Realizations of Polylogarithms [Paperback ed.] 3540624600, 9783540624608

Table of contents :
1 Mixed structures on fundamental groups......Page 1
2 The canonical construction of mixed sheaves on mixed shimura varieties......Page 58
3 Polylogarithmic extensions on mixed shimura varieties. Part I Construction and basic properties......Page 122
4 Polylogarithmic extensions on mixed shimura varieties. Part II The classifical polylogarithm......Page 170
5 Polylogarithmic extensions on mixed shimura varieties. Part III The elliptic polylogarithm......Page 215

Citation preview

Mixed structures on fundamental groups

J¨org Wildeshaus Mathematisches Institut der Universit¨at M¨ unster Einsteinstr. 62 D–48149 M¨ unster∗

∗

current address: Dept. of Math. Sciences, South Road, GB–Durham DH1 3LE

Introduction Let X be a smooth, separated, geometrically connected scheme over a number field k. In [D4], Deligne constructed a mixed system of realizations on the completed groups rings of the topological fundamental groups of X in the various (Betti, de Rham, ...) contexts. He did so under the very restrictive hypothesis that the H 1 (Xσ , Q) l be of Hodge type (1, 1) for any embedding σ : k ,→ C. l In his attempt to understand and generalize Beilinson’s work on polylogarithms ([B]), the present author soon realized that it would be of vital importance to further develop Deligne’s program – excluding the crystalline version – by not only removing the above hypothesis but also by considering the relative situation, i.e., the case of a suitably regular morphism X → Y . Indeed, the central, if elementary insight is already contained in Proposition 1.4. In the applications we have in mind it states that the injectivity of the natural morphism of the pro–unipotent envelope of the fundamental group to the Tannaka dual of some category of mixed sheaves on X is equivalent to the existence of a mixed sheaf structure on the completed group ring itself. So the desire to describe a part of this Tannaka dual necessitates the study of mixed structures on fundamental groups. The main results of this work are those of § 3, and Theorem 4.3. We chose to discuss the absolute situation separately (§ 2), hoping that this would clarify the exposition by focusing on the actual construction of the mixed structures, which in the relative case are then defined fibrewise. Also, this reflects the development of ideas: the rather general results 1.4 and 1.5 were modelled after the classification theorem for admissible unipotent variations of Hodge structure ([HZ1], Theorem 1.6), which we recall in 2.6. So the main results of § 2 (2.9, 2.12) should be seen as analogues of this theorem in the λ–adic and “mixed realizations” settings, while the results of § 3 are the respective relative versions. Likewise, Theorem 4.3 should be regarded as a generalization of [CH], Theorem 12.1. As an immediate consequence of the classification theorems for relatively unipotent sheaves (Corollaries 3.2.ii) and 3.4.ii), Theorem 3.6.c)), we establish a certain universal property of our “generic relatively unipotent sheaf” (Theorems 3.5 and 3.6.d)). It is this universal property, which will turn out to be a most usei

ful device both in the study of the functor “canonical construction” of sheaves on Shimura varieties from representations of the group underlying the Shimura data, and the development of the general theory of polylogarithmic extensions. In the case of a family of elliptic curves, the result appears already in [BL], 1.2.8.b). We conclude by an application of 4.3 in a rather special case (Corollary 4.4). While this may appear as an anticlimax at first sight, this computation is again motivated directly by the author’s interest in polylogarithms. In his attempt to generalize the work of Beilinson, Deligne and Levin ([BD], [BL]) to the context of mixed Shimura varieties ([W]), the computation in 4.4 constitutes the first step toward the definition of the polylogarithmic extension. Details will be published separately. In our work, motivic sheaves are hardly ever mentioned although they certainly motivate our study to a large extent. In fact, as a very first step in the direction of a K-theoretical construction of polylogarithms, one should convince oneself that the “generic relatively unipotent sheaves” of § 3 (renamed “logarithmic sheaves” in the context of Shimura varieties) are actually of motivic, or geometric origin. In the absolute case (§ 2), this follows (with a suitable definition of the term “of geometric origin”) from work of Wojtkowiak ([Wo1], [Wo2]). This article is a revised version of the first chapter of my doctoral thesis ([W]). It is a pleasure to thank C. Deninger for introducing me to Beilinson’s ideas and for his generosity and constant support. I am obliged to A. Beilinson for kindly supplying me with copies of [BLpp] and [BLp], thus enabling me to work with the material long before is was published. I tried occasionally to indicate the impact that the concepts developed in [B], [BLpp] and [BLp] had on this work. I am grateful to F. Oort and J. Stienstra for the invitation to the “intercity seminarium algebra en meetkunde” in March 1992, and to J. Coates for the invitation to Cambridge in Spring 1993. Both occasions proved to be very stimulating. Also, I would like to thank T. Scholl for the invitation to Durham in 1994, where all results concerning the de Rham version of the generic sheaf were found.

ii

Last, but not least, I am most deeply indebted to Mrs. G. Weckermann for transforming my manuscript into an excellent TEX–file.

iii

§ 1 Review of neutral Tannakian categories and pro–unipotent groups We follow the treatment of [DM], §§ 1–2. Let F be a field, P/F an affine group scheme. Then the category C := Rep F (P ) of finite–dimensional representations of P over F is a neutral Tannakian category over F , i.e.: i) it is abelian. ii) it is a tensor category, i.e., there exists a functor ⊗ : C × C −→ C together with associativity and commutativity constraints subject to the following condition: for X, Y, Z ∈ C ∼

∼

X ⊗ (Y ⊗ Z) −→ (X ⊗ Y ) ⊗ Z −→ Z ⊗ (X ⊗ Y )   oy

∼

∼

 

oy

X ⊗ (Z ⊗ Y ) −→ (X ⊗ Z) ⊗ Y −→ (Z ⊗ X) ⊗ Y is commutative. Furthermore, there is a left neutral element U for ⊗. iii) it is rigid, i.e., there is a functor Hom : C × C −→ C representing T 7→ HomC (T ⊗ − , − ) , such that every object of C is reflexive, i.e., canonically isomorphic to its bidual and such that Hom(X1 , Y1 ) ⊗ Hom(X2 , Y2 ) = Hom(X1 ⊗ X2 , Y1 ⊗ Y2 ) , Xi , Yi ∈ C . iv) F = EndC (U ), and there is a fibre functor, i.e., an exact faithful F –linear tensor functor ω : C −→ VecF . † Here, VecF is the category of finite–dimensional vector spaces over F . †

ω is not part of the data!

1

The Main Theorem of the Tannakian formalism ([DM], Theorem 2.11) gives a converse of the above construction: Theorem 1.1: Let C be a neutral Tannakian category over F . Then every fibre functor ω defines an equivalence of tensor categories ∼

C −→ RepF (P ) for some affine group scheme P over F . We refer to P as the Tannaka dual of C with respect to ω. P represents a certain functor Aut⊗ (ω) of affine F –schemes. This shows that any F –linear tensor functor C → C 0 of neutral Tannakian categories compatible with fixed choices of fibre functors ω and ω 0 comes from a morphism P 0 → P of the associated Tannaka duals. π

It is possible to recover properties of P and of morphisms P 0 −→ P from those of RepF (P ) and of π ∗ : RepF (P ) → RepF (P 0 ): Proposition 1.2: ([DM], Proposition 2.21.) a) π is faithfully flat if and only if π ∗ is fully faithful and every subobject of π ∗ (X), X ∈ RepF (P ), is isomorphic to the image of a subobject of X. b) π is a closed immersion if and only if every object of RepF (P 0 ) is isomorphic to a subquotient of an object of the form π ∗ (X), X ∈ RepF (P ). As any morphism of F –Hopf algebras factors uniquely into a surjection followed by an injection, any morphism of affine group schemes over F factors uniquely into a faithfully flat morphism followed by a closed immersion. In particular, a morphism of affine group schemes over F is a closed immersion if and only if it is injective on points. Quite often, we shall have to study morphisms of groups with pro–unipotent kernel: π

Lemma 1.3: Let P −→ → G be a faithfully flat morphism, W := ker(π). Then W is pro–unipotent if and only if every nonzero object X ∈ Rep F (P ) has a nonzero subobject of the form π ∗ (Y ), where Y ∈ RepF (G). Proof: If W is pro–unipotent then 0 6= X W for 0 6= X ∈ RepF (P ). 2

Conversely, assume every X ∈ RepF (P ) has a finite filtration 0 = X 0 ⊂ X1 ⊂ . . . ⊂ X n = X , whose graded objects are trivial as W –modules. Then the same is true for any W –subquotient of X. From 1.2.b) we conclude that every nonzero X ∈ RepF (W ) has non–trivial invariants.

q.e.d.

Proposition 1.4: In the situation of 1.3, assume W = ker(π) is pro–unipotent. η

Let W −→ W be a morphism of pro–unipotent groups. a) η is faithfully flat if and only if the following holds: for any X ∈ Rep F (P ) , X W is the largest subobject of X lying in π ∗ (RepF (G)). b) Assume π admits a right inverse, η is faithfully flat and char(F ) = 0. Then ˆ (LieW ) such that η is an isomorphism if and only if there is an action of P on U ˆ (LieW ). the associated action of LieP extends the multiplication by LieW ⊂ U ˆ (LieW ) is defined as follows: Here, U wj as the projective limit of its finite–dimensional quotient write LieW = lim ←− j

Lie algebras. ˆ (LieW ) := lim U ˆ (wj ), where U ˆ (wj ) is the completion of the universal Then U ←− j

enveloping algebra of wj with respect to the augmentation ideal aj . The action of P is supposed to be such that the algebraic P –representation ˆ (wj )/anj | j ∈ J , n ∈ lN).† spaces are cofinal in the projective system (U Proof: a) Nothing changes if we replace W by η(W ), which is a closed subgroup of W . Also, we may suppose that everything is algebraic. By one of Chevalley’s Theorems ([Hum1], Theorem 11.2) there is a representation X of P and a one–dimensional subspace L ⊂ X such that W = StabP (L) . Since W is unipotent, W acts trivially on L, i.e., L ⊂ X W . By hypothesis, W also acts trivially on L. So W and W must be equal. b) Since char(F ) = 0, there is an equivalence of categories ([DG], IV, § 2, Corollaire 4.5.b)) ∼

, RepF (W ) −→ Modfin ˆ U(LieW ) †

By definition, lN is the set of positive integers, and lN0 is the set of non-negative integers.

3

ˆ (LieW )–modules such the latter denoting the category of finite–dimensional U that the action is continuous with respect to the discrete topology on the module ˆ (LieW ). The algebra U ˆ (LieW ) acting on itself and the inverse limit topology on U by multiplication is a pro–object of RepF (W ), and every X ∈ RepF (W ) is a ˆ (LieW ). So if this representation subquotient of a finite number of copies of U of W can be extended to P , we may apply 1.2.b) to conclude that W → P is a closed immersion. ˆ (LieW ) by conjugation, using the Conversely, if W = W , we let G act on U section of π. This, together with the above action of W , defines the desired action of P = W × G.

q.e.d.

ˆ (LieW ), but the Remark: There is always the canonical action “Ad” of P on U associated action of LieP extends the action “ad” of LieW , which we shall have to distinguish carefully from the multiplication by LieW . “ad” is a representation of Lie(W/Z(W )), which in general won’t generate Modfin . ˆ U(LieW ) Lemma 1.5: Assume char(F ) = 0 and P = W × G with a pro–unipotent group W . Then RepF (P ) is equivalent to the category of finite–dimensional ˆ (LieW )–modules X equipped with a representation of G such that discrete U the morphism ˆ (LieW ) −→ EndF (X) U ˆ (LieW ) by conjugation. is G–equivariant, G acting on U Proof: straightforward.

q.e.d.

In the rest of this paragraph, we shall discuss several methods of calculating Yoneda–Ext groups of representations of a pro–unipotent group scheme over a field of characteristic 0. The results won’t be needed until § 4, and the reader is invited to ignore them until then. So let char(F ) = 0, W = lim Wj ←−

j∈lN

w = lim wj ←−

a pro–unipotent group scheme over F , which is a countable projective limit of unipotent group schemes Wj , its Lie–algebra,

j

U(wj )

the universal enveloping algebra,

ˆ (wj ) U

the completion with respect to the augmentation ideal aj ,

4

ˆ (w) U

ˆ (wj ) = lim U(wj )/anj = lim U(wn )/ann , := lim U ←−

←−

j

a

←−

j,n

n

ˆ (w). ˆaj the augmentation ideal of U := lim ←− j

Following [Ho], we define Rat(W ) := Ind–RepF (W ) = ˆ (w) with continuous action} . {discrete modules under U Proceeding dually to [H1], § 3, we define R(W ) as the full subcategory of Rat(W ) of modules of countable dimension. So R(W ) = {X ∈ Rat(X) | X is the union of countably many algebraic representation spaces of W } . ˆ (w), R(W ) and Rat(W ) are closed under ⊗F , and the topological dual of U c

ˆ (w))∨ := HomF,cont. (U ˆ (w), F ) = lim(U(wj )/anj )∨ (U −→ j,n

ˆ (w) on itself is given by multiplication. is an object of R(W ). Here, the action of U Consider the functor *: ModW (F ) → Rat(W ) : M 7→ the maximal rational submodule of M . The functor *, being the right adjoint to the forgetful functor, which is exact, maps injectives to injectives. Proposition 1.6: a) If M ∈ ModW (F ) is projective, then M ∨ := HomF (M, F ) ∈ ModW (F ) is injective. b) R(W ) and Rat(W ) are abelian subcategories of ModW (F ) with sufficiently many injectives. c) The inclusions RepF (W ) ,→ R(W ) and R(W ) ,→ Rat(W ) respect Yoneda– Ext groups. In particular, for any X ∈ RepF (W ), Hochschild cohomology H · (W, X) = Ext·Rat(W ) (F, X) coincides with Ext·RepF (W ) (F, X). Proof: a) straightforward. b) “abelian”: easy. 5

“enough injectives” for Rat(W ): (compare [Ho], Proposition 2.1.) ModW (F ) has enough injectives. Given X ∈ Rat(W ), choose a monomorphism X ,→ I into an injective object I ∈ ModW (F ) . This map must factor through I ∗. “enough injectives” for R(W ): (compare [H1], Proposition 4.8, Lemma 4.9.) 1) In R(1) every element is injective. 2) The forgetful functor V : R(W ) → R(1) has the right adjoint c

ˆ (w))∨ ⊗F Y : R(1) → R(W ), Y 7→ (U Let X ∈ R(W ), Y ∈ R(1), X =

S

i∈lN

Xi such that dimF Xi < ∞. Then c

c

ˆ (w))∨ ) ⊗F Y ) ˆ (w))∨ ⊗F Y ) = lim(HomR(W ) (Xi , (U HomR(W ) (X, (U ←− i

ˆ (w), Xi∨ ) ⊗F Y ) (U = lim(HomU(w),cont. ˆ ←− i

= lim HomF (Xi , Y ) ←− i

= HomR(1) (V (X), Y ) . Since V is exact, its adjoint maps injectives to injectives. 3) For all X ∈ R(W ), the adjunction morphism c

ˆ (w))∨ ⊗F V (X) X −→ (U is injective. c) For a quite general study of the question, “when are we allowed to calculate cohomological derived functors on a category in its Ind–category?”, see [Hub], § 2, especially Theorem 2.6. For the special cohomological derived functor “Yoneda–Ext· ”, we refer to the following lemma.

q.e.d.

Lemma 1.7: Let A1 ⊂ A2 ⊂ A3 be fully faithful, exact inclusions of abelian categories such that i) A1 is the category of noetherian objects of A3 . ii) Every object in A2 is a countable direct limit of objects of A1 , and every countable direct system in A1 has a limit in A2 . 6

iii) Every object in A3 is a direct limit of objects in A1 . Then the inclusions A1 ,→ A2 and A2 ,→ A3 respect Yoneda–Ext groups. Proof: (sketch) surjectivity: the essential point is that if A and B are objects of A1 and A3 respectively, and if 0→B→E→A→0 is exact in A3 , then this sequence is the push–out of a sequence in A1 . Using this, for any exact sequence 0 → B → E 1 → . . . → En → A → 0 in A3 with A, B in A1 , one constructs a sequence 0 → B → E10 → . . . → En0 → A → 0 in A1 and a morphism of this n–extension into the above, i.e., an elementary equivalence. (∗) This proves the surjectivity for the relative situation A1 ,→ A3 . Using this, the same procedure shows the analogous statement for A2 ,→ A3 . injectivity: the main step in the proof is the following observation: if E → E1 ← E2 are elementary equivalences of n–extensions of A by B in A3 with A, B, E in A1 , then there are elementary equivalences E → E10 ← E20 → E2 with E10 , E20 in A1 : using (∗), we may assume E2 = E20 is in A1 . Now choose E10 ,→ E1 in A1 such that the components of E10 contain the image of the components of E and E20 . One proceeds similarly for A2 ,→ A3 .

q.e.d.

Corollary 1.8: For any X ∈ RepF (W ), H · (W, X) = lim H · (Wj , X) . −→ j

Proof:

H · (W, X) = Ext·RepF (W ) (F, X) = lim Ext·RepF (Wj ) (F, X) −→ j

= lim H · (Wj , X) . −→ j

q.e.d. 7

Lemma 1.9: If W is algebraic, then for any X ∈ RepF (W ), the natural map H · (W, X) −→ H · (w, X) is an isomorphism. Note that the left hand side is calculated via injective resolutions in Rat(W ), while the right hand side is calculated via injective resolutions in ModU(w) . Proof: By filtering X, we may suppose X = F . By filtering W , we may suppose W = G| a . Then the cohomology groups are well known and coincide.

q.e.d.

It is possible to describe the category R(W )opp explicitly as follows: ˆ (w)–modules M such that let T (W ) be the category of topological U a) the topology is given by a countable descending filtration M = M0 ⊃ M1 ⊃ . . . ˆ (w)–modules of finite codimension. of U ∼

b) M −→ lim M/M i . ←− i

Then the two functors c

∨

: T (W )opp −→ R(W ) , M 7−→ HomF,cont. (M, F ) ,

∨

: R(W )opp −→ T (W ) , X 7−→

HomF (X, F )

ˆ (w) itself lies in define an identification of T (W ) and R(W )opp . Note that U T (W ). Dualizing 1.6.b) and its proof, one obtains Corollary 1.10: (compare [H1], Theorem 3.4.) T (W ) is an abelian category with surjective epimorphisms, injective monomorphisms and enough projectives. More precisely, for all M ∈ T (1) the completed ˆ (w)⊗ ˆ F M is projective in T (W ). tensor product U Proof: The functor

∨

transforms injective maps into surjective maps and vice

versa.

q.e.d.

The category RepF (W ) is contained in T (W ). Because of 1.6.b), we get for objects X, Y ∈ RepF (W ): c

c

c

c

Ext·RepF (W ) (X, Y ) = Ext·RepF (W ) (Y ∨ , X ∨ ) = Ext·R(W ) (Y ∨ , X ∨ ) = Ext·T (W ) (X, Y ) .

