote the order of H 0 (Γ, H i (∆, Ep∞ )), we conclude from the above exact sequence that #(H i (Σ, Ep∞ )) = hi hi−1 ∀ i ≥ 1.

(A7)

Now assume that p is such that Σ has no element of order p. By results of Lazard and Serre (cf. [Se3]), the p-cohomological dimension of Σ is finite and equal to the dimension d of Σ as a p-adic Lie group. We can then define d Y i χ(Σ, Ep∞ ) = (# H i (Σ, Ep∞ ))(−1) . i=0

It is clear from (A7) that χ(Σ, Ep∞ ) = h0 · (h0 h1 )−1 · (h1 h2 ) · · · = 1.

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Appendix

To complete the proof of Theorem 3.19, it remains to show that H d (Σ, Ep∞ ) = 0. Choose n so large that pn annihilates H d (Σ, Ep∞ ). Taking cohomology of the exact sequence 0

/ Epn

n / Ep∞ p / Ep∞

/0

and recalling that H d+1 (Σ, Ep∞ ) is zero, it follows immediately that H d (Σ, Ep∞ ) = 0, as required.

A.3. Tate parametrization of elliptic curves The main references for this are [R] and [Ta2]. Suppose E is an elliptic curve over C. It is well-known that there is an analytic isomorphism between the group E(C) and C/Λ, where Λ is a lattice in C (i.e., a discrete subgroup of C which contains an R-basis). Let now E be a curve over a finite extension Fv of Qp . When considering p-adic elliptic curves, there is no such parametrization of E(Fv ) as a quotient of the additive group of Fv . On the other hand there is the normalized exponential map exp : C/Λ → C∗ /q Z where if Λ = Z + τ Z, q = exp(τ ). Now discrete subgroups of Fv∗ abound. Tate showed that if q ∈ Fv∗ is such that 0 6=| q |< 1, then Fv∗ /q Z =: Eq is an elliptic curve over Fv . More precisely, Tate showed the following: Let q be any non-zero element of Fv with ordv (q) > 0. Define a4 = −5 Σ n3 q n /(1 − q n ), n≥1

a6 = 1/12 Σ (7n5 + 5n3 )q n /(1 − q n ). n≥1

Then Eq : y 2 + xy = x3 + a4 x + a6 is an elliptic curve over Fv which we call the Tate curve attached to q. We call q the Tate period of Eq . It can be shown that ordv (j(Eq )) < 0 and that Eq has split multiplicative reduction. Theorem A.3.1. Let q be any element of Fv∗ with ordv (q) > 0. Then there is an analytic isomorphism φ : Fv∗ /q Z ' Eq (Fv ).

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95

Conversely, if E is any elliptic curve over Fv with ordv (jE ) < 0 and split multiplicative reduction, then E is isomorphic over Fv to a Tate curve Eq for some q in Fv with ordv (q) > 0. This parametrization was used in Chapter 5 in the explicit calculations of universal norms for the curves of conductor 11 and 294.

A.4. Multiplicative Kummer generators In this section, we record some well-known results on the Weil pairing and multiplicative Kummer theory. All this is folklore (Weil, Cassels, Tate, . . .), and the results below have been used in the cup-product computation of Chapter 5. Let F be any field of characteristic zero, and E/F an elliptic curve. Let m be an integer > 1, and Em the group of m-division points in E(F ). We assume that Em ⊂ E(F ). A.4.1. Lemma. There is a canonical isomorphism m

φ : H 1 (F, Em ) ' Hom(Em , F ∗ /F ∗ ). Proof. Put GF = G(F /F ). For simplicity, we shall write ‘=’ to denote a canonical, obvious isomorphism. Since Em ⊂ E(F ), we have H 1 (F, Em ) = Hom(GF , Em ). But the Weil pairing gives a canonical isomorphism Em = Hom(Em , µm ). Hence H 1 (GF , Em ) = Hom(GF , Hom(Em , µm )) = Hom(Em , Hom(GF , µm )). (A8) But µm ⊂ F because Em ⊂ E(F ), and so, multiplicative Kummer m theory gives the identification F ∗ /F ∗ = Hom(GF , µm ), and the proof is complete.  The next corollary spells out how we get the multiplicative Kummer generators corresponding to a given α ∈ H 1 (F, Em ). A.4.2. Corollary. Assume that Em ⊂ E(F ). Let α be any element of H 1 (F, Em ) = Hom(GF , Em ), and let Lα be the fixed field of Ker α. Then Lα = F (φ(α)(u)1/m , u ∈ Em ). (A9)

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Proof. Since α factors through Lα , it is clear that the image of α under the chain of canonical isomorphisms in (A8) lands in the group Hom(Em , Hom(G(Lα /F ), Em )). Hence φ(α)(u) must be contained in the Kummer group of Lα for all u ∈ Em . Thus the right hand side of (A9) is contained in the left hand side. Conversely, it is clear that α must factor through the field on the right hand side of (A9), and so this field must contain Lα . 

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