8

In particular, by 1.6.c), for any X ∈ RepF (W ), Hochschild cohomology H · (W, X) coincides with Ext·T (W ) (F, X). So if L/F is a field extension, the natural map H · (W, X) ⊗F L −→ H · (WL , X ⊗F L) is an isomorphism: apply 1.10 and use the fact that base change by L is exact ˆ (w) to U ˆ (wL ). and maps U Now consider the right exact functor ˆ U(w) F. Λ : T (W ) → T (1), M 7→ H0 (W, M ) = M/aM = M ⊗ ˆ Using projective resolutions, first in T (W ) and then in T (W )×T (W ), one shows that the left derivatives Hk (W, − ) exist and can be calculated by resolving either F or the module in question. Lemma 1.11: If W is algebraic, then for any M ∈ RepF (W ), the natural map H· (W, M ) −→ H· (w, M ) is an isomorphism. Note that the right hand side is calculated via projective resolutions in ModU(w) . Proof: as in 1.9.

q.e.d.

Proposition 1.12: a) Assume Hk (W, F ) is finite–dimensional for all k ∈ lN0 . This is true in particular if W is algebraic. Then Hk (W, M ) = lim Hk (W, M/M i ) ←− i

i

for all k ∈ lN0 and M = lim M/M ∈ T (W ) . ←− i

b) Assume H k (W, F ) is finite–dimensional for all k ∈ lN0 . This is true in particular if W is algebraic. Then H k (W, M/M i ) = ExtkT (W ) (F, M ) H k (W, M ) := lim ←− i

for all k ∈ lN0 and M = lim M/M i ∈ T (W ). ←− i

9

Proof: By filtering the M/M i , we get the finiteness condition for the Hk (W, M/M i ) or H k (W, M/M i ) as well. The statement follows from a standard application of the Mittag–Leffler criterion.

q.e.d.

Corollary 1.13: Let W1 ≤ W2 be algebraic unipotent groups, d1 := dim W1 . Then

  0,

ˆ (w2 )) = H k (W1 , U 

k 6= d1

ˆ (w2 ))⊗ ˆFΛ H0 (W1 , U

d1

w∨ 1

, k = d1

.

Proof: By 1.9, 1.11, 1.12.a) and [K], Theorem 6.10, we have canonically ˆ (w2 )) = Hd1 −k (W1 , U ˆ (w2 ))⊗ ˆ F Λd1 w∨1 . H k (W1 , U ˆ (w2 )) is zero unless n = 0. Letting So we have to show that Hn (W1 , U d2 := dim W2 , this will be achieved by induction, first on d2 − d1 and then on d1 : ˆ (w1 ) ∈ T (W1 ) is projective. If d1 = 0, If d1 = d2 , then W1 = W2 , and so U then the claim is trivial. In the general case, if Z(W2 ) ∩ W1 is non–zero and unequal to W1 , we use the Hochschild–Serre spectral sequence together with the isomorphism ˆ (w2 )) ∼ ˆ (w2 /(z(w2 ) ∩ w1 )) . H0 (Z(W2 ) ∩ W1 , U =U If Z(W2 ) ∩ W1 is zero, choose a one–dimensional subgroup W 0 of Z(W2 ). Then W 0 × W1 is a subgroup of W2 of dimension d01 = d1 + 1. So d2 − d01 = d2 − d1 − 1, and ˆ (w2 )) = 0 unless m = 0 . Hm (W 0 × W1 , U Now use the Hochschild–Serre spectral sequence and the fact that Hp (W 0 , − ) is zero for p > 1 to conclude that ˆ (w2 ))) = 0 unless n = 0 . H0 (W 0 , Hn (W1 , U But any M ∈ T (W 0 ) with trivial co–invariants is itself trivial. It remains to consider the case where W1 ≤ Z(W2 ). Clearly our claim follows if ˆ (w2 ) ∈ T (W1 ) is projective. Choose a filtration we manage to show that U 0 = w(k+1) ⊂ . . . ⊂ w(1) = w2 of sub–Lie algebras satisfying [w(i) , w(j) ] ⊂ w(i+j) and w1 = w(k) . Choose a basis (w1 , . . . , wd1 , wd1 +1 , . . .) of w2 respecting this filtration. 10

By the Poincar´e–Birkhoff–Witt theorem ([Hum2], 17.3, Corollary C), n

(w1n1 · . . . · wd2d2 | n1 , . . . , nd2 ∈ lN0 ) ˆ (w2 ) is equivalent to the is a basis of U(w2 ). Hence the a–adic filtration of U filtration F · by degree, where we set deg(wj ) := max{m | wj ∈ w(m) } . So another application of the Poincar´e–Birkhoff–Witt theorem gives an isomorphism ˆ (w2 ) ∼ ˆ (w1 )⊗ ˆFM U =U of objects of T (W1 ), where M ∈ T (1) is the completion with respect to the degree of the vector space with basis n

n

1 +1 (wd1d+1 · . . . · wd2d2 | nd1 +1 , . . . , nd2 ∈ lN0 ) .

Corollary 1.10 then concludes the proof.

q.e.d.

§ 2 The generic pro–sheaf We begin this paragraph by recalling the notion of the pro–unipotent envelope of an abstract or profinite group. We then use the results of § 1 for a reinterpretation of Chen’s construction of a pro–mixed Hodge structure on the completed group ring of the fundamental group of a smooth complex algebraic variety X ([C]; see also [H2]) together with the classification theorem for admissible unipotent variations of Hodge structure on X ([HZ1], Theorem 1.6; see also [HZ2], § 2) in Tannakian terms (Corollary 2.7). This result is a prototype of those that will follow in this and the next paragraph: the analogous statement for mixed lisse λ–adic sheaves is also true and follows from results in [SGA1]. The universal property 2.6 of what we call the generic pro–variation suffices to generate what amounts to a descent datum for the weight and Hodge filtration of the underlying vector bundle, thereby defining the de Rham version of the generic sheaf. We remark that for X = lP1 \{0; 1; ∞}, the main results of this paragraph (2.7, 2.9, 2.12) occur implicitly in [B] in the form: “L(X)g [which coincides with Lie(Wx ) in our notation] is the free Lie algebra in two variables” ([B], 1.3.1.ii)). 11

So let π be an abstract, finitely generated group, Q[π] l its group ring, Q[π] l ∧ its completion with respect to the augmentation ideal a, W (π) the Tannaka dual of the category of unipotent representations of π over Q, l i.e., the pairs (X, ρ), where dimQl X < ∞ , ρ : π → GL(X)(Q) l a group homomorphism such that X has a filtration of π–submodules whose graded objects are trivial π–modules, w(π) := Lie(W (π)) .

By 1.3, W (π) is pro–unipotent, and by the explicit construction of W (π) (as reviewed in [D4], § 9), it is a countable projective limit of unipotent groups, i.e., it satisfies the hypothesis set up before 1.6. Observe that RepQl (W ) is the category of finite–dimensional discrete Q[π] l ∧ –modules. Definition: W (π) is called the pro–unipotent envelope of π. w(π) is called the Malcev–Lie algebra of π.

Example: Let W/Q l be a unipotent algebraic group, π ≤ W (Q) l an arithmetic subgroup. By [D4], 9.5, W is the pro–unipotent envelope of π. ˆ (w(π)) are canonically isomorphic. Lemma 2.1: Q[π] l ∧ and U Proof: Both are objects of T (W (π)). Whenever an object of T (W (π)) together with an element is given, there exists a unique morphism of Q[π] l ∧ into this ˆ (w(π)) has the same universal property. object sending 1 to the given element. U q.e.d. Proposition 2.2: a) The functor {finitely generated groups} −→ {pro–unipotent groups}, π 7−→ W (π) is right exact. ˆ Ql F is the Tannaka dual of the b) For any field F of characteristic 0 , W (π)⊗ category of unipotent representations of π over F . i

p

Proof: a) Let π 0 → π → π 00 → 1 be an exact sequence of finitely generated 12

groups, i.e., p surjective, im (i) = ker(p), i

p

W 0 −→ W −→ W 00 −→ 1 the sequence of the Tannaka duals. That p is faithfully flat follows from 1.2.a). Next, we obviously have p ◦ i = 0. By 1.4.a), im (i) = ker(p). b) Let W := W (π). There is a natural map π → W (Q). l It induces a functor RepF (W ) −→ {unipotent representations of π over F } . Conversely, if π → GLF (X) is a unipotent representation, we get a map ˆ (w⊗ ˆ Ql F ) → EndF (X) F [π]∧ = U (by 2.1), i.e., a rational representation of W over F .

q.e.d.

Now let π ˆ be a profinite group, which is topologically finitely generated. Let l ∈ lN be a prime and define Wl (ˆ π ) as the Tannaka dual of the category of continuous unipotent representations of π ˆ over Q l l , and let wl (ˆ π ) := Lie(Wl (ˆ π )). Here, we consider the l–adic topology on finite–dimensional Q l l –vector spaces. We shall frequently allow ourselves to refer to continuous representations on Q l l –vector spaces simply as l–adic representations. Statements analogous to 2.2 hold, where in b) we only consider finite extensions of Q l l ; in particular Wl (ˆ π ) is a quotient of Wl (ˆ π 0 ) for a group π ˆ 0 , which is the profinite completion of some finitely generated group. By the next proposition, Wl (ˆ π ) also satisfies the hypothesis set up before 1.6. Proposition 2.3: Let π be a finitely generated group, π ˆ its profinite completion, l ∈ lN prime, F/Q l l a finite extension. a) resππˆ induces an equivalence of categories ∼

{continuous unipotent representations of π ˆ over F } −→ ∼

−→ {unipotent representations of π over F } . ˆ Ql Q b) Wl (ˆ π ) = W (π)⊗ l l. Proof: a) Any unipotent representation of π over F stabilizes some lattice over oF , hence is continuous with respect to the profinite topology on π. b) follows from a) and 2.2.b).

q.e.d. 13

Definition: A finitely generated group π is called pseudo–nilpotent, if H · (W (π), X) −→ H · (π, X) is an isomorphism for X = Q. l This is then automatically true for all X ∈ RepQl (W (π)). H · (π, X) is ordinary group cohomology, calculated via injective resolutions in ModQ[π] . l Remarks: a) As in 1.9, one shows using the construction of W (π) for nilpotent groups (compare [D4], § 9) and the usual filtration argument, that nilpotent finitely generated groups are pseudo–nilpotent. b) As 1.8 shows, this definition coincides with the one given in [H1], 5.3. More precisely, continuous cohomology of the fundamental group is the same as Hochschild cohomology of its pro–unipotent envelope. This viewpoint can be employed e.g. for a simplification of the proof of the following result, which however won’t be needed in the sequel. Lemma 2.4: (compare [H1], Theorem 5.1.) Let π be a finitely generated group. Then the natural maps H k (W (π), Q) l → H k (π, Q) l are isomorphisms for k = 0, 1, and injective for k = 2. Proof: We interpret the cohomology groups as Yoneda–Ext groups in the category RepQl (W (π)) = {unipotent representations of π over Q} l as we may by 1.6.c), and the category ModQ[π] of arbitrary π–modules respectively. l The claim for k = 0 and k = 1 is trivial. So take a two–extension 0→Q l → E 1 → E2 → Q l →0 with unipotent representations E1 and E2 of π over Q, l that becomes trivial in (Q, l Q). l This means precisely that there is a π–module E together with Ext2ModQ[π] l a filtration 0 ⊂Q l ⊂ E1 ⊂ E by π–submodules such that E/Q l = E2 . Thus E is necessarily of finite dimension, and since it has a filtration with unipotent graded objects, it is itself unipotent. 14

q.e.d.

We won’t define quasi–nilpotency for profinite topologically finitely generated groups. If a quasi–nilpotent abstract group is the fundamental group of a topological space X, and X is a K(π, 1), then the cohomology of X can be calculated in the Tannaka category of unipotent representations of π. It is in this property that we shall be interested. We shall formulate it also in the l–adic context (compare § 4).

For a nice overview concerning admissible graded–polarizable unipotent variations of mixed Hodge structure, see [HZ2], §§ 1–2. Let X be a smooth complex variety. Assume X is connected. Fix x ∈ X(C) l and l x := x. There is a canonical mixed graded–polarizable Q– l write X := X(C), Hodge structure (Q–M l HS) on the truncated group ring Q[π l 1 (X, x)]/an , n ∈ lN ([C], [H2]). Thus, if we let G denote the Tannaka dual of the category M HQl of graded– polarizable Q–M l HS with respect to the forgetful functor M HQl → VecQl , we get a pro–algebraic action of G on ˆ (w(π1 (X, x))) =: U ˆx , lim Q[π l 1 (X, x)]/an = Q[π l 1 (X, x)]∧ = U ←−

2.1

n

where, as before, w(π1 (X, x)) is the Malcev Lie–algebra of the finitely generated group π1 (X, x). ˆ x is a morphism of M HS, and the unit 1 is an element Multiplication within U ˆ x ). of (W0,Ql ∩ F 0 )(U Now let V be an admissible graded–polarizable unipotent variation of Q–M l HS on X and let ˆ x −→ EndQl Vx ρx : U be the monodromy representation. Both sides carry pro–Q–M l HS, and we have Theorem 2.5: ([HZ1], Theorem 7.2 or [HZ2], Theorem 2.2.) ρx is a morphism of pro–M HS.

15

Theorem 2.6: ([HZ1], Theorem 1.6 or [HZ2], Theorem 2.6.) The functor V 7→ ρx is an equivalence of categories     admissible graded–             polarizable uni  potent     

variations

of Q–M l HS on X

      

−→

  graded–polarizable       l HS H together  Q–M  

with

a morphism of Q–M l HS

    ˆx  U     

−→ End(H) respecting

the algebra structure

          

.

         

Let Px be the Tannaka dual of the category U VarQl (X) of admissible graded– polarizable unipotent variations of Q–M l HS on X with respect to the functor “fibre at x” : U VarQl (X) −→ VecQl . We have natural morphisms π

−→ Px ←− G , x

belonging to x∗ := “fibre at x” , π ∗ := “associated constant variation” . Clearly π ◦ x = idG , and by 1.3, Wx := ker(π) is pro–unipotent. The tensor functor “monodromy representation” induces a morphism W (π1 (X, x)) −→ Px , which factors over Wx . Corollary 2.7: The natural morphism η

x W (π1 (X, x)) −→ Wx

is an isomorphism. Proof: We have to check the criteria of 1.4: W (π (X,x))

a) Let V ∈ U VarQl (X). Then Vx 1 is a sub–M HS of Vx : observe that we ˆ x ) as follows from the explicit construction of the M HS on U ˆx. have a = W−1 (U (See [H2], Remark 5.4.) By 2.5, we have a morphism of M HS Vx → HomQl (a, Vx ), v 7→ (α 7→ ρx (α) · v) . 16

W (π1 (X,x))

Now Vx

is the kernel of this morphism, i.e., it is a sub–M HS of Vx . ˆ x . By 2.6, it corresponds to a subobject It carries the trivial representation of U

of V, which is clearly the largest subvariation, that is constant. ˆ x yields representations b) Multiplication within U ˆ x → EndQl (U ˆ x /an ) , n ∈ lN , U which, as remarked earlier, are morphisms of M HS. 2.6 gives the desired action ˆx. q.e.d. of Px on U ˆ x , together with the action of Px of the proof of 2.7, defines a Remark: U pro–object of U VarQl (X). This is not the tautological variation of [HZ2], § 1. ˆ x (compare [D1], I, The latter corresponds to the adjoint action of Px on U ˆy , Proposition 1.6) and is independent of x. Its fibre at y ∈ X is Q[π l 1 (X, y)]∧ = U 0 ∧ ] together with its natural while the fibre of the above representation is Q l [Ωx,y 0 ˆ x –module structure. Here, Ωx,y denotes the set of homotopy classes of paths U

connecting x and y. This is the canonical variation with base point x of [HZ1], § 1. We suggest another terminology: ˆ x with the mixed Definition: The pro–object of U VarQl (X) corresponding to U Hodge–representation given by left multiplication is called Genx , the generic pro–unipotent variation with basepoint x on X. This terminology can be justified as follows: the finite–dimensional subquotients of Genx , together with the constant variations, generate U VarQl (X) as a full Tannakian subcategory, that is closed under formation of subobjects. (Proof: by 1.2.a), the Tannaka dual of the subcategory is a quotient of Px = Wx × G. But G is still a quotient, and Wx still injects.) The dependence of Genx on the basepoint is as follows: any path connecting x and x0 or, more generally, any isomorphism of fibre functors ∼

“fibre at x” −→ “fibre at x0 ” ∼

on U VarQl (X) defines an isomorphism Genx0 −→ Genx . Note that if we apply 1.5 to the result on the shape of Px in 2.7, we get back 2.6. Statements 2.5–2.7 remain correct, possibly after applying ⊗Ql F , if we replace Q l by an arbitrary coefficient field F contained in lR.

17

Now for the λ–adic situation: fix a prime number l; let F/Q l l be finite, λ ∈ F a prime element. Let X be a quasi–compact scheme over a number field k, and assume that X := X ⊗k k is connected and that X(k) 6= ∅. Let x ∈ X(k). By [SGA1], Exp. IX, Th´eor`eme 6.1, there is an exact sequence of algebraic fundamental groups π

1 −→ π1 (X, x) −→ π1 (X, x) −→ Gk −→ 1 , and the point x gives a splitting of π. Here, Gk := Gal(k/k). Via this splitting, we write π1 (X, x) = π1 (X, x)× Gk . This already resembles the Hodge–theoretic situation to a large extent. In fact, it is easy to see that the category of λ–adic representations of π1 (X, x) is equivalent to the category of Galois–equivariant λ–adic representations of π1 (X, x). However, this is not quite what we want. We need to consider λ–adic sheaves on X, that are mixed in the sense of [D3], VI: Definition: Let F be a finite extension of Q l l , and let X/k be separated and of finite type. i) Etl,m F (X) is the full subcategory of the category of constructible F –sheaves on X consisting of objects V satisfying a) V is lisse. b) There is a finite set S ⊂ Spec(ok ) containing the primes dividing l, a separated scheme X → Spec(oS ) of finite type and a lisse constructible F –sheaf V on X such that α) X = X ⊗oS k, V = V ⊗oS k. β) V is mixed in the sense of [D3], D´efinition 1.2.2. In particular, there is a weight filtration W· of V. l,m ii) U Etl,m F (X) is the full subcategory of those V ∈ Et F (X) admitting a

filtration, whose graded objects are geometrically trivial. It is not difficult to see that if X is smooth and if such a filtration exists, the weight filtration will have the same property. (Use [D3], Th´eor`eme 6.1.2, generic base change ([SGA4 1/2], Th. finitude, Th´eor`eme 1.9) and [D3], Corollaire 3.3.5 to see that a pure sheaf in U Etl,m F (X) is geometrically trivial.) 18

In terms of continuous representations, property i)b) implies that π1 (X, x) −→ GLF (Vx ) factors through π1 (X, x) for a suitable extension X of X. Property ii) means precisely that the induced representation of π1 (X, x) is unipotent. It is not hard to see that U Etl,m F (X) is a neutral Tannakian category over F . We let π denote the morphism X −→ Spec(k) . By functoriality of the Malcev Lie algebra, the point x defines a continuous action of Gk on ˆ (wl (π1 (X, x))) ⊗Ql l F =: U ˆx . U Note that by the comparison theorem ([SGA1], Exp. XII, Corollaire 5.2), π1 (X, x) is topologically finitely generated. As in the Hodge–theoretic setting, the first thing we have to make sure is that ˆ x itself belongs to pro–Etl,m U F (Spec(k)): Theorem 2.8: Assume X is smooth. ˆ x is mixed. Then the representation of Gk on U Proof: Etl,m F (Spec(k)) is closed under formation of quotients in the category of constructible F –sheaves on Spec(k) ([J], Lemma 6.8.1.b)). Take an affine neighbourhood U of x. As X is normal and U is dense in X, π1 (X, x) is a quotient of π1 (U , x). So we may assume X is affine. By [Hi], Main Theorem I and Corollary 3 of Main Theorem II, X can be embedded as an open dense subvariety in a smooth projective variety Z/k such that Z \X is a divisor with normal crossings. There is a finite set S ⊂ Spec(ok ) containing the primes dividing l such that there is a smooth extension $

X −→ Spec(oS )

of X with geometrically connected fibres, which is the complement of a relative divisor with normal crossings in a smooth, projective scheme over Spec(oS ) and such that x extends to a section X

Spec(oS ) −→ X . 19

Now we may apply [SGA1], Exp. XIII, Proposition 4.3 and Exemples 4.4 to conclude that the natural sequence $

−→ 1 −→ π1l (X, x) −→ π10 (X, x) ←− π1 (Spec(oS ), π(x)) −→ 1 X

is split exact. Here, π1l (X, x) is the largest pro–l quotient of π1 (X, x), and π10 (X, x) is the quotient of π1 (X, x) by N , where ker($)/N is the largest pro–l quotient of ker($). So the action of Gk on π1l (X, x) factors through π1 (Spec(oS ), π(x)). Now observe that wl (π1 (X, x)) = wl (π1l (X, x))

since every unipotent λ–adic representation of π1 (X, x) factors through π1l (X, x). ˆ x factors through π1 (Spec(oS ), π(x)). So the representation of Gk on U ˆ x . It will automatically be compatible We now define the weight filtration on U ˆx. with multiplication on U ˆ x := U ˆ x , W−1 U ˆ x := a. Let a denote the augmentation ideal. Set W0 U ∼

Now observe that a/a2 −→ H 1 (X, F (0))∨ = (R1 $∗ FX (0))∨gen.pt. , which is mixed of weights −1 and −2. Writing p : a −→ → a/a2 , ˆ x := p−1 (W−2 (a/a2 )). we set W−2 U Now let W· be the filtration “generated” by W−1 and W−2 . ˆ x for all n ∈ lN0 , and W−n (U ˆ x /an ) is spanned by More explicitly, an ⊂ W−n U ˆ x , βj ∈ W−2 U ˆx products of the form α1 · . . . · αm · β1 · . . . · βl , where αi ∈ W−1 U and m + 2l ≥ n. With this definition, the canonical surjection H 1 (X, F (0))⊗(−n) −→ → an /an+1 respects the weight filtration for all n ∈ lN0 . It follows that the graded parts are pure.

q.e.d.

As before, let X/k be a smooth, separated, geometrically connected scheme of finite type, x ∈ X(k). Define a lisse pro–F –sheaf Genx on X, i.e., a continuous representation of π1 (X, x) = π1 (X, x)× Gk as follows: ˆ x , π1 (X, x) acts by multiplication, and Gk the underlying vector space is U acts as in 2.8. By construction, the induced representation of π1 (X, x) is pro– ˆ x defines a filtration W· Genx by lisse sub– unipotent, and the filtration W· on U pro–sheaves. 20

Using the same techniques as in the proof of 2.8, it follows that Genx is an object of pro–U Etl,m F (X) if X is quasi–projective, the graded objects being sums of quotients of π ∗ (H 1 (X, F (0)))⊗(−n) , n ∈ lN0 . In order to obtain the statement for general X, we have to study the dependence of Genx on x: let x0 ∈ X(k) be another point, and denote by ζ the generic point of X. Choose specialization maps ζ → x, ζ → x0 (compare [SGA4,II], Exp. VIII, 7.2). These give an isomorphism of fibre functors ∼

“fibre at x” −→ “fibre at x0 ” on U EtlF (X) , the category of unipotent lisse F –sheaves on X. Any such isomorphism induces an isomorphism ∼ P˜x0 −→ P˜x

of the corresponding Tannaka duals, which in particular yields an isomorphism ∼

Genx0 −→ Genx , Wl (π1 (X, x)) ⊗Ql l F and Wl (π1 (X, x0 )) ⊗Ql l F being the respective kernels of ˜, ˜ , P˜x0 −→ P˜x −→ →G →G ˜ is the Tannaka dual of Etl (Spec(k)). where G F

Theorem 2.9: Let X be a smooth, separated, geometrically connected scheme of finite type over a number field k, and let x ∈ X(k). a) Genx is an object of pro–U Etl,m F (X). b) Let Px be the Tannaka dual of U Etl,m F (X) with respect to “fibre at x” : U Etl,m F (X) −→ VecF , and let

π

−→ Px ←− G x

be the morphisms induced by π ∗ and “fibre at x”, where G is the Tannaka dual of Etl,m F (Spec(k)). Then the natural morphism η

x Wl (π1 (X, x)) ⊗Ql l F −→ Wx := ker(π)

is an isomorphism. 21

c) The functor

U Etl,m F (X) −→

  l,m   objects H of Et (Spec(k))   F          together with a morphism       

of Galois modules

    ˆx  U     

−→ End(H) respecting

the algebra structure

,

         

V 7−→ Vx is an equivalence of categories. Proof: We prove a), leaving b) and c) to the reader. (Hint: use 1.4 and 1.5!) Let X =

Sn

i=1

Ui be an affine covering. Using [EGAIV,3], Th´eor`eme 8.8.2 and

the unicity of the weight filtration of a mixed sheaf, it is not hard to see that we may replace k by a finite extension. So we may assume that every Ui contains a k–rational point xi . As remarked before, GenUi ,xi ∈ U Etl,m F (Ui ). But Genx is isomorphic to Genxi , whose restriction to Ui is a quotient of GenUi ,xi . So Genx |Ui is mixed for all i. Again using [EGAIV,3], Th´eor`eme 8.8.2, one sees that after possibly enlarging S, the extensions of Ui to Spec(oS ) glue together to give an extension X of X. The extensions of Genx | Ui , being lisse and coinciding on a non–empty open subset of X, necessarily glue together to form an extension of Genx , which is mixed.

q.e.d.

l l –sheaves instead of F –sheaves. 2.9 remains correct if we consider Q by a full Tannakian subcategory Remark: If one replaces U VarF or U Etl,m F C closed under formation of subobjects, then 2.7 and 2.9.b) will be false in general. However, as the proofs show, we obtain a correct statement once we replace ˆ Ql (l)F by its largest quotient WC,x such that W(l) (π1 (X, x))⊗ ˆ Ql (l) F → WC,x ) ker(W(l) (π1 (X, x))⊗ is normal in Px and the corresponding quotient of Genx is contained in pro– C(X).† For example, we could take C to be the category of objects whose graded parts †

By definition, the subscript (l) can take the two values blank and l, depending on whether

one considers the Hodge or l-adic setting.

22

are of Tate type. Then π

WC,x = ker(PC,x −→ GC ) ˆ Ql (l) F . It coincides with is the largest pro–Tate quotient of W(l) (π1 (X, x)) ⊗ ˆ Ql (l) F if and only if H 1 (X, Q W(l) (π1 (X, x))⊗ l (l) (0)) is of Tate type (compare the proof of 2.8), e.g. if X = lP1 \{x1 , . . . , xn }. Another possibility is to let C be the category of objects “of geometric origin”. It seems to be reasonable to expect Genx to be of geometric origin, but I have no proof. But see the remark preceding Theorem 2.12. As a first approximation to the definition of a category of smooth mixed motivic sheaves, we might let ourselves be inspired by Jannsen’s or Deligne’s definition of mixed realizations ([J], § 2, [D4], § 1): Definition: i) Let k be a number field, X/k smooth, separated and of finite type. M SQsl (X), the category of mixed systems of smooth sheaves on X consists of families (Vl , VDR , V∞,σ , Il,σ , IDR,σ , I∞,σ | l ∈ lN prime, σ : k ,→ C, l σ : k ,→ C) l , where l,m a) Vl ∈ EtQ l l (X),

b) VDR is a vector bundle on X, equipped with a flat connection ∇, which is regular at infinity in the sense of [D1], II, remark following D´efinition 4.5. Further parts of the data are an ascending weight filtration W· by flat subbundles and a descending Hodge filtration F · by subbundles. c) V∞,σ is a variation of Q–M l HS on Xσ (C), l which is admissible in the sense of [Ka]. More precisely, in the notation established there, we require V∞,σ to be admissible with respect to any compactification of Xσ . By the proof of [Ka], Proposition 1.10.1, this definition does not depend on the choice of this compactification. The underlying local system of V∞,σ , together with its weight filtration is supposed to come from a local system over ZZ. This hypothesis 23

is automatically satisfied if V∞,σ is unipotent. It implies that the local system, tensored with Q l l , can be interpreted as a lisse l–adic sheaf on Xσ (compare [FK], I, § 11). d) Il,σ is an isomorphism FO (V∞,σ |k ) ⊗Ql Q l l −→ σ ∗ Fl (Vl ) of weight–filtered l–adic sheaves on Xσ . Here, FO and Fl are suitably defined forgetful functors. e) IDR,σ is a horizontal isomorphism FO0 (V∞,σ ) −→ VDR ⊗k,σ Cl of bifiltered vector bundles on Xσ (C). l Again, FO0 is a suitable forgetful functor. It follows that the filtrations in b) are finite and that the Hodge filtration in b) satisfies Griffiths–transversality: ∇F p ⊂ F p−1 ⊗OX Ω1X/k

for all p ∈ ZZ .

f) Let c : Cl → Cl denote complex conjugation. For any σ : k ,→ C, l conjugation defines a diffeomorphism cσ : Xσ (C) l −→ Xc ◦ σ (C) l . For a variation of Q–M l HS W on Xc ◦ σ (C), l we define a variation c∗σ (W) on Xσ (C) l as follows: the local system and the weight filtration are the pull backs via cσ of the local system and the weight filtration on W, and the Hodge filtration is the pull back of the conjugate of the Hodge filtration on W. c∗σ preserves admissibility. I∞,σ is an isomorphism of variations of Q–M l HS V∞,σ −→ c∗σ (V∞,c ◦ σ ) −1 such that c∗c ◦ σ (I∞,σ ) = I∞,c . ◦σ

For ρ ∈ Gk we suppose that Il,σρ = σ ∗ (canρ ) ◦ Il,σ . Here, canρ denotes the isomorphism Fl (Vl ) −→ ρ∗ Fl (Vl ) given by the fact that Fl (Vl ) comes 24

from X. Furthermore, we require the following: For each σ, let c∞,σ be the antilinear involution of Fdiff. (V∞,σ ), the C ∞ –bundle underlying V∞,σ , given by complex conjugation of coefficients. Likewise, let cDR,σ be the antilinear isomorphism Fdiff. (V∞,σ ) −→ c−1 σ (Fdiff. (V∞,c ◦ σ )) given by complex conjugation of coefficients on the right hand side of the isomorphism in e). Our requirement is the validity of the formula ◦ Fdiff. (I∞,σ ) = cDR,σ ◦ c∞,σ = c−1 σ (c∞,c ◦ σ ) cDR,σ .

Example: In order to make conditions d)–f) transparent, we consider the mixed system Q(1) l on Spec(k): The data a)–c) are given by the usual constructions. For d), note that the underlying integral structures are given by the “local system on Spec(C)” l 2πi·ZZ on the left hand side and the projective system ZZl (1) of ln –th roots of unity in Q. l The embedding σ : Q l ,→ Cl maps a 2πi

topological generator γ of ZZl (1) to the projective system ((e ln )n∈ZZ )m for some number m ∈ ZZ∗l , and the isomorphism ∼

(2πi · ZZ/ln ZZ)n∈ZZ −→ (µln )n∈ZZ z

is given by sending z to γ m . The isomorphisms I∞,σ are given by the identity. ⊗n For n ∈ ZZ, we let Q(n) l := Q(1) l . Also, for a scheme X/k as above, we

denote by Q(n) l the pull–back via the structural morphism of the mixed system Q(n) l on Spec(k). Definition (continued): The last condition we impose is the existence of a system of polarizations: there are morphisms W GrW l l (−n) , l ∈ lN prime , n ∈ ZZ, l l Grn Vl → Q n Vl ⊗Q W GrW l DR (−n) , n ∈ ZZ n VDR ⊗OX Grn VDR → Q

of l–adic sheaves and flat vector bundles on X, and polarizations W GrW l , σ : k ,→ C, l n ∈ ZZ l Grn V∞,σ → Q(−n) n V∞,σ ⊗Q

25

of variations of Q–M l HS such that the Il,σ , IDR,σ and I∞,σ and the corresponding morphisms of the mixed system Q(−n) l form commutative diagrams. ii) U M SQsl (X) is the full subcategory of objects admitting a filtration, whose graded objects come from Spec(k). If X is geometrically connected, then M SQsl (X) and U M SQsl (X) are Tannakian categories. As we shall see, results analogous to 2.6, 2.7 and 2.9 hold. Remarks: a) Observe that a polarization on a pure Hodge structure induces a polarization on any subquotient. In particular, any subquotient of a pure polarizable Hodge structure enjoys the property of “self–duality up to twist”. We don’t expect a concept similar to graded–polarizability to exist for Galois modules. However, the definition of graded–polarizability, that we introduced for mixed systems, is well behaved. As in the Hodge–theoretic context it ensures that there are no non–trivial extensions of pure objects of the same weight. b) As suggested by the notation, the category M SQsl (X) does not depend on the base field k: if we consider the Grothendieck restriction X −→ Spec(k) −→ Spec(Q), l we get a smooth, separated scheme Z of finite type over Q l and a canonical isomorphism between M SQsl (X) and M SQsl (Z). c) For an extension K/k of number fields, we get a natural faithful forgetful functor M SQsl (X) −→ M SQsl (XK ). It identifies M SQsl (X) with the category of descent data in M SQsl (XK ). In order to be able to define the de Rham–version of Genx , we prove the following result: Lemma 2.10: Let L/K be an extension of fields of characteristic zero, S/K a locally noetherian scheme, F a coherent sheaf on S and G a coherent subsheaf of FL := F ⊗K L, that is invariant under all automorphisms of L over K. Then G descends to a coherent subsheaf of F .

26

Proof: Because of the coherence of G, and by usual Galois descent and induction, we may assume that L = K(X). In order to get a descent datum for G, we need to show that under the identification of p∗1 FK(X) and p∗2 FK(Y ) , where p1 and p2 are the morphisms SK(X)⊗K K(Y ) p1 .

& p2

SK(X)

SK(Y ) &

,

. S

the sheaves p∗1 G and p∗2 G correspond. Observe that K(X) ⊗K K(Y ) is the ring obtained by localizing K[X, Y ] at the multiplicative subset {f · g | f ∈ K[X], g ∈ K[Y ]}. It is integral, noetherian of dimension one, and its fraction field is K(X, Y ). Let R := K(X) ⊗K K(Y )alg , where K(Y )alg is the algebraic closure of K(Y ). Let Z denote the reduced closed subscheme of the points s of SR where we have (p∗1 G)s 6= (p∗2 G)s . Now observe that the maximal ideals of R are in bijective correspondence with the non–constant functions in K(Y )alg . It follows from our hypothesis that Z must be contained in the generic fibre of SR → Spec(R), and hence that Z is empty.

q.e.d.

Recall ([D1], II, Th´eor`eme 5.9) that any local system over Cl on the set U (C) l of C–valued l points of a smooth complex variety U , i.e., any vector bundle with a flat connection, is canonically equipped with an algebraic structure. The algebraic connection is regular at infinity. The following result is essentially a consequence of Schmid’s Nilpotent Orbit Theorem ([Sch], Theorem 4.9): Theorem 2.11: ([Ka], Proposition 1.11.3.) Let U be a smooth complex variety, V an admissible variation of Hodge structure on U . Then the Hodge filtration is a filtration by subbundles, that are algebraic with respect to the canonical algebraic structure on the local system underlying V. We are now in a position to define the de Rham–version of Genx : fix an embedding σ of k into C. l By [D1], II, Th´eor`eme 5.9 and the previous theorem, both the weight and the Hodge filtrations of Genx,∞,σ are algebraic. In [D4], 27

10.36–10.43, it is proven that the pro–vector bundle underlying Genx,∞,σ and its flat connection carry a canonical k–structure. It is given by the fact that the base change by Cl of the Tannaka dual of the category of unipotent vector bundles with integrable connection on X coincides with the Tannaka dual of the analogous category of bundles on Xσ , i.e., the Tannaka dual of the category of unipotent representations of π1 (Xσ (C), l x), which by 2.2.b) coincides with ˆ Ql C. W (π1 (Xσ (C), l x))⊗ l By the same process as in the proof of 2.8, it is possible to describe the k–structure of the weight filtration. So it makes sense to speak of Genτx,∞,σ , the variation on Xσ (C) l conjugate to Genx,∞,σ under τ ∈ Aut(C/k). l Its underlying local system and weight filtration coincide with those of Genx,∞,σ . Clearly Genτx,∞,σ is pro–unipotent and admissible. In order to show that the Hodge filtration coincides with that of Genx,∞,σ , ˆ x of Genx,∞,σ at x induces a observe that by 2.6 the element 1 of the fibre U natural isomorphism of functors on U VarQl (Xσ ) ∼

Hom(Genx,∞,σ , ) −→ (V 7→ (W0,Ql ∩ F 0 )Vx ) . If we apply this observation to the element 1 of the fibre of Genτx,∞,σ at x, we see that the isomorphism 1 7→ 1 of the underlying local systems of Genx,∞,σ and Genτx,∞,σ respects the Hodge filtrations as well. By 2.10, the Hodge filtration of Genx,∞,σ descends to k. Arguments similar to the ones used above show that the object Genx,DR thus defined is independent of the choice of σ. Remarks: a) The fact that the weight and Hodge filtrations of Genx,∞,σ descend to the base field was already observed by Wojtkowiak: see [Wo1], Theorem E. b) In fact, Wojtkowiak shows ([Wo1], [Wo2]) that the Hodge, l–adic and de Rham versions of Genx appear as relative cohomology objects for a morphism of certain smooth, simplicial schemes. So if we follow the definition of the category of sheaves “of geometric origin” proposed in the introduction of [Wo1], the generic sheaves Genx in fact belong to that category. It remains to show that the various versions of Genx fit together to form a pro–mixed system of smooth sheaves on X. The compatibility of the weight filtrations follows from an observation similar to the above: they are compatible with the multiplicative structures of the fibres at x, and they correspond on the cohomology groups (see [J], § 3). 28

Similar arguments show that the morphisms induced by complex conjugation give rise to isomorphisms I∞,σ , which behave as required. So up to the existence of a system of polarizations, we have checked all of the axioms. Theorem 2.12: Let X be a smooth, separated, geometrically connected scheme of finite type over a number field k, and let x ∈ X(k). a) Genx := (Genx,l , Genx,DR , Genx,∞,σ , Il,σ , IDR,σ , I∞,σ | l, σ, σ) is a pro–object of U M SQsl (X). b) Let Px be the Tannaka dual of U M SQsl (X) with respect to “fibre at x”

◦

(projection to the (∞, σ0 )–component):

U M SQsl (X) −→ U VarQl (Xσ0 ) −→ VecQl for some choice of σ0 : k ,→ C, l π

−→ Px ←− G x

the morphisms induced by π ∗ and “fibre at x”, where G is the Tannaka dual of M SQsl (Spec(k)) with respect to “forget”

◦

(projection to the (∞, σ0 )–component).

Then the natural morphism η

x Wx := ker(π) W (π1 (Xσ0 (C), l x)) −→

is an isomorphism. c) The functor  s   objects H of M SQl (Spec(k))      together with a morphism   

          

   

   

U M SQsl (X) −→  of mixed systems on Spec(k)  ,      ˆ x −→ End(H) respecting     U the algebra structure

V 7−→ Vx

is an equivalence of categories. ˆ x carries the mixed structure of the fibre at x of Genx . Here, as usual, U

29

Proof: We prove a), leaving b) and c) to the reader. It suffices to show graded–polarizability of the restriction of our system to a dense open subset U of X: any polarization will automatically extend to the whole of X because the respective fundamental groups of U surject onto those of X and because of 2.2.a), 2.6, 2.9.c) and [D4], Corollaire 10.43. The remark following the definition of U M SQsl (X) allows us to assume that X is affine. Then we have a mixed system on Spec(k) 1 H 1 (X, Q(0)) l := (H 1 (X, Q l l (0)), HDR (X), H 1 (Xσ (C), l Q(0)) l | l, σ) ,

where the polarizations on the graded parts are constructed as follows: choose a smooth projective compactification Z/k of X such that Z\X is a divisor with normal crossings. The Leray spectral sequences for X ,→ Z, for X ,→ Z and the Xσ ,→ Zσ give an interpretation of the graded parts of the above system as subquotients of cohomology systems of smooth projective varieties. For any such, choose a hyperplane section. It defines an algebraic correspondence, which in particular defines a compatible decomposition of the cohomology system into primitive components. On these, polarizations are given by the composition of the successive intersection with the class of the hyperplane section and cup product. For details, see [D2], 3.2 and 2.2.6. The weight–graded objects of Genx are subquotients of direct sums of tensor ∨ powers of π ∗ H 1 (X, Q(0)) l : the map

π∗

M

∨ ⊗m ∨ ⊗l 1 l ) ⊗Ql (W−2 H 1 (X, Q(0)) l ) → Grw (Grw −n Genx −1 H (X, Q(0))

2l+m=n

is well–defined and surjective. So again by the remark following the definition of U M SQsl (X), the system Genx is graded–polarizable.

q.e.d.

30

§ 3 The generic pro–sheaf: the relative case In this paragraph, we aim for statements analogous to 2.7, 2.9.b) and 2.12.b) in the relative case, i.e., the case of a suitably regular morphism π:X →Y . While the results of [SGA1], Exp. XIII are strong enough to let the proofs of 2.8 and 2.9 carry over almost verbatim, the situation in the Hodge–theoretic setting requires a bit more work. The main step, as suggested by the proof of 2.7, is to show that the relative ˆ x underlies an admissible pro–variation of M HS on X. We conclude version of U the paragraph by a characterization of the relative version of Genx by a universal property (3.5, 3.6.d)), which we regard as the central result of this work. Again, let l be a prime number, either F/Q l l finite or F = Q l l , k a number field, π : X → Y a morphism of type (S) of schemes over k, which we define to be a smooth morphism with geometrically connected fibres between smooth, separated, geometrically connected schemes of finite type over k, π being compactifiable in such a way that X is the complement of a relative divisor with normal crossings in a smooth, projective Y –scheme. Following [BL], 1.1.1, we define: l,m Definition: π–U Etl,m F (X) is the full subcategory of those V ∈ EtF (X) admit-

ting a filtration, whose graded objects lie in π ∗ (Etl,m F (Y )). Again, since π is smooth, if such a filtration exists, the weight filtration will have the same property, as follows from [D3], Th´eor`eme 6.1.2, generic base change ([SGA4 1/2], Th. finitude, Th´eor`eme 1.9) and [D3], Corollaire 3.3.5. Objects in π–U Etl,m F (X) will be called π–unipotent, or relatively unipotent, mixed lisse F –sheaves. Until Theorem 3.1, we shall also assume that π admits a section i : Y ,→ X. Afterwards, this assumption will be weakened slightly. Fix y ∈ Y (k) and let x := i(y). Because π is of type (S), we may apply [SGA1], Exp. XIII, Proposition 4.3 and Exemples 4.4 to conclude that there is a split exact sequence π

−→ 1 −→ π1 (X y , x) −→ π1 (X, x) ←− π1 (Y, y) −→ 1 . i

31

So if a mixed lisse F –sheaf on X is relatively unipotent then the induced representation of π1 (X y , x) is unipotent. It will be shown in the proof of 3.2 that the converse also holds. Again, π–U Etl,m F (X) is a neutral Tannakian category over F . The splitting i defines a continuous action of π1 (Y, y) on ˆ (wl (π1 (X y , x)))⊗ ˆ π,x . ˆ Ql l F =: U U ˆ π,x carries the structure of a lisse pro–F –sheaf on Y . So U As before, we define a continuous representation of π1 (X, x) = π1 (X y , x)× π1 (Y, y) as follows: ˆ π,x , the fundamental group π1 (X y , x) acts by the underlying vector space is U multiplication, and π1 (Y, y) acts as above. This defines a lisse pro–F –sheaf Geni on X. By construction, the induced representation of π1 (X y , x) is pro–unipotent. Theorem 3.1: Geni is an object of pro–π–U Etl,m F (X). Proof: The assumptions on π are sufficient to construct extensions $

−→ ←−

X

Y

ı

&

. Spec(oS )

of π and i for a suitably chosen finite set S ⊂ Spec(ok ) containing the primes dividing l, such that [SGA1], Exp. XIII, Proposition 4.3 and Exemples 4.4 are applicable. So as in the proof of 2.8, the representation of π1 (X, x) factors through π10 (X, x) := π1 (X, x)/N , where ker($ : π1 (X, x) → π1 (Y, y))/N is the largest pro–l quotient of ker($). ˆ π,x as defined in the proof of 2.8 is stable under the The weight filtration on U action of π1 (X, x), and the graded quotients are pure subquotients of direct ⊗(−n)

sums of the H 1 (X y , F (0))⊗(−n) = (R1 $∗ FX (0))y

.

The representation of π1 (X, x) on these quotients corresponds precisely to the 32

representation of π1 (X, x) on $ ∗ (R1 $∗ FX (0))⊗(−n) , so the corresponding sheaves are mixed. As their stalks at x are pure, they are pure altogether.

q.e.d.

For the independence of Geni of the choice of y, see Theorem 3.5.iii). Corollary 3.2: Assume that after a finite ´etale covering Y 0 → Y, π admits a section i : Y 0 → X 0 := X ×Y Y 0 . Fix y 0 ∈ Y 0 (k), and let x0 := i(y 0 ), x := pr1 (x0 ) and y := π(x). Let Px be the Tannaka dual of π–U Etl,m F (X) with respect to “fibre at x” : π–U Etl,m F (X) −→ VecF , Gy the Tannaka dual of Etl,m F (Y ) with respect to “fibre at y” : Etl,m F (Y ) −→ VecF , π : Px → Gy the morphism induced by π ∗ . i) The natural morphism ηx ˆ Ql l F −→ Wl (π1 (X y , x))⊗ Wx := ker(π)

is an isomorphism. ii) If Y 0 = Y , then i defines a section of π : Px → Gy , and the functor

π–U Etl,m F (X) −→

 l,m    objects of EtF (Y )     together with a

       

   π1 (Y, y)–equivariant            ∗

:

action of i Geni

∗

V 7−→ i V is an equivalence of categories. Proof: The sequence η

π

x ˆ Ql l F −→ Px −→ Gy −→ 1 Wl (π1 (X y , x))⊗

is exact: π is an epimorphism by 1.2.a), and im (ηx ) = Wx by 1.4.a): in order to see that a representation V of π1 (Y, y) factors over some π1 (Y, y) if π ∗ V factors 33

over some π1 (X, x), we choose a fixed extension $ of π as in the proof of 3.1 and apply [SGA1], Exp. XIII, Proposition 4.3 and Exemples 4.4, together with [EGAIV,3], Th´eor`eme 8.8.2. So if Y 0 = Y , the statements follow from 3.1, 1.4.b) and 1.5. In the general case, we assume Y 0 to be a geometrically connected Galois covering of Y . If Px0 , Gy 0 and Wx0 denote the corresponding objects on the level of X 0 , we have an isomorphism ∼

ˆ Ql l F −→ Wx0 . Wl (π1 (X y , x))⊗ It remains to show that Px0 → Px is a monomorphism: given a representation of π1 (X 0 , x0 ), we convince ourselves that the induced representation of π1 (X, x) inherits relative unipotency, mixedness and graded–polarizability. So we may apply 1.2.b).

q.e.d.

We now describe the Hodge–theoretic situation. Let F ⊂ lR be a field, π : X → Y a morphism of type (S) of schemes over C. l π : X → Y will denote the map on topological spaces underlying πan . Definition: i) VarF (X) is the full subcategory of those objects of the category of graded– polarizable variations of F –M HS on X, that are admissible in the sense of [Ka]. ii) π–U VarF (X) is the full subcategory of those V ∈ VarF (X) admitting a filtration, whose graded objects lie in π ∗ (VarF (Y )). From [Ka], § 0, we recall that admissibility of a graded–polarizable variation can be checked via the curve test. It follows from [SZ], Corollary A.10 that the above categories are neutral Tannakian. Now assume that π admits a section i : Y ,→ X. Fix y ∈ Y (C) l and let x := i(y). Because π is of type (S), the continuous map π : X → Y is locally trivial (compare [D1], II, 6.17). In particular, it is a weak fibration ([Sp], II, § 7, Corollary

34

14 and VII, § 2, Definition after Corollary 4), so we have a split exact sequence π

−→ 1 −→ π1 (X y , x) −→ π1 (X, x) ←− π1 (Y , y) −→ 1 . i

We define a representation of π1 (X, x) = π1 (X y , x)× π1 (Y , y) as follows: the underlying vector space is ˆ π,x := U ˆ (w(π1 (X y , x)))⊗ ˆ Ql F , U π1 (X y , x) acts by multiplication, and π1 (Y , y) acts by conjugation. This defines a pro–local system of F –vector spaces on X. By construction, the induced representation of π1 (X y , x) is pro–unipotent. Given y 0 ∈ Y (C), l there is a canonical isomorphism ∼

ˆ π,x ) |X −→ For(Geni(y0 ) ) , (U 0 y

where the right hand side denotes the pro–local system underlying the generic pro–unipotent variation on Xy0 with basepoint i(y 0 ). This isomorphism allows ˆ π,x , at least fibrewise. us to define weight and Hodge filtrations on U Theorem 3.3: The above data define an object Geni of pro–π–U VarF (X). Proof: Our data provide an example of the path space variations considered in [HZ1], § 4. In their notation, g : Y −→ S is p1 : X ×Y X → X in ours, and the two sections σ0 and σ1 are taken to be x 7→ (x, i ◦ π(x)) and x 7→ (x, x) respectively. Then our data are precisely the ones called {H0 (Pσ0 (s),σ1 (s) Ys ; C)} l s∈S in [HZ1], Proposition 4.20.ii), as can be seen from [HZ1], Definition 4.21.ii). By [HZ1], Proposition 4.20.ii), they define a graded–polarizable variation of Q-M l HS. We now check the conditions of [Ka], (1.8) and (1.9). If the image of a morphism f : ∆∗ → X of the punctured unit disc into X is contained in a single fibre, then f ∗ (Geni ) is admissible because Geni(π ◦ f (∆∗ )) is 35

admissible. Else we may replace Y by ∆∗ , assuming that there is another section f of π. But this is exactly the situation studied in [HZ1], § 6. There, conditions (1.8.3) and (1.8.4) of [Ka] are proven. Condition (1.8.2) of [Ka], i.e., quasi– unipotency at infinity, which in [SZ], (3.13) was not yet formulated, follows l But this is a consequence of from the corresponding statement for R1 π ∗ (C). Brieskorn’s Monodromy Theorem ([D1], III, Th´eor`eme 2.3).

q.e.d.

Corollary 3.4: Assume that after a finite ´etale covering Y 0 → Y , π admits a section i : Y 0 −→ X 0 := X ×Y Y 0 . Fix y 0 ∈ Y 0 (C), l and let x0 := i(y 0 ) , x := pr1 (x0 ) and y := π(x). Let Px be the Tannaka dual of π–U VarF (X) with respect to “fibre at x ” : π–U VarF (X) −→ VecF , Gy the Tannaka dual of VarF (Y ) with respect to “fibre at y ” : VarF (Y ) −→ VecF , π : Px → Gy the morphism induced by π ∗ . i) The natural morphism ηx ˆ Ql F −→ W (π1 (X y , x))⊗ Wx := ker(π)

is an isomorphism. ii) If Y 0 = Y , then i defines a section of π : Px → Gy , and the functor   admissible graded–polarizable variations        W of F –M HS on Y together with  

π–U VarF (X) −→  a morphism i∗ Geni → EndF (W) of pro–       variations on Y respecting the algebra   

structure

V 7−→ i∗ V

          

,

         

is an equivalence of categories. Proof: left to the reader.

q.e.d. 36

As in § 2, it is only a formal matter to write down statements analogous to 3.1–3.4 for mixed systems. Assume that π is of type (S) and admits a section i sending y to x. We have to study the dependence of Geni on y: as usual, the choice of a path in the Hodge–theoretic setting or a chain of specialization maps connecting y and y 0 in the λ–adic setting gives an isomorphism ∼

Geni,y0 −→ Geni,y sending 1 ∈ Γ(Y, i∗ Geni,y 0 ) to 1 ∈ Γ(Y, i∗ Geni,y ). Somewhat surprisingly, this isomorphism is in fact independent of our choices, as follows from the next result, which holds in the Hodge theoretic as well as in the l–adic context: Theorem 3.5: i) The natural transformation of functors from π–U VarQl (X) to VarQl (Y ) l,m l,m (resp. from π–U EtQ l l (X) to EtQ l l (Y ))

ev : π∗ Hom(Geni , − ) −→ i∗ , ϕ 7−→ (i∗ ϕ)(1) is an isomorphism. Observe that the direct system cd(π∗ Hom(Geni /an , V))n∈lN , where a denotes the augmentation ideal of Geni , becomes constant for any l,m V ∈ π–U VarQl (X) (resp. π–U EtQ l l (X)). This constant value is denoted

by π∗ Hom(Geni , V) . ii) The natural transformation of functors from the category of relatively unipotent local systems (resp. l–adic lisse sheaves) on X to the category of local systems (resp. l–adic lisse sheaves) on Y ∗

ForQl (l) (ev) : π ∗ Hom(ForQl (l) (Geni ), − ) −→ i , ∗

ϕ 7−→ (i ϕ)(1) is an isomorphism. iii) The pair (ForQl (l) (Geni ), 1) admits no non–trivial automorphisms. 37

Proof: Let P = W × G be a semidirect product of pro–algebraic groups over Q l (l) . Assume that W is a countable projective limit of unipotent group schemes, ˆ (LieW ). and let 1 be the unit element of the completed universal envelope U Then the natural transformation of functors from RepQl (l) (P ) to RepQl (l) (G) ˆ (LieW ), − ) −→ resG ev : HomW (U P , ϕ 7−→ (resG P (ϕ))(1) is an isomorphism.

q.e.d.

Remark: 3.5.i) still holds in the l–adic context when we consider the categories l l π–U EtQ l l (X) and EtQ l l (Y ), i.e., remove the mixedness assumption.

Now, again assume k is a number field and π : X → Y is of type (S). Definition: π–U M SQsl (X) is the full subcategory of M SQsl (X) of objects admitting a filtration, whose graded objects lie in π ∗ (M SQsl (Y )). The de Rham version of Geni is constructed in a manner analogous to that of § 2, Theorem 2.6 being replaced by 3.4.ii). In order to apply 2.10, we need to know in advance that the pro–vector bundle underlying Geni,∞,σ and its connection carry a canonical k–structure. This is provided by [D4], Corollaire 10.42.ii), applied to F = {R1 π∗ Ω·X/Y }, the vector bundle R1 π∗ Ω·X/Y being equipped with the Gauß–Manin connection, which is flat and regular at infinity. Remark: If we consider the categories VB(Y ) and π–U VB(X) of flat vector bundles on Y and π–unipotent flat vector bundles on X, whose connection is regular at infinity, then the flat vector bundle underlying Geni,DR , which is a pro–object of π–U VB(X), together with the section 1 of i∗ Geni,DR , has a universal property similar to that of 3.5.i). Theorem 3.6: Assume that after a finite ´etale covering Y 0 → Y , π admits a section i : Y 0 −→ X 0 := X ×Y Y 0 . Fix y 0 ∈ Y 0 (k), and let x0 := i(y 0 ) , x := pr1 (x0 ) and y := π(x). a) If Y 0 = Y , then Geni := (Geni,l , Geni,DR , Geni,∞,σ , Il,σ , IDR,σ , I∞,σ | l, σ, σ) is a pro–object of π–U M SQsl (X). 38

b) Let Px be the Tannaka dual of π–U M SQsl (X) with respect to “fibre at x”

◦

(projection to the (∞, σ0 )–component)

for some choice of σ0 : k ,→ C, l Gy the Tannaka dual of M SQsl (Y ) with respect to “fibre at y”

◦

(projection to the (∞, σ0 )–component),

and π : Px −→ Gy the morphism induced by π ∗ . Then the natural morphism η

x W (π1 (Xy,σ0 (C), l x)) −→ Wx := ker(π)

is an isomorphism. c) If Y 0 = Y , then i defines a section of π : Px → Gy , and the functor

π–U M SQsl (X) −→

  objects W of M SQsl (Y )       together with a morphism   algebras in M SQsl (Y )      i∗ Gen → End (W) i

Q l

∗

V 7−→ i V

       of 

,

      

is an equivalence of categories. d) If Y 0 = Y , then the natural transformation of functors from π–U M SQsl (X) to M SQsl (Y ) ev : π∗ Hom(Geni , − ) −→ i∗ , ϕ 7−→ (i∗ ϕ)(1) is an isomorphism. Proof: left to the reader.

q.e.d.

39

Remarks: a) Note that in the case of relative elliptic curves, the pro–sheaf Geni coincides with the “logarithmic sheaf” of [BL], 1.2. (Compare [BLp], § 1 for the case of arbitrary curves, that are “unipotent K(π, 1)s”, i.e., unequal to lP1 .) This follows from 3.5.i) and [BL], 1.2.6. Note that [BL], 1.2.10.v) coincides with 3.2.ii), 3.4.ii) and 3.6.c) here. b) (compare [BL], 1.2.10.) ˆ Ql (l) Geni and the section 1⊗1, ˆ If we apply 3.5.i) to the pro–object Geni ⊗ we get a comultiplication ˆ Ql (l) Geni , Geni −→ Geni ⊗ which is coassociative and cocommutative as follows from another application of 3.5.i). If follows that Geni carries the structure of a cocommutative coalgebra, the counit being given by the augmentation morphism. Furthermore, i∗ Geni carries the structure of an algebra, the unit being given by 1. Both structures are compatible, i.e., i∗ Geni is equipped with a natural Hopf algebra structure. This is exactly the Hopf algebra structure corresponding to the tensor structure of   smooth   



sheaves W on Y together with     ∗ RepF (i Geni ) := a morphism of sheaves of algebras        i∗ Gen −→ End (W)  i F given by 3.2.ii), 3.4.ii) and 3.6.c) respectively. c) Given a morphism π : X −→ Y of type (S), one may form the Poincar´e groupoid sheaf, i.e., the pr1 –unipotent sheaf Gen∆ on X ×Y X associated to the “universal section” ∆. It is certainly a more canonical object than the Geni associated to sections of π, and the main results of this section easily follow from the corresponding results for Gen∆ .

40

§ 4 Families of unipotent K(π, 1)s In this paragraph, we shall be concerned with the following problem: when is it possible to compute higher direct images of mixed smooth relatively unipotent sheaves within the category of such sheaves, i.e., by Hochschild cohomology? In doing so, we have to make use of certain categories of perverse sheaves (see [BBD], § 2) and of Saito’s theory of algebraic mixed Hodge modules ([S1], [S2]). Whenever we speak of perverse sheaves they will be formed with respect to the middle perversity. This means that we shall always have to deal with a shift of degree when comparing “usual” and perverse cohomology. We chose not to introduce any specific notation for the inclusion of the category of smooth sheaves into the category of all sheaves. However, this leads to the following slight complication: if for example V is an admissible variation of Hodge structure on X, then we consider V as an algebraic mixed Hodge module (see [S2], Theorem 3.27 and the remark following it). Its underlying perverse sheaf on X(C) l is the complex For(V)[dim X], where For(V) denotes the local system underlying V. We hope that these conventions won’t lead to too much confusion. Definition: a) Let X be a pathwise connected topological space, x ∈ X such that π1 (X, x) is finitely generated. X is called a unipotent K(π, 1) if the natural map H · (W (π1 (X, x)), Q) l → H · (X, Q) l is an isomorphism. (So if X is a K(π, 1), then this is the case if and only if π1 (X, x) is pseudo–nilpotent.) b) Let X be a pathwise connected scheme ([SGA4,III], Exp. IX, D´efinition 2.12) over an algebraically closed field k of characteristic 0, x a geometric point such that π1 (X, x) is topologically finitely generated. X is called a unipotent l–K(π, 1) if the natural map H · (Wl (π1 (X, x)), Q l l ) −→ He´· t (X, Q l l) is an isomorphism.

41

Remark: We chose not to follow the terminology of [CH] and [H1], where a topological space as in a) is called a rational K(π, 1). Lemma 4.1: If k can be embedded into C, l and if X/k is connected and of finite l x) is finitely generated, type, then for any embedding k ,→ Cl such that π1 (X(C), X is a unipotent l–K(π, 1) if and only if X(C) l is a unipotent K(π, 1). Proof: By [SGA1], Exp. XII, Corollaire 5.2, π1 (X, x) is the profinite completion l x). So by 2.3 and the remark following 1.10, the left hand sides of of π1 (X(C), a) and b) coincide after tensoring a) with Q l l. Similarly, one uses [FK], Theorem 11.6 for the right hand sides.

q.e.d.

Since all the schemes occurring in this paragraph are of the type considered in the lemma, we shall also simply speak of unipotent K(π, 1)s, the condition being checked at any prime number l. From now on, it will happen frequently that definitions, theorems or proofs are “formally identical” in the Hodge theoretic and the λ–adic setting. In order to make the writing style more economical, and also to make clear that the conclusions really are purely formal once we have a theory of sheaves satisfying a certain set of axioms, we fix the following rules: whenever an area of paper is divided by a vertical bar:

(usually)

or

(rarely)



the text on the left of it will concern the Hodge–theoretic setting, while the text on the right will deal with the λ–adic setting. This understood, we let k := C, l

k := a number field, l := a fixed prime number,

F ⊂ lR a subfield,

F/Q l l finite or F = Q l l,

π : X −→ Y a morphism of type (S) between schemes over k, 42

X := X(C), l

X := X ⊗k k,

Y := Y (C) l as topological spaces,

Y := Y ⊗k k,

π :X →Y,

π : X → Y,

x ∈ X(k), y := π(x). We continue to assume that after a finite ´etale covering Y 0 → Y , π admits a section i : Y 0 → X 0 := X ×Y Y 0 sending a pre–image of y to one of x. The hypothesis that Y , hence also X, be geometrically connected is not really necessary. It just serves to make applicable the Tannakian formalism. Shs (Y )

:= Etl,m F (Y ),

VarF (Y ),

Shs (Y )

Shsπ (X) :=

π–U VarF (X),

Shsπ (X) := π–U Etl,m F (X),

Shs (Y )

the category of local

Shs (Y )

:= :=

systems of F –vector

constructible

spaces on Y , Shsπ (X)

:=

:= the category of lisse F –sheaves on Y ,

Shsπ (X)

the category of

:= the category of

π–unipotent local

π–unipotent lisse

systems of F –vector

constructible

spaces on X.

F –sheaves on X.

Each of these categories is naturally contained in one of the following: := M HMF (Y ),

Sh(Y )

:= Perv m F (Y ),

Sh(X) := M HMF (X),

Sh(X)

:= Perv m F (X),

Sh(Y )

:= Perv F (Y ),

Sh(Y )

:= Perv F (Y ),

Sh(X)

:= Perv F (X).

Sh(X)

:= Perv F (X).

Sh(Y )

Here, M HMF denotes the category of algebraic mixed F –Hodge modules ([S2], § 4). Perv F denotes the category of perverse sheaves on the topological space underlying a complex manifold ([BBD], 2.1) or on a smooth scheme over an algebraically closed field of characteristic zero ([BBD], 2.2).

43

In order to define Perv m F (Y ), we proceed as follows: a constructible F –sheaf V on Y is called mixed if it can be extended to a separated scheme of finite type Y → Spec(oS ) such that the extension is mixed in the sense of [D3], D´efinition 1.2.2. Etm F (Y ) is the category of b mixed constructible sheaves on Y . We define Dm (Y, F ) to be the full sub-

category of Dcb (Y, F ) of those complexes whose usual cohomology objects b lie in Etm F (Y ). By [D3], VI, the categories Dm (− , F ) are stable under the

usual six functors. Since only these functors are used to define the perverse t–structure on Dcb (Y, F ) (see in particular [BBD], Proposition 2.1.3 and Th´eor`eme 1.4.10), we may proceed as in [BBD], 5.1 and define Perv m F (Y ) as the heart of b (Y, F ). We have perverse cohomology functors Dm b H q : Dm (Y, F ) → Perv m F (Y ).

In particular we have the perverse higher direct images b Hq π∗ : Perv m F (X) ,→ Dm (X, F )

π∗ restricted to

−→

b (X,F ) Dm

Hq

† b −→ Dm (Y, F ) −→ Perv m F (Y ).

Because π is of type (S), up to a shift of degree, Hq π∗ restricted to m Etl,m F (X) ⊂ Perv F (X), can be computed via the ordinary higher direct

image, which is in fact what we are always going to do. Note that this definition should only be seen as a very modest approximation of what one might consider to be “the right one”. As suggested by Saito’s definition ([S2], 2.1) one should start by using a filtered category of complexes of constructible sheaves such that the filtration induces up to a shift the weight filtration on the mixed cohomology objects. However, the aim of this paragraph is only to show that under a condition on the fibres of π, the higher direct image Hq π∗ , when restricted to Shsπ (X), can be calculated within Shsπ (X). For this, we just need a “surrounding triangulated category” of Shsπ (X) in Dcb (X, F ). We could even have chosen Dcb (X, F ). But we definitely feel that for less special π as considered here, one should use perverse (as opposed to usual) higher direct images. In any case, as soon as the correct definition of Perv m F is found, Theorem 4.3 below will hold for Shsπ (X) ∩ Perv m F (X) if we manage to show that Geni lies in †

As in [BBD], we denote by π∗ , π ∗ , Hom etc. the respective functors on the derived

category of mixed sheaves.

44

pro–Perv m F (X) and if the natural functor b Perv m F (X) −→ Dc (X, F )

is compatible with π∗ . We call sheaves on Y and X topological sheaves while referring to sheaves on Y and X simply as “sheaves”. So we have natural forgetful functors associating to a sheaf its underlying topological sheaf. In the case of Sh(Y ) and Sh(X), they are compatible with Hq π∗ and Hq π ∗ : Sh(X) −→ Sh(X)   q yH π ∗

 

H q π∗ y

Sh(Y ) −→ Sh(Y )

commutes, as follows from [S2], Theorem 4.3.

smooth base change ([SGA4,III], Exp. XVI, Corollaire 1.2).

An analogous statement holds for Shs (Y ) and Shsπ (X): Lemma 4.2: Ind–Shsπ (X) −→ Ind–Shπs (X)  

  q s yR π ∗

Rq π∗s y

Ind–Shs (Y ) −→ Ind–Shs (Y ) commutes. Here, Rq π∗s is the q–th higher direct image, computed in Ind–Shsπ (X), which, being the ind–category of a neutral Tannakian category, has enough injectives (compare [Ho], § 2). Similarly for Rq π s∗ . Proof: Rq π∗s is calculated using cohomology of Wx , Rq π s∗ is calculated using cohomology of W(l) (X y , x). The groups coincide by Corollary 3.4.i).

Corollary 3.2.i). q.e.d.

In particular, if H · (Wx , Q l (l) ) is finite–dimensional, we have the same commutative diagram without the prefixes “Ind”. Slightly generalizing the notion recalled at the beginning of the paragraph, we define: 45

Definition: Let X be a pathwise

connected

topological

pathwise connected scheme over an

space, x ∈ X such that π1 (X, x) is

algebraically closed field k of charac-

finitely generated.

teristic 0, x a geometric point such that π1 (X, x) is topologically finitely generated.

X is called a unipotent (l–)K(π, 1, ≤ q0 ) if the natural map H q (W (π1 (X, x)), M ) → H q (X, M ) H q (Wl (π1 (X, x)), M ) → He´qt (X, M ) is an isomorphism for any q ≤ q0 , and any unipotent local system of Q–vector l spaces

lisse constructible Q l l –sheaf

M on X. So a sufficient condition is that the above map for M = Q l (l) be bijective for q ≤ q0 and injective for q = q0 + 1. 

By [Hub], Theorem 2.6, the system (Rq π∗s ) |Shsπ (X)



q∈lN0

is the cohomological

derived functor of π∗s |Shsπ (X) . This guarantees the existence of the transformation of functors in the following Theorem 4.3: Assume in addition to the hypothesis on π already made, that the fibres of π are unipotent K(π, 1, ≤ q0 )s. Let q ≤ q0 , d := dim X − dim Y . Then the natural transformation of functors on Shsπ (X) Rq π∗s −→ (Hq−d π∗ ) |Shsπ (X) is an isomorphism. Proof: Since the forgetful functors are exact and faithful, it suffices to show that the assertion holds on the level of topological sheaves. Remember that because π is of type (S), perverse higher direct images coincide, up to shift, with usual higher direct images, when evaluated on smooth topological sheaves. Rq π ∗ transforms smooth topological sheaves into smooth topological sheaves and satisfies base change. So the result follows from the definition of unipotent K(π, 1, ≤ q0 )s.

q.e.d.

It should be possible to define a category M SQl (X) of mixed systems of constructible sheaves on a separated scheme X of finite type over a number field 46

k, and prove an analogue of 4.3 in this setting. Of course we are confident that once the “real” category of mixed motivic sheaves is found, the above proof will carry over without difficulties. Theorem 4.3 allows us to compute π∗ (Geni ) under rather restrictive hypotheses on the fibres of π. Note however that these assumptions are fulfilled if we consider the projection of a mixed Shimura variety (see [P]) to the underlying pure Shimura variety. Corollary 4.4: Let d := dim X − dim Y be the relative dimension of π. Assume that Wx is algebraic of dimension N and that the fibres of π are unipotent K(π, 1)s. Consider the augmentation morphism ε : Geni −→ →Q l (l) (0). π∗ (ε) factors over an isomorphism ∼

π∗ (Geni ) −→ ΛN (Lie(Wx ))∨ [−N + d], which is the unique map making the following diagram commutative: HN −d π∗ (Q l (l) (0))[−N + d] −→ π∗ (Q l (l) (0)) o↓

↑ π∗ (ε) ∼

ΛN (Lie(Wx ))∨ [−N + d] ←− π∗ (Geni ) More precisely, write (n)

Geni = lim Geni ←−

n∈lN

(n)

where the Geni

are smooth quotients of finite rank of Geni , and let ε(n) , n  0

be the augmentations. For any q, consider the projective system (n)

(Hq π∗ (Geni ))n∈lN . Then, for q 6= N − d, this system is ML–zero, i.e., for any n there is a positive integer m(q, n) ∈ lN such that the transition morphism (n+m(q,n))

Hq π∗ (Geni

(n)

) −→ Hq π∗ (Geni )

47

is zero. For n  0, the morphism (n)

HN −d π∗ (ε(n) ) : HN −d π∗ (Geni ) −→ HN −d π∗ (Q l (l) (0)) is surjective. The projective system (ker(HN −d π∗ (ε(n) )))n0 is ML–zero. Observe that the map HN −d π∗ (Q l (l) (0))[−N + d] −→ π∗ (Q l (l) (0)) exists because Hq π∗ (Q l (l) (0)) = 0 for q > N − d by 4.3. The Tannaka dual Gy of Shs (Y ) acts on Lie(Wx ) via the section i and conjugation. The left vertical arrow is induced by the canonical isomorphism of 1.13. Remarks: a) Readers irritated by the “wrong” shifts should recall that the Hq π∗ correspond to perverse higher direct images of π. b) While N equals the cohomological dimension of Wx , which can be defined whenever the fibres of π are unipotent K(π, 1)s, the statement “HN −d π∗ (Geni ) is of rank one” is in general false without the algebraicity assumption on Wx . For Y = Spec(k), X an incomplete curve over k not containing G| m,k , we have N = d = 1, but the vector space underlying l (l) (0)) H0 π∗ (Q l (l) (0)) = H 1 (X, Q is of finite dimension greater or equal to two. Furthermore, H0 π∗ (Geni ) surjects onto H0 π∗ (Q l (l) (0)). (n)

Proof of Corollary 4.4: By 4.3, we have to consider (H q (Wx , Geni ))n∈lN . But this is what we already did in 1.12.b) and 1.13. Note that in our situation, the isomorphism of 1.13, being canonical, is an isomorphism of Gy –modules. q.e.d.

48

Index of Notations RepF (P )

1

I∞,σ

24

VecF ˆ (LieW ) U

1

c∞,σ

25

3

cDR,σ

25

Rat(W )

5

U M SQsl (X)

26

R(W )

5

Genx,∞,σ

27

H (W, X)

5

Genx,DR

28

T (W )

8

Genx,l

29

H· (W, M )

9

H 1 (X, Q(0)) l

30

Q[π] l

12

π–U Etl,m F (X)

31

Q[π] l ∧

12

ˆ π,x U

32

W (π)

12

Geni

32

w(π)

12

π

34

Wl (ˆ π)

13

VarF (X)

34

wl (ˆ π)

13

π–U VarF (X)

34

X

15

π–U M SQsl (X)

38

Q–M l HS

15

Geni,∞,σ

38

M HQl ˆx U

15

Geni,DR

38

15

Geni,l

38

U VarQl (X)

16

s

Sh (Y )

43

Genx

17

Shsπ (X)

43

·

Etl,m F (X) U Etl,m F (X)

18

Sh (Y )

43

18

Shπs (X)

43

Q l (l)

22

Sh(Y )

43

M SQsl (X)

23

Sh(Y )

43

s

24

q

H π∗

44

IDR,σ

24

q

H π∗

45

cσ

24

Rq π∗s

45

24

Rq π s∗

45

Il,σ

c∗σ

49

References [B]

A.A. Beilinson, “Polylogarithm and Cyclotomic Elements”, typewritten preprint, MIT 1989 or 1990.

[BBD]

A.A. Beilinson, J. Bernstein, P. Deligne, “Faisceaux pervers”, in B. Teissier, J.L. Verdier, “Analyse et Topologie sur les Espaces singuliers” (I), Ast´erisque 100, Soc. Math. France 1982.

[BD]

A.A. Beilinson, P. Deligne, “Motivic Polylogarithm and Zagier Conjecture”, preprint, 1992.

[BL]

A.A. Beilinson, A. Levin, “The Elliptic Polylogarithm”, in U. Jannsen, S.L. Kleiman, J.–P. Serre, “Motives”, Proc. of Symp. in Pure Math. 55, Part II, AMS 1994, pp. 123–190.

[BLp]

A.A. Beilinson, A. Levin, “Elliptic Polylogarithm”, typewritten preliminary version of [BL], preprint, MIT 1992.

[BLpp]

A.A. Beilinson, A. Levin, “Elliptic Polylogarithm”, handwritten preliminary version of [BLp], June 1991.

[C]

K.–T. Chen, “Iterated path integrals”, Bull. AMS 83 (1977), pp. 831–879.

[CH]

J.A. Carlson, R.M. Hain, “Extensions of Variations of Mixed Hodge Structure”, in D. Barlet, H. Esnault, F. El Zein, J.L. Verdier, E. Viehweg, “Actes du Colloque de Th´eorie de Hodge, Luminy 1987”, Ast´erisque 179–180, Soc. Math. France 1989, pp. 39–65.

[D1]

P. Deligne, “Equations Diff´erentielles a` Points Singuliers R´eguliers”, LNM 163, Springer–Verlag 1970.

[D2]

P. Deligne, “Th´eorie de Hodge, II”, Publ. Math. IHES 40 (1971), pp. 5–57.

[D3]

P. Deligne, “La Conjecture de Weil II”, Publ. Math. IHES 52 (1981), pp. 313–428.

50

[D4]

P. Deligne, “Le Groupe Fondamental de la Droite Projective Moins Trois Points”, in Y. Ihara, K. Ribet, J.–P. Serre, “Galois Groups over Q”, l Math. Sci. Res. Inst. Publ. 16, Springer–Verlag 1989, pp. 79–297.

[DG]

M. Demazure, P. Gabriel, “Groupes Alg´ebriques”, Tˆome 1, North–Holland Publ. Comp. 1970.

[DM]

P. Deligne, J.S. Milne, “Tannakian Categories”, in P. Deligne, J.S. Milne, A. Ogus, K.–y. Shih, “Hodge Cycles, Motives, and Shimura varieties”, LNM 900, Springer–Verlag 1982, pp. 101–228.

[EGAIV,3]

A. Grothendieck, J. Dieudonn´e, “Etude locale des Sch´emas et des Morphismes de Sch´emas”, Troisi`eme Partie, Publ. Math. IHES 28 (1966).

[FK]

E. Freitag, R. Kiehl, “Etale Cohomology and the Weil Conjecture”, Erg. der Math. und ihrer Grenzgeb., Band 13, Springer– Verlag 1988.

[H1]

R.M. Hain, “Algebraic Cycles and Extensions of Variations of Mixed Hodge Structure”, in J.A. Carlson, C.H. Clemens, D.R. Morrison, “Complex Geometry and Lie Theory”, Proc. of Symp. in Pure Math. 53, AMS 1991, pp. 175–221.

[H2]

R.M. Hain, “The Geometry of the Mixed Hodge Structure on the Fundamental Group”, in S.J. Bloch, “Algebraic Geometry – Bowdoin 1985”, Proc. of Symp. in Pure Math. 46, AMS 1987, pp. 247–282.

[Hi]

H. Hironaka, “Resolutions of singularities of an algebraic variety over a field of characteristic zero”, Ann. of Math. 79 (1964), pp. 109–326.

[Ho]

G. Hochschild, “Cohomology of Algebraic Linear Groups”, Illinois Jour. of Math. 5 (1961), pp. 492–519.

[Hub]

A. Huber, “Calculation of Derived Functors via Ind–Categories”, Jour. of Pure and Appl. Algebra 90 (1993), pp. 39–48. 51

[Hum1]

J.E. Humphreys, “Linear Algebraic Groups”, GTM 21, Springer– Verlag 1987.

[Hum2]

J.E. Humphreys, “Introduction to Lie Algebras and Representation Theory”, GTM 9, Springer–Verlag 1980.

[HZ1]

R.M. Hain, S. Zucker, “Unipotent variations of mixed Hodge structure”, Inv. math. 88 (1987), pp. 83–124.

[HZ2]

R.M. Hain, S. Zucker, “A Guide to Unipotent variations of mixed Hodge structure”, in E. Cattani, F. Guill´en, A. Kaplan, F. Puerta, “Hodge Theory. Proceedings, Sant Cugat, Spain, 1985”, LNM 1246, Springer–Verlag 1987, pp. 92–106.

[J]

U. Jannsen, “Mixed Motives and Algebraic K–Theory”, LNM 1400, Springer–Verlag 1990.

[K]

A.W. Knapp, “Lie groups, Lie algebras, and Cohomology”, Mathematical Notes 34, Princeton Univ. Press 1988.

[Ka]

M. Kashiwara, “A Study of Variation of Mixed Hodge Structure”, Publ. RIMS, Kyoto Univ. 22 (1986), pp. 991–1024.

[P]

R. Pink, “Arithmetical compactification of Mixed Shimura Varieties”, thesis, Bonner Mathematische Schriften 1989.

[S1]

Morihiko Saito, “Modules de Hodge Polarisables”, Publ. RIMS, Kyoto Univ. 24 (1988), pp. 849–995.

[S2]

Morihiko Saito, “Mixed Hodge Modules”, Publ. RIMS, Kyoto Univ. 26 (1990), pp. 221–333.

[Sch]

W. Schmid, “Variation of Hodge Structure: The Singularities of the Period Mapping”, Inv. math. 22 (1973), pp. 211–319.

[SGA1]

A. Grothendieck et al., “Revˆetements Etales et Groupe Fondamental”, LNM 224, Springer–Verlag 1971.

[SGA4,II]

M. Artin, A. Grothendieck, J.L. Verdier et al., “Th´eorie des Topos et Cohomologie Etale des Sch´emas”, Tˆome 2, LNM 270, Springer– Verlag 1972. 52

[SGA4,III]

M. Artin, A. Grothendieck, J.L. Verdier et al., “Th´eorie des Topos et Cohomologie Etale des Sch´emas”, Tˆome 3, LNM 305, Springer– Verlag 1973.

[SGA4 1/2]

P. Deligne et al., “Cohomologie Etale”, LNM 569, Springer–Verlag 1977.

[Sp]

E.H. Spanier, “Algebraic Topology”, Springer–Verlag 1966.

[SZ]

J. Steenbrink, S. Zucker, “Variation of mixed Hodge structure, I”, Inv. math. 80 (1985), pp. 489–542.

[W]

J. Wildeshaus, “Polylogarithmic Extensions on Mixed Shimura varieties”, Schriftenreihe des Mathematischen Instituts der Universit¨at M¨ unster, 3. Serie, Heft 12, 1994.

[Wo1]

Z. Wojtkowiak, “Cosimplicial objects in algebraic geometry”, in “Algebraic K–theory and Algebraic Topology”, Proceedings of the Lake Louise conference 1991 or 1992, Kluver Academic Publishers 1993, pp. 287–327.

[Wo2]

Z. Wojtkowiak, “Cosimplicial objects in algebraic geometry II”, preprint.

53

The canonical construction of mixed sheaves on mixed Shimura varieties

J¨org Wildeshaus Mathematisches Institut der Universit¨at M¨ unster Einsteinstr. 62 D–48149 M¨ unster∗

∗

current address: Dept. of Math. Sciences, South Road, GB–Durham DH1 3LE

e–mail: [email protected]

Introduction Given a pure Shimura variety M L (G, H), it is rather well known how to construct functors associating to a representation V of G an l–adic sheaf µL,l (V) on M L (G, H) and a variation of Hodge structure µL,∞ (V) on M L (G, H)(C). l Milne’s results on canonical models of standard principal bundles ([M], III, §§ 4, 5) allow one to show that the vector bundle underlying µL,∞ (V) has a model over the reflex field E(G, H), and that the flat connection and the weight and Hodge filtrations descend to this model, giving rise to a bifiltered flat vector bundle µL,DR (V) on M L (G, H). In this article, we study the analogous functors in the context of mixed Shimura varieties as defined in [P1]. Let W denote the unipotent radical of the underlying group P . The universal envelope of LieW , completed with respect to ˆ (LieW ). Since RepQl (P ) is generated the augmentation ideal, is denoted by U ˆ (LieW ), it apby RepQl (P/W ) and the finite–dimensional subquotients of U pears natural to consider the values of the “canonical construction” functors ˆ (LieW ). For example, admissibility ([Ka]) of the variations of Hodge strucon U ture coming from representations of P/W follows automatically from Schmid’s Nilpotent Orbit Theorem ([Sch], Theorem 4.9), since all these variations are ˆ (LieW )) merely direct sums of their weight–graded parts. By contrast, µK,∞(U is as mixed as one can get by applying the canonical construction. The proof of admissibility of this pro–variation is one of the main results of this work. ˆ (LieW )) coThe central observation, that will simplify our task, is that µK,− (U incides with the generic pro–sheaf of [W2], § 3 for the relative situation given by the projection [π] : M K (P, X) −→ M π(K) (P/W, H) to the underlying pure Shimura variety. It therefore has a lot of desirable properties, which will enable us to show that the canonical construction is just as well behaved as in ˆ (LieW )) and call it the logarithmic the pure case. We decided to rename µK,− (U pro–sheaf. The motivation for reserving this name of the generic pro–sheaf for the context of Shimura varieties is the following: the simplest non–trivial case of a mixed Shimura variety is given by the trivial torus G| m,Ql over the “pure Shimura variety” Spec(Q). l The entries of the period matrix of the logarithmic pro–variation of Hodge structure are essentially powers of the multivalued function

1 ·log 2πi

on G| m (C). l Since its values at roots of unity are rational numbers,

i

ˆ (LieW )) at such roots of unity are canonically equal to the the fibres of µK,∞ (U direct product of their weight–graded objects. Now G| m,tors is precisely the union of the pure sub–Shimura varieties of G| m , and the above “splitting principle” of the logarithmic pro–sheaf over this union is in fact prototypical for all mixed Shimura varieties. In this sense, we like to think of the splitting principle as being a generalization of the fact that log(G| m (C) l tors ) ⊂ 2πi · Q l . On the other hand, we don’t expect the generic pro–sheaf for arbitrary morphisms to split over a Zariski–dense subset unless the fibres of the morphism in question are of a specific shape. § 1 starts with a collection of results of [P1], which we hope is self–contained enough to provide non–experts with an idea of the basic concepts underlying the theory of mixed Shimura varieties. We then recall the Hodge version of the canonical construction. While the definition is rather straightforward, by far the best part of § 2 is taken up by the ˆ (LieW )) is the generic sheaf for proof of the fact mentioned earlier, that µK,∞(U [π] (Theorem 2.1). The proof of admissibility of all variations µK,∞ (V) is then a rather formal matter (Theorem 2.2). In § 3, we define the de Rham version of µK . Theorem 2.1 allows us to use the results of [W2], § 3 and hence reduce ourselves to the pure case, which is covered by [M], III, §§ 4, 5. § 4 treats the λ–adic component of µK . Again, the definition of the λ–adic sheaves poses no problem. We state a conjecture analogous to [P2], Conjecture 5.4.1, which amounts to saying that the sheaves µK,λ(V) are mixed in the sense of [D2], VI (Conjecture 4.2). By proving the λ–adic version of Theorem 2.1 (Theorem 4.4), we are able to show that 4.2 holds for the mixed Shimura variety if and only if it holds for the underlying pure Shimura variety (Theorem 4.6). For a functor with values in mixed systems of smooth sheaves ([W2], § 2), we need to define an admissible variation of Hodge structure not just for the canonical embedding σ0 of E(P, X) into C, l but for any such embedding. This forces us to generalize Milne’s and Shih’s results on conjugates of pure Shimura varieties ([M], II, §§ 4, 5, 7) to the mixed case (§ 5). For the sake of completeness, we also include a description of complex conjugation on the C–valued l points of a ii

Shimura variety, whose reflex field is real (Lemma 5.11, Corollary 5.12). In § 6, we associate to a representation V of P its conjugates representations of the groups

τ,x

V, which are

τ,x

P obtained by the process of twisting defined in

§ 5. We show that if 4.2 holds then the canonical constructions of all the

τ,x

V fit

together to define a mixed system of smooth sheaves µK,M S (V) on M K (P, X). In §§ 2–4, we included results on the compatibility of µK,− and µπ(K), − with higher direct images of [π]∗ and group cohomology of W (Theorems 2.3, 3.5 and 4.7). This article is a revised and extended version of § 5 of my doctoral thesis ([W1]). I would like to thank C. Deninger for his generosity and constant support, and T. Scholl for valuable comments. I am obliged to the organizers of the Oberwolfach Arbeitstagung on Shimura varieties in Spring 1992. What I learned while preparing myself for that conference stimulated the study of the canonical construction for mixed Shimura varieties. Finally, I am most grateful to Mrs. G. Weckermann for skillfully TEXing my manuscript.

iii

§ 1 Mixed Shimura data and mixed Shimura varieties We recall the definition and basic properties of mixed Shimura varieties. Our exposition follows [P1]. Let P/Q l be a connected algebraic group, W := Ru (P ) its unipotent radical, G := P/W , π : P −→ → G, U ≤ W a normal subgroup of P , | m,C | m,IR −→ S the weight, S := ResC/IR G l l the Deligne torus, w : G

X a homogeneous space under P (IR)·U (C), l

h : X −→ Hom(SCl , PCl ) a P (IR)·U (C)–equivariant l map with finite fibres. Write hx for h(x). Let V := W/U , πm : P −→ → P/U . Definition: ([P1], Definition 2.1.) (P, X) is called mixed Shimura data if the following holds for some (hence all) x ∈ X: i) πm ◦ hx : SCl −→ (P/U )Cl is already defined over IR. | m,IR −→ GIR is a cocharacter of the center Z(G)IR of GIR . ii) π ◦ hx ◦ w : G

iii) AdP ◦ hx induces on Lie P a mixed graded–polarizable Q–Hodge l structure (Q–M l HS) of type {(−1, 1), (0, 0), (1, −1)} ∪ {(−1, 0), (0, −1)} ∪ {(−1, −1)} . iv) the weight filtration on Lie P is given by

Wn (Lie P ) =

       

0

, n ≤ −3

Lie U , n = −2

  Lie W     

, n = −1

.

Lie P , n ≥ 0

√ v) int (π(hx ( −1))) induces a Cartan involution on Gad IR . vi) Gad l IR has no nontrivial factors of compact type, that are defined over Q. vii) Z(G) acts on U and on V through a torus, that is an almost direct product of a Q–split l torus with a torus of compact type defined over Q. l 1

Because of weight reasons, the algebraic group V is abelian, and U is contained in Z(W ). If W = 1 then (P, X) is called pure. Actually, in order to be able to define the canonical construction, we shall restrict ourselves to those mixed Shimura data satisfying vii)’ Z(G)0 is an almost direct product of a Q–split l torus with a torus of compact type defined over Q. l This condition implies that any real cocharacter of Z(G) is defined over Q. l Again, because of weight reasons, π : P −→ G is injective on Z(P ), so Z(P )0 is a torus of the same type. As explained in [P1], § 1, these axioms imply Theorem 1.1: Let F ⊂ IR be a field. a) There is a canonical P (IR)·U (C)–invariant l complex structure on X. b) There is a tensor functor RepF (P ) −→

  graded–polarizable  of



variations 

F –Hodge structure on X

.



c) For every irreducible V ∈ RepF (P ), which is pure of some weight n, √ there is a representation of P on F (−n) := (2π −1)−n F ⊂ Cl and a P –equivariant bilinear form Ψ : V × V −→ F (−n) such that for all x ∈ X either Ψ or −Ψ is a polarization of the corresponding M HS on V. Here, V is called pure of weight n if for some (hence all) x ∈ X, | m,C hx ◦ w : G l −→ PC l

acts on VCl by z 7−→ (multiplication by z −n ) . Proof: [P1], 1.18.

q.e.d.

2

The functor RepF (P ) −→ is as natural as it could be:

  graded–polarizable  of



variations 

F –Hodge structure on X



x ∈ X gives a map hx : SCl −→ PCl , i.e., a ZZ2 –grading on VCl for any representation V of P . By definition, the underlying local system is constant, and x 7−→ hx defines the weight and Hodge filtrations. More precisely, for x ∈ X, we have Wn,x (V)Cl =

M

Hxp,q (V), Fxp (VCl ) =

p+q≤n

M

0

Hxp ,q (V),

p0 ≥p

where Hxp,q (V) is the eigenspace of the cocharacter (z1 , z2 ) 7−→ z1−p z2−q of SCl under the action of SCl on VCl given by hx . The complex structure on X is unique with respect to the requirement that the Hodge filtration of any V ∈ RepF (P ) vary holomorphically ([P1], Proposition 1.7.a)). Griffiths transversality is a direct translation of axiom iii), and graded– polarizability follows from 1.1.c). Whenever we have a normal subgroup P0 ≤ P , we can define quotient mixed Shimura data (P, X)/P0 , whose underlying algebraic group is P/P0 and which have a universal property ([P1], Proposition 2.9). In particular, we write (G, H) := (P, X)/W . As in the classical case, one defines mixed Shimura varieties, or rather, their topological spaces of C–valued l points, as follows: let IAf denote the ring of finite adeles over Q, l and let K ≤ P (IAf ) be open and compact. Set M K (C) l := M K (P, X)(C) l := P (Q)\( l X × (P (IAf )/K)) where P (Q) l acts on both factors from the left. We have Proposition 1.2: S S a) M K (C) l = ni=1 Γ(pf,i )\(X0i × pf,i K/K) = ni=1 Γ(pf,i )\ X0i ,

where the union is finite and disjoint, X0i denotes a connected component of X, −1 0 l is an pf,i ∈ P (IAf ), and Γ(pf,i ) := StabP (Q) l (Xi ) ∩ pf,i · K · pf,i ≤ P (Q)

arithmetic subgroup. 3

b) For any i, the group Γ(pf,i ) acts properly discontinuously on X0i . M K (C) l is a normal complex space, whose singularities are at most quotient singularities by finite groups. c) If K is neat, then for any i, the group Γ(pf,i ) acts freely on X0i , so M K (C) l is a complex manifold.† Proof: [P1], 3.2 and Proposition 3.3 including its proof. There, it is shown that b) and c) are true modulo Γ(pf,i ) ∩ Z(P )(Q). l But Z(P ) injects into Z(G) since Lie W is of weight ≤ −1, and by vii)’, any arithmetic subgroup of Z(G)(Q) l is finite. Note that by convention [P1], 0.4, the usage of the term “properly discontinuous” in [P1] differs from the usual one. So we cannot quote [P1], Proposition 3.3 directly.

q.e.d.

Remark: We note that in order to get the conclusion of 1.2.c) for a fixed K, we need only assume that any subgroup of P (Q) l of the shape 0 −1 StabP (Q) l (X ) ∩ pf ·K ·pf

is neat. The conclusions of this article, in particular 1.3, 1.4, 4.1 and the calculation of the Galois group preceding 4.1 remain valid under this weaker assumption as the proofs of the relevant results of [P1] (Lemma 3.11, Corollary 3.12.a)) and [P2] (Proposition 3.3.3) run through without any problems. A morphism ϕ : (P1 , X1 ) −→ (P2 , X2 ) of mixed Shimura data consists of a morphism ϕ : P1 −→ P2 and a P1 (IR) · U1 (C)–equivariant l map ψ : X1 −→ X2 such that X1

ψ −→

  h1 y

X2   yh2

ϕ∗ Hom(SCl , P1,Cl ) −→ Hom(SCl , P2,Cl ) commutes. If ψ is injective and ϕ is a closed immersion, then the morphism is called an embedding. †

For the definition of neatness, see [P1], 0.6 or [P2], 3.2.

4

If Ki ≤ Pj (IAf ), i = 1, 2, satisfy ϕ(K1 ) ≤ K2 , then there is a canonical map [ϕ](C) l = [ϕ]K1 ,K2 (C) l : M K1 (P1 , X1 )(C) l −→ M K2 (P2 , X2 )(C) l , which is holomorphic ([P1], 3.4.b)). Similarly, if pf ∈ P (IAf ) and K 0 ≤ pf ·K ·p−1 f , we have 0

[·pf ](C) l = [·pf ]K 0 ,K (C) l : M K (P, X)(C) l −→ M K (P, X)(C), l which is holomorphic, finite and surjective ([P1], 3.4.a)). We now turn to two of the main results of [P1]: By [P1], Corollary 8.14 and § 9, every M K (P, X)(C) l is the set of C–valued l points of a quasi–projective variety M K (P, X)Cl over C. l By [P1], Theorem 11.18, each M K (P, X)Cl admits a canonical model M K (P, X), which is a normal quasi–projective variety over a number field E(P, X), the so–called reflex field of (P, X), which is given together with fixed embeddings σ0 : E(P, X) ,→ Cl and σ0 := σ0 |E(P,X) . (For a definition of both the reflex field and the canonical model, see [P1], Definitions 11.1 and 11.5.) By [P1], Definition 11.5.a) and Proposition 11.10, all the above holomorphic maps [ϕ](C) l and [·pf ](C) l come from algebraic morphisms [ϕ]Cl and [·pf ]Cl , that descend to the reflex field of the source. These morphisms will be denoted by the symbols [ϕ] and [·pf ] respectively. If K is neat, then [·pf ] is ´etale. By [P1], Corollary 3.12.a), 3.14, 3.22 and Corollary 3.12.b), up to an error obtained by dividing out the action of a finite group, we may think of [π] : M K (P, X) −→ M π(K) (G, H) as a torus–torsor over an abelian scheme over M π(K) (G, H). The abelian scheme is of relative dimension

1 2

dim V , the torus–torsor is of rel-

ative dimension dim U over the abelian scheme. We need to be more precise since we want to show that, possibly up to the geometrical connectedness of M K (P, X), we are in the situation studied in [W2], § 3. 5

Fix once and for all a Levi section i : G −→ P of π. It is not difficult to see that H = (W (IR) · U (C))\ l X: see the remark following [P1], Proposition 2.9. Or look at the proof of [P1], Proposition 2.9 and use [P1], Lemma 1.17 and Corollary 2.12, together with the connectedness of the topological group W (IR)·U (C). l Next, if x ∈ X, then i ◦ π ◦ hx and hx are conjugate under W (IR) · U (C): l namely, the map W (IR)·U (C) l −→

  Levi



decompositions of PCl , that, 

 modulo

UCl , are defined over IR 

,

p 7−→ p·CentPCl (hx ◦ w)·p−1

is a bijection; see the proof of [P1], Proposition 1.16.b). Since Lie(W ) has negative weights, the group CentPCl (i ◦ π ◦ hx ◦ w) defines such a decomposition, hence CentPCl (i ◦ π ◦ hx ◦ w) = p·CentPCl (hx ◦ w)·p−1 = CentPCl (int(p) ◦ hx ◦ w) for some p ∈ W (IR)·U (C). l But i ◦ π ◦ hx and int(p) ◦ hx both lift π ◦ hx . Since they land in the same Levi subgroup, they are equal. This, together with [P1], Corollary 2.12 and the connectedness of W (IR)·U (C) l shows that i : G −→ P can be extended to an embedding i : (G, H) −→ (P, X) which is uniquely determined by the following properties: a) π ◦ i = id(G,H) . b) for any x ∈ X, the element i ◦ π(x) lies in the same connected component as x, i.e., i ◦ π(x) and x are conjugate under W (IR)·U (C). l † Remark: Note that the existence of such a splitting i on the level of Shimura data shows that the following holds for any V ∈ RepF (P ) and x ∈ X: the decomposition Wn,x (V)Cl =

M

Hxp,q (V) , Fxp (VCl ) =

p+q≤n †

M

0

Hxp ,q (V)

p0 ≥p

Note that by [P1], Proposition 2.17.b), there is exactly one morphism i with property a).

That it has property b) can also be seen from the proof there.

6

corresponding to the action of SCl on VCl given by hx is the unique decomposition satisfying H q,p ≡ H p,q mod

M

0

0

H p ,q .

p0

2

where we let ϕ(d) := ](ZZ/dZZ). ii) Show that d(k) of the [iu ]∗ pol(−1)k suffice to generate the Q–vector l space generated by all [iu ]∗ pol(−1)k . iii) Show that the [iu ]∗ pol(−1)k generate the Q l 2 –vector space Hg1 .

For i), observe that by [J3], proof of Lemma 1 and [M], II, Proposition 2.9, the dimension of Hg1 (Q(µ l d ), Q l 2 (k)) equals the corank of the ´etale cohomol

ogy group H´e1t. Spec(oQ(µ l d))

h i 1 2



,Q l 2 /ZZ2 (k) . By [So3], 1.2 and [So3], Proposi-

tion 2, this corank is greater or equal to d(k). Furthermore, equality holds if 

H´e2t. Spec(oQ(µ l d) )

h i 1 2



,Q l 2 /ZZ2 (k) is torsion. This in turn follows from [So4], The-

orem 2, whose proof can be modified to give an analogous statement for l = 2, with k! possibly replaced by 2m k!. From the formula in 4.5, it is easy to conclude that we need only consider those pairs (N, u) with N = 1 and f = d, the b forming a set of representatives of (ZZ/dZZ)∗. Furthermore, we have h

ib

d

i∗

h

pol(−1)k = (−1)k−1 i− b

d

i∗

pol(−1)k .

This shows ii). Finally, observe that since the part of [So1] from page 384 on

wards works for arbitrary l once we know that H´e2t. Spec(oQ(µ l d)) 36

h i 1 l

,Q l l /ZZl (k)



is torsion, all we have to show is the validity of [So2], Th´eor`eme 3 for arbitrary primes. The proof of this result can actually be simplified: [G], Theorem 3.1 and the remark following it, together with elementary class field theory show that the characteristic ideals of the Iwasawa modules E/C and Gal(L∞ /F∞ )+ (in Soul´e’s notation) coincide up to a power of l. This reduces us to showing [So2], Th´eor`eme 2 for arbitrary primes. This in turn follows from [S], § 5, Corollary 4 and § 6, Lemma 1, which are also valid for l = 2 and totally imaginary number fields.† For F = Q(µ l d ) = Q, l the group Gal(L∞ /F∞ )∧ (i)G (in the notation of [So2], page 247) is finite since already the invariants of Gal(L∞ /F∞ )∧ (i) under a subgroup of G of index 2 are finite. This shows iii), and proves our claim.

§ 5 Remarks on the Tamagawa number conjecture for Tate motives Assume we are prepared to accept the existence of a motivic formalism, i.e., a theory of mixed motivic sheaves admitting the usual six functors on the level of derived categories, Hodge– and l–adic realization functors into the categories of algebraic mixed Hodge modules and l–adic mixed perverse sheaves (see [W2], § 4) compatible with the six functors, an isomorphism between motivic cohomology and Ext–groups of mixed motivic sheaves and the compatibility of the realization functors with the regulators. Assume also that a decent motivic analogue of the canonical construction ([W3]) is available. Then the same proofs as in [W4], § 1 yield a motivic version of the polylogarithm, and because of the motivic rigidity principle, its realizations must be the Hodge and l–adic versions of pol described in §§ 3 and 4. Because of the motivic splitting principle, the elements in 3.11 and 4.5 must be the respective regulators of the same elements in motivic cohomology. Observe that this is precisely what is needed to complete the proof of [BK], Theorem 6.1, i.e., the Tamagawa number conjecture modulo powers of 2 for Tate motives Q(k) l with k ≥ 3 odd. Thanks to Kato’s work in [BK], § 6, it can be shown that the conjecture holds if we replace the rational structure Φ of [BK], (5.11) by the rational structure †

The assumption on the number fields is needed in order to have cohomological 2–

dimension of the absolute Galois group equal to two.

37

Φpol,k given as follows: by [W4], Corollary 2.2, there is a mixed system of smooth sheaves pol, whose Hodge– and l–adic components are those of §§ 3 and 4. Fix b run through all possible combinad > 1 and k ≥ 2, and let N and u = f f ·N 1 . Let [iu ]Lu,N ,KN : Spec(Q(µ l d )) ,→ lPQ tions satisfying d = l \{0, 1, ∞} gcd(b, N ) be as before. Define Φpol,k ⊂ Ext1M S s (Spec(Q(µ l Q(k)) l to be the Q–vector l l d ))) (Q(0), Q l

space of one–extensions in the category of mixed systems of smooth sheaves ([W3], § 6) on Spec(Q(µ l d )) generated by the k–th components of the classes of all [iu ]∗Lu,N ,KN pol(−1). We define d(k) as before, i.e.,

d(k) =

        

0

, d = 2, k even

1

, d = 2, k odd .

1 ϕ(d) 2

, d>2

Theorem 5.1: Φpol,k has dimension d(k). Proof: Since the image of the Hodge component defines a Q–structure l of L

ζ∈µprim. d

k C/(2πi) l lR

+

(see the remark at the end of § 3), which is of dimension

d(k) over lR, the dimension of Φpol,k is at least d(k). Alternatively, we can use

the remark at the end of § 4. In order to see that the dimension is at most d(k), we again have to see that the pairs (N, u) with N = 1 and f = d, the b forming a set of representatives of (ZZ/dZZ)∗, generate Φpol,k , and that h

ib

d

i∗

h

pol(−1)k = (−1)k−1 i− b

d

i∗

pol(−1)k .

The second claim follows from the fact that pol is mapped to −pol under the map t 7→ t−1 : see the remark at the end of § 1.

b such that f f ·N b b0 u 0 0 0 0 , set N := 1 , f := d , b := and u := 0 = . So d= gcd(b, N ) gcd(b, N ) f N −1 Lu0 ,N 0 = i−1 (K ) = i (K ) = L , and we have a commutative diagram 0 1 N u,N u u The proof of the first claim runs along similar lines: for N and u =

38

M Lu,N  Q

 Q

Q

Q

[iu ]

M KN

-

o [ϕN ]

Q

[iu0 ] Q

Q

Q

Q

Q

Q s Q

?

M K1 Here, ϕN is the automorphism of (P0 , X0 ) defined at the end of § 1. So from the identification of [ϕN ]∗ pol(0, i, K1 ) and N ·pol(0, i, KN ) we conclude that [iu ]∗ pol(−1)k = N k−1 ·[iu0 ]∗ pol(−1)k .

q.e.d.

As shown by Kato, Theorems 3.11 and 4.5 then imply Theorem 5.2: If k ≥ 3 is odd, then the Tamagawa number conjecture is true modulo a power of 2 for the motivic pair 



0 V = H 0 ((SpecQ)( l C), l Q(k)) l , D = HDR (SpecQ) l ,

equipped with the Q–structure l Φpol,k for d = 2. Proof: [BK], § 6. Observe the relation ck (1) =

39

2k−1 ck (−1). 1 − 2k−1

q.e.d.

Index of Notations P0

1

ForQl

9

X0

1

9

G0

1

πe1

α0

9

U0

1

α1

9

1

f [π]

10 11 11

H0 π

3

Forhol. M KN (C) l

i

3

ek

3

LN

KN KN

p,q

12

3

H (Log(i, KN ))

12

M L (G0 , H0 )

3

f

13

M KN (P0 , X0 )

3

Λk

13

µKN ,l

4

Lik

14

iu

4

14

Lu,N

4

e0 X

p

14

µprim. d,C l

5

PN

16

pol(0, i, KN )

6

M HQl

20

ϕN ˆ (Lie U0 ) U

7

Cl

28

7

Rl

28

Log

7

Tl

30

7

Ql

30

e

7

Hg1 (Q(µ l d ), Q l l (k))

35

γ

7

Φpol,k

38

µKN ,∞

9

M

q

(C) l

1 N

40

References [B1]

A.A. Beilinson, “Higher regulators and values of L–functions”, Jour. Soviet Math. 30 (1985), pp. 2036–2070.

[B2]

A.A. Beilinson, “Polylogarithm and Cyclotomic Elements”, typewritten preprint, MIT 1989 or 1990.

[B3]

A.A. Beilinson, “Notes on absolute Hodge cohomology”, in S.J. Bloch, R.K. Dennis, E.M. Friedlander, M.R. Stein, “Applications of Algebraic K–Theory to Algebraic Geometry and Number Theory”, Proceedings of the 1983 Boulder Conference, Part I, Contemp. Math. 55 (1986), pp. 35–68.

[BD1]

A.A. Beilinson, P. Deligne, “Motivic Polylogarithm and Zagier Conjecture”, preprint, 1992.

[BD2]

A.A. Beilinson, P. Deligne, “Interpr´etation motivique de la conjecture de Zagier reliant polylogarithmes et r´egulateurs”, in U. Jannsen, S.L. Kleiman, J.–P. Serre, “Motives”, Proc. of Symp. in Pure Math. 55, Part II, AMS 1994, pp. 97–121.

[BL]

A.A. Beilinson, A. Levin, “The Elliptic Polylogarithm”, in U. Jannsen, S.L. Kleiman, J.–P. Serre, “Motives”, Proc. of Symp. in Pure Math. 55, Part II, AMS 1994, pp. 123–190.

[BLp]

A.A. Beilinson, A. Levin, “Elliptic Polylogarithm”, typewritten preliminary version of [BL], preprint, MIT 1992.

[BLpp]

A.A. Beilinson, A. Levin, “Elliptic Polylogarithm”, handwritten preliminary version of [BLp], June 1991.

[BK]

S. Bloch, K. Kato, “L–functions and Tamagawa Numbers of Motives”, in P. Cartier et al., “The Grothendieck Festschrift”, Volume I, Birkh¨auser 1990, pp. 333–400.

[D]

P. Deligne, “Equations Diff´erentielles a` Points Singuliers R´eguliers”, LNM 163, Springer–Verlag 1970.

41

[G]

C. Greither, “Class groups of abelian fields, and the main conjecture”, Ann. de l’Inst. Fourier 42 (1992), pp. 449–499.

[GS]

P. Griffiths, W. Schmid, “Recent developments in Hodge theory: A discussion of techniques and results”, in W.L. Baily, Jr. et al., “Discrete subgroups of Lie groups and applications to moduli”, Proceedings of the 1973 Bombay Colloquium, Oxford Univ. Press 1975, pp. 31–127.

[J1]

´ U. Jannsen, “Continuous Etale Cohomology”, Math. Ann. 280 (1988), pp. 207–245.

[J2]

U. Jannsen, “Mixed Motives and Algebraic K–Theory”, LNM 1400, Springer–Verlag 1990.

[J3]

U. Jannsen, “On the l–adic cohomology of varieties over number fields and its Galois cohomology”, in Y. Ihara, K. Ribet, J.–P. Serre, “Galois Groups over Q”, l Math. Sci. Res. Inst. Publ. 16, Springer–Verlag 1989, pp. 315–360.

[Ka]

M. Kashiwara, “A Study of Variation of Mixed Hodge Structure”, Publ. RIMS, Kyoto Univ. 22 (1986), pp. 991–1024.

[M]

J.S. Milne, “Arithmetic Duality Theorems”, Perspectives in Mathematics 1, Academic Press 1986.

[N]

J. Neukirch, “The Beilinson Conjecture for Algebraic Number Fields”, in M. Rapoport, N. Schappacher, P. Schneider, “Beilinson’s Conjectures on Special Values of L–Functions”, Perspectives in Mathematics 4, Academic Press 1988, pp. 193–247.

[P]

R. Pink, “Arithmetical compactification of Mixed Shimura Varieties”, thesis, Bonner Mathematische Schriften 1989.

[S]

¨ P. Schneider, “Uber gewisse Galoiscohomologiegruppen”, Math. Zeitschrift 168 (1979), pp. 181–205.

[So1]

C. Soul´e, “On higher p–adic regulators”, in E.M. Friedlander, M.R. Stein, “Algebraic K–Theory”, Proceedings of the 1980

42

Evanston Conference, LNM 854, Springer–Verlag 1981, pp. 372– 401. [So2]

C. Soul´e, “El´ements cyclotomiques en K–th´eorie”, in “Journ´ees arithm´etiques de Besan¸con 1985”, Ast´erisque 147/148, Soc. Math. France 1987, pp. 225–257, 344.

[So3]

C. Soul´e, “The rank of ´etale cohomology of varieties over p–adic and number fields”, Comp. Math. 53 (1984), pp. 113–131.

[So4]

C. Soul´e, “K–th´eorie des anneaux d’entiers de corps de nombres et cohomologie ´etale”, Inv. math. 55 (1979), pp. 251–295.

[SZ]

J. Steenbrink, S. Zucker, “Variation of mixed Hodge structure, I”, Inv. math. 80 (1985), pp. 489–542.

[Wa]

L.C. Washington, “Introduction to Cyclotomic Fields”, LNM 83, Springer–Verlag 1982.

[W1]

J. Wildeshaus, “Polylogarithmic Extensions on Mixed Shimura varieties”, Schriftenreihe des Mathematischen Instituts der Universit¨at M¨ unster, 3. Serie, Heft 12, 1994.

[W2]

J. Wildeshaus, “Mixed structures on fundamental groups”, preprint, 1994.

[W3]

J. Wildeshaus, “The canonical construction of mixed sheaves on mixed Shimura varieties”, preprint, 1994.

[W4]

J. Wildeshaus, “Polylogarithmic Extensions on Mixed Shimura varieties. Part I: Construction and basic properties”, preprint, 1994.

43

Polylogarithmic Extensions on Mixed Shimura varieties. Part III: The elliptic polylogarithm

J¨org Wildeshaus Mathematisches Institut der Universit¨at M¨ unster Einsteinstr. 62 D–48149 M¨ unster [email protected]

Introduction The subject of this article is the presentation of the Hodge–de Rham version of the (small) elliptic polylogarithm pol. Our procedure will be analogous to the one in [W5], §§ 1–3: § 1 contains the geometric set–up: we define the Shimura data and varieties underlying our construction and identify the representations of the algebraic groups that will be used. In § 2, we describe the extension of local systems underlying pol (Theorem 2.3). This is a relatively easy matter since it only involves cohomology of the fundamental groups. § 3 starts with a description of the Hodge version of the logarithmic sheaf Log (Proposition 3.3). We then exhibit multivalued functions “trivializing” the monodromy of pol (3.4–3.11). Here, we were lucky enough to find these functions “up to uninteresting terms” in [BL], 4.8. We copied the computation of their monodromy from [BLpp], 4.5 and 4.9. However, to get a precise result (Lemma 3.11), we need to compute the error terms. We end up with a matrix (PNW )−1 , whose inverse PNW identifies the weight–filtered local system underlying pol as a subobject of a C ∞ –bundle with trivial filtration. This matrix should be seen as an analogue of the matrix 

PN =

             



1

0

0

0

0 ...

0

1

0

0

1 Li 2πi 1 N − (2πi) 2 Li2 2 N Li3 (2πi)3

N − 2πi log

1

0

 0 ...  

N − 2πi log

1

0 ...  

N log − 2πi .. .

1 ...    .. .

.. .

1 2! 1 3!



N − 2πi log



N − 2πi log .. .

2 3

1 2!



N − 2πi log .. .

2

  

0 ...   

of [W5], § 3. Its entries Lim,n are therefore called the elliptic higher logarithms, as opposed to the elliptic Debye polylogarithms Λm,n , which occur as entries of the period matrix. We give a complete description of the Hodge version of pol in Theorem 3.14. Contrary to the case of the classical polylogarithm, it involves a second matrix PNF , which identifies the Hodge–filtered holomorphic bundle underlying pol. Since we deal with variations that are not of Tate type, PNF cannot be expected to be, and in fact is not trivial. Norm compatibility is then translated into a distribution property of the elliptic i

higher logarithms (Corollary 3.16). As a by–product, we get a rather extravagant proof of the distribution property of the Bernoulli polynomials. As is remarked at the end of § 3, norm compatibility for Li0,1 already implies the distribution law for the Siegel function, whose precise shape was unknown until recently ([K], § 2). Before getting there, values at Levi sections are discussed at length (3.18–3.36). In [BL], § 2, this is done by calculating what is called their “residues at infinity”, which determines them uniquely. Our approach is more “down to earth” in that we basically specify the functions Lim,n to Levi sections, i.e., modular curves embedded into the scheme of torsion of the universal elliptic curve. They describe one–extensions in the category of so–called variations of Hodge–de Rham structure HDRs , and are given, in explicit form, as modular forms and non– vanishing algebraic functions (Theorems 3.24 and 3.31). Their modular and algebraic properties (Corollaries 3.25 and 3.32) follow automatically from the rigidity of the extension class of pol in Ext1HDRQsl . In order to compare our result to [BL], Proposition 2.2.3, we calculate the residues of our functions (Corollaries 3.26 and 3.34). This also allows to conclude that just as in the classical case (see Remark b) at the end of [W5], § 3) the classes given by evaluating pol at Levi sections actually generate the whole of the group of extensions in question (Theorems 3.29 and 3.36). Still, let us insist that in order to calculate these classes, it is not necessary to compute the whole of pol and then specialize it to Levi sections: as is indicated after Proposition 3.20, we actually have a rigidity principle for the values at Levi sections of pol themselves. In order to identify them, one may first calculate the underlying topological extensions (as in [BL], § 2). Rigidity then allows to make competent guesses about the mixed structure. Concerning variations of Hodge–de Rham structures, let us remark that in retrospect, a more systematic treatment of the theory might seem desirable. Here, it is developed according to our needs (Lemma 3.27, Proposition 3.35, Corollaries 4.3 and 4.4). This reflects the author’s development of ideas: originally, we liked to think of Deligne cohomology as Ext–groups in a category of “algebraic Hodge modules, fixed under complex conjugation”. However, Corollary 3.26 made us realize that this was actually wrong. (See the remark thereafter.) For the time being, we dream of a category of “algebraic Hodge–de Rham modules”, whose smooth objects are the variations of Hodge–de Rham structure. As shown in ii

4.3, on finite schemes over Q, l their Ext1 –groups cannot be distinguished from the Ext1 –groups of the a priori coarser category of Hodge–structures over lR. One last remark: the cup–products of the extensions on modular curves of § 3 give two–extensions, which are closely related to the elements in Deligne cohomology appearing in the proof of Beilinson’s conjecture for modular curves. (For details, see forthcoming work of Scholl.) Rather obviously, these cup–products appear as values of Levi sections of the polylogarithmic (two–)extension associated to the product of the universal elliptic curve E with itself, with the subscheme (0 × E) ∪ (E × 0) removed. We don’t know of any other example of a higher–dimensional polylogarithm with non–trivial values at Levi sections. In § 4, we specialize further to CM –elliptic curves occurring as fibres of the universal elliptic curve of § 3. The main result is Theorem 4.8, which gives a description of the values at torsion points of pol as classes in Ext 1HDRs of an lR

abelian extension of the reflex field K. We conclude by comparing these classes to the ones constructed in [De1]. They differ by a rational constant and hence, by Deninger’s results, are of motivic origin. The main theorems of [De1] and [De2] show that again these classes generate all of Ext 1HDRs . lR

This article is a revised version of § 9 of my doctoral thesis ([W1]). I thank C. Deninger for introducing me to Beilinson’s ideas and initiating my interest in this beautiful area of mathematics – the connection of the elements of § 4 to the classes constructed in [De1] finally confirms his hope that an interpolation process similar to the classical polylogarithm should be possible for these classes as well. Let me once more express my gratitude to A. Beilinson for supplying me with copies of [BLpp] and [BLp]. Some of the results there (like the ones mentioned further above) have regrettably not been included in the final version ([BL]), yet certainly eased my task considerably. Also, the rather formal approach of [BLp], § 2 should be seen as the nucleus of [W4], and provided further understanding of the classical situation. I am obliged to T. Scholl for the invitation to come to Durham in 1994/95, where this article was finally completed. Last, but not least, my hearty thanks go to G. Weckermann for her excellent TEX ing.

iii

§ 1 The Shimura data (P2,a , X2,a ) Throughout this article, the notation is as follows: G2 := CSp2,Ql = GL2,Ql , | 2 V2 ∼ =G a,Q l is the standard representation of G2 , P2,a := V2 × G2 . We think of P2,a as the subgroup of GL3,Ql of matrices of the shape     

1 0 0 a α β b γ δ

    

.

There are pure Shimura data (G2 , H2 ) ([P], 2.7): H2 := Cl \ lR is the disjoint union of the upper half plane and the lower half plane, and G2 (lR) acts as follows:  

α β γ δ

 

∈ G2 (lR) sends τ ∈ H2

to

ατ + β . γτ + δ

h : H2 −→ Hom(S, G2,lR ) is given by sending τ to hτ , where hτ : S −→ G2,lR ,





i  τ z1 − τ z2 −|τ |2 (z1 − z2 )  (z1 , z2 ) 7−→ . 2Im τ z1 − z 2 −τ z1 + τ z2

So on lR–valued points, 

hi (z) = 

Re (z)

Im (z)

−Im (z) Re (z)

 

for z ∈ Cl ∗ .

h is G2 (lR)–equivariant. Lemma 1.1: Let τ ∈ H2 . Then the Hodge structure on V2 induced by hτ is given as follows: W−2 (V2 ) = 0, W−1 (V2 ) = V2 , F 1 (V2,Cl ) = 0, ! τ iCl , F 0 (V2,Cl ) = h 1 F −1 (V2,Cl ) = V2,Cl . Proof: H 0,−1 (V2,Cl ) is the eigenspace belonging to the character (z1 , z2 ) 7−→ z2 . The claim follows from a calculation, which we leave to the reader. 1

q.e.d.

It follows from [P], Proposition 1.7.a) that H2 carries the usual complex structure. (P2,a , X2,a ) is the unipotent extension ([P], Proposition 2.17) of (G2 , H2 ) by V2 . By 1.1, LieV2 is of Hodge type {(0, −1), (−1, 0)}. Therefore, X2,a = {(k, τ ) ∈ Hom(S, P2,a,lR ) × H2 | hτ = πa ◦ k} ,

where we denote by πa the projection P2,a −→ G2 . Lemma 1.2: The following diffeomorphism is P2,a (lR)–equivariant: ∼

lR × lR × H2 −→ X2,a , (r1 , r2 , τ ) 7−→ (kτr1 ,r2 , τ ) , where kτr1 ,r2 : S −→ P2,a,lR is defined by 

1

0 0



1 0 0





  kτr1 ,r2 :=  r1 1 0   0   r2 0 1 0

hτ

    

1

0 0

−r1 1 0 −r2 0 1

    

.

On the left side,     

1 0 0

  

∈ P2,a (lR) a α β   b γ δ

acts by sending (r1 , r2 , τ ) to ατ + β αr1 + βr2 + a, γr1 + δr2 + b, γτ + δ

!

.

Proof: left to the reader.

q.e.d.

However, the complex structure on X2,a turns out not to be compatible with the product decomposition in 1.2: Lemma 1.3: Let V be the standard representation of P2,a ≤ GL3,Ql , xa = (r1 , r2 , τ ) ∈ X2,a . Then the Hodge structure on V induced by kxa is given as follows: 2

W−2 (V) = 0,

0

 

0

 

   

 



W−1 (V) = h 1  ,  0 iQl ,     1 0 W0 (V) = V, F 1 (VCl ) = 0,

0

     

 

1

  

F 0 (VCl ) = h τ  ,  r1 iCl ,     r2 1 F −1 (VCl ) = VCl . 

1

 

  

Proof: A computation shows that  r1  is an eigenvector for the trivial action   r2 of SCl . The rest of the claim is already contained in 1.1. q.e.d. Corollary 1.4: a) The following diffeomorphism is P2,a (lR)–equivariant: ∼

Cl × H2 −→ lR × lR × H2 , ! Re (τ )·Im (z) Im (z) ,− ,τ , (z, τ ) 7−→ Re (z) − Im (τ ) Im (τ ) its inverse being given by (−r2 τ + r1 , τ ) ←−| (r1 , r2 , τ ) . On the left side,     

1 0 0

  

∈ P2,a (lR) a α β   b γ δ

acts by sending (z, τ ) to !

z ατ + β ατ + β (αδ − βγ)· + −b +a , γτ + δ γτ + δ γτ + δ

!

.

b) The composition of the diffeomorphisms in a) and 9.2 gives an isomorphism ∼

Cl × H2 −→ X2,a of complex structures.

3

Proof: a) left to the reader.

 

0

 

  τ  and b) By 1.3, the Hodge filtration step F 0 over (z, τ ) is generated by    1     

1

  

, where r1   r2

  

1

  

r1 := Re (z) −   

0

  



Re (τ )·Im (z) Im (τ )

r2 := −

,

Im (z) . Im (τ )



1

 

 

But  r1  − r2  τ  =  z , so F 0 depends holomorphically on (z, τ ).       r2 1 0 Now apply [P], Proposition 1.7.a).

q.e.d.

We let r1 and r2 denote the functions r1 : Cl × H2 −→ lR,

(z, τ ) 7−→ Re (z) −

r2 : Cl × H2 −→ lR,

(z, τ ) 7−→ −

Re (τ )·Im (z) , Im (τ )

Im (z) . Im (τ )

Also we let cH2 and cCl denote the natural projections Cl × H2 −→ H2

and Cl × H2 −→ Cl .

We fix the following Levi section of πa : ia : G2 −→ P2,a ,  

α β γ δ

 



1 0 0

 

  

7−→  0 α β  .   0 γ δ

c Let L be a neat open compact subgroup of G2 (IAf ) contained in G2 (Z Z), the

subgroup of automorphisms of the standard lattice c V2 ( Z Z) :=

(

!

)

a c ∈ V2 (IAf ) | a, b ∈ Z Z . b

Observe that the inclusion of the upper half plane H2+ −→ H2 induces an isomorphism ∼

G2 (Q) l + \(H2+ × (G2 (IAf )/L)) −→ M L (G2 , H2 )(C) l . 4

Here, G2 (Q) l + is the subgroup of G2 (Q) l of matrices of positive determinant. c By strong approximation, G2 (IAf ) = G2 (Q) l + · G 2 (Z Z). Therefore, we have an

isomorphism

∼

c SL2 (ZZ)\(H2+ × (G2 (Z Z)/L)) −→ M L (G2 , H2 )(C) l ,

and any connected component of M L (C) l is of the shape Γ(gf )\H2+ c for some gf ∈ G2 (Z Z), where we let

Γ(gf ) := SL2 (ZZ) ∩ gf ·L·gf−1 ≤ SL2 (ZZ) . Let N ∈ lN, and define an open compact subgroup c Ka,N := (N ·V2 (Z Z))× L .†

The connected component of M Ka,N (P2,a , X2,a )(C) l mapping to Γ(gf )\H2+ is Λ(pf )\ X+ 2,a , where X+ l × H2+ , 2,a := C

P2,a (Q) l + := {p ∈ P2,a (Q) l | det(p) > 0} , 

1 0 0



 pf :=  0 

0

gf

    

and Λ(pf ) := P2,a (Q) l + ∩ pf ·Ka,N ·p−1 l + . Observe that we have f ≤ P2,a (Q) −1 c pf ·Ka,N ·p−1 f = (N ·V2 (ZZ))× (gf ·L·gf ).

Write M L (G2 , H2 )C0l and M Ka,N (P2,a , X2,a )C0l for the algebraic varieties over Cl underlying the respective connected components. So M L (G2 , H2 )C0l (C) l = Γ\H2+ , where Γ := Γ(gf ) = SL2 (ZZ) ∩ gf ·L·gf−1 is a neat arithmetic subgroup of SL2 (ZZ). The algebraic structure of Γ\H2+ is obtained by the classical process of adding †

By definition, lN is the set of positive integers, and lN0 is the set of non–negative integers.

5

finitely many cusps, thereby embedding Γ\H2+ into a compact Riemann surface, which is the same as the analytic space of C–valued l points of a smooth projective curve over C. l M Ka,N (P2,a , X2,a )C0l (C) l = Λ\ X+ 2,a , where 

1

0 0



 Λ := Λ(pf ) =  N ·ZZ 

Γ

N ·ZZ

    

.

M Ka,N (P2,a , X2,a )C0l is a family of elliptic curves over M L (G2 , H2 )C0l . Observe that the universal covering map prN : X+ → M Ka,N (P2,a , X2,a )C0l (C) l 2,a −→ factors over X+ → (N ·V2 (ZZ))\ X+ 2,a −→ 2,a ,

where the object on the right is a family of one–dimensional tori over H2+ , which by 1.2 is trivial in the C ∞ –category. By [W3], Theorems 4.6.a) and 4.3.b), Conjecture 4.2 of [W3] holds, and hence the image of the l–adic canonical construction functor µKa,N ,l lies in the category of mixed sheaves. We proceed to study Levi sections: the morphism of Shimura data covering 

ia : G2 −→ P2,a ,



α β γ δ

 



1 0 0

 



 7−→   0 α β    0 γ δ

sends τ ∈ H2 to (0, τ ) ∈ X2,a . !

v1 ∈ V2 (Q), l we have the morphism of Shimura data For v = v2 ia,v : (G2 , H2 ) −→ (P2,a , X2,a ) . On the level of groups, it is given by 

ia,v 

α β γ δ

 



= int(v) ◦ ia  6

α β γ δ

 

,

which equals     

1 (1 − α)v1 − βv2 −γv1 + (1 − δ)v2

We therefore have



ia,v (g) = 



0 0

 

. α β   γ δ

1

0 0

(1 − g)v

g

 

.

On the level of homogeneous spaces, ia,v (τ ) = (v1 , v2 , τ ) = (−v2 τ + v1 , τ ) . So the embedding [ia,v ] : M Lv,N (G2 , H2 ) −→ M Ka,N (P2,a , X2,a ) , c where Lv,N := (ia,v )−1 (Ka,N ) = {g ∈ L | (1 − g)v ∈ N ·V2 (Z Z)}, maps M Lv,N into

the scheme of d–torsion of M Ka,N , where d ∈ lN is such that d·v ∈ N ·V2(ZZ).

By [P], Proposition 3.9, there are [det(L) : det(Lv,N )] connected components of L

K

MCl v,N mapping to (MCl a,N )0 . They are indexed by any set of representatives of G2 (Q) l + \(ia,v )−1 (P2,a (Q) l + pf Ka,N )/Lv,N = G2 (Q) l + \(G2 (Q) l + gf L)/Lv,N = Γ\gf L/Lv,N . If gf0 ∈ gf L, then the corresponding component is given by Γ(gf0 ) \ H2+ , where Γ(gf0 ) = SL2 (ZZ) ∩ gf0 ·Lv,N ·(gf0 )−1 . Lemma 1.5: There is a commutative diagram [ia,v ](C) l M Lv,N (C) l  6

6




Γ(gf0 )\H2+

M Ka,N (C) l

-






Λ(pf )\ X+ 2,a

where the lower horizontal map is given by sending the class of τ ∈ H2+ to the class of (−v20 τ + v10 , τ ) ∈ X+ 2,a . c Here, v 0 ∈ V2 (Q) l is chosen such that v 0 − gf0 v ∈ N ·V2 (Z Z).

Proof: left to the reader.

7

q.e.d.

Set W 0 := 0. So with the notations of [W4], § 1, (P 0 , X0 ) coincides with (G2 , H2 ) and the embedding k coincides with ia . Moreover, π 0 and i0 are both equal to the identity on (G2 , H2 ). We have h−1,−1 = h

00 −1,−1

= 0 , h0,−1 = h

00 0,−1

= 1,

hence d = d00 = 1 , N = N 00 = 2 . [ia ](M L (G2 , H2 )C0l ) is the zero section of M Ka,N (P2,a , X2,a )C0l , so the open immersion fKa,N (P , X )0 −→ M Ka,N (P , X )0 jKa,N : M 2,a 2,a C 2,a 2,a C l l

is the inclusion of the complement of the zero section. Because of [W4], § 1, Corollary 2.2 and § 4, we expect a projective system pol(0, ia , Ka,N ) of extensions of relatively unipotent mixed systems of smooth fKa,N (P , X ), uniquely determined by the underlying system of sheaves on M 2,a 2,a

extensions of relatively unipotent smooth topological sheaves ([W4], Theorem

2.3.a)). The restriction to modular curves embedded into some scheme of d– torsion of M Ka,N (P2,a , X2,a ) will give one–extensions of sheaves of finite rank ([W4], § 6). Observe that for N ∈ ZZ\{0} there is an automorphism ϕN of the Shimura data (P2,a , X2,a ), which is trivial on (G2 , H2 ) and which on group level is given by     

1 0 0





 

 

7−→  a α β    b γ δ

1 1 a N 1 b N

0 0

  

. α β   γ δ

∼ ˆ (LieV2 ) −→ ˆ (LieV2 ) given by multiplication by N on U The isomorphism ϕ∗N U

LieV2 identifies [ϕN ]∗ Log(ia , Ka,1 ) and Log(ia , Ka,N ) as well as [ϕN ]∗ pol(0, ia , Ka,1 ) and N ·pol(0, ia , Ka,N ).

8

§ 2 The topological extension underlying pol Let



0 0 0









0 0 0

 



   e1 :=   1 0 0  , e2 :=  0 0 0  ∈ LieV2 .     0 0 0 1 0 0

Then γ1N := exp(N e1 ), γ2N := exp(N e2 ) ∈ V2 (Q) l are the generators of the group of covering transformations of 00

00

X2 + −→ → (N ·V2 (ZZ))\ X2 +

given by (z, τ ) 7−→ (z + N, τ ) and (z, τ ) 7−→ (z − N τ, τ ) respectively. ˆ (LieV2 ) with the ring Q[[e We identify U l 1 , e2 ]] of power series in the commuting variables e1 and e2 . In [W4], § 1, we defined a pro–algebraic action of ˆ (LieV2 ): G2 acts by conjugation, and V2 acts by multiplicaP2,a = V2 × G2 on U tion. Lemma 2.1: Under this action     

1 0 0

  

∈ P2,a (Q) l a α β   b γ δ

l 1 , e2 ]] to the power series maps ek1 el2 ∈ Q[[e (αe1 + γe2 )k (βe1 + δe2 )l exp(ae1 ) exp(be2 ) . Proof: left to the reader.

q.e.d.

ˆ (LieV2 )) and the logarithBy [W3], Theorem 2.1, we may identify µKa,N ,∞ (U mic pro–variation Log(ia , Ka,N ). So the pro–local system ForQl (Log(ia , Ka,N )) is given by the restriction of the above pro–representation to Λ. In particular, γjN maps ek1 el2 to ek1 el2 exp(N ej ), for j = 1, 2.

9

In the notation of [W4], § 1, we have bm (0, i) = 0 for m ≥ 0. For m ≤ −1, we ∗

g] Sym−m µ (V ). So pol(0, i , K ) is the one–extension in have bm (0, i) = [π a L,− 2 a a,N

Ext1Sh(Me Ka,N (P

∗

2,a ,X2,a

g] µ (V ), j ∗ Log(i , K )(1)) ([π a L,− 2 a a,N ))

corresponding to the inclusion µL,− (V2 ) −→ morphism of [W4], Theorem 1.5.b).

Q

l≥1

Syml µL,− (V2 ) under the iso-

By [W4], Theorem 2.3.a), pol(0, ia , Ka,N ) is uniquely determined by the underlyg]–unipotent local systems. We think of it as a pro–local ing one–extension of [π a

system sitting in an exact sequence

∗

g] µ 0 −→ ForQl (j ∗ Log(ia , Ka,N )(1)) → ForQl (pol) −→ ForQl ([π a L,∞ (V2 )) −→ 0 .

Let fKa,N (P , X )0 (C) e 0 −→ pa,N : X →M 2,a 2,a C 2,a l l

be the universal covering morphism. If factors over

fKa,N (P , X )0 (C) →M prN |(lR×lR\N ZZ×N ZZ)×H+ : (lR × lR \ N ZZ × N ZZ) × H2+ −→ 2,a 2,a C l l . 2

e 0 be a base point mapping to (r 0 , r 0 , i) = (−r 0 i + r 0 , i) under Let xe ∈ X 2,a 1 2 2 1

(X00 )0 −→ (lR × lR \ N ZZ × N ZZ) × H2+ ,

such that 0 < r10  1, and −1  r20 < 0. fKa,N (C), We want to describe πe1 := π1 (M l pa,N (xe)). πe1 sits in an exact sequence fKa,N (C) 1 −→ π1 (M l i , pa,N (xe)) −→ πe1 −→ Γ −→ 1 .

In terms of the partial covering prN |(lR×lR\N ZZ×N ZZ)×H+ , we choose generators α1N 2

−1 −1 fKa,N (C) and α2N of π1 (M l i , pa,N (xe )), that are images of prN (α1N ) and prN (α2N )

respectively:

N r r

(r10 , r20 + N )

6 −1 prN (α2N )

0

N r

r -

r

(r10 , r20 )

r

−1 prN (α1N )

10

(r10 + N, r20 )

fKa,N (C) π1 ( M l i , pa,N (xe)) is free in α1N and α2N , and under the inclusion Ka,N

f M i

Ka,N

−→ Mi

,

αjN is mapped to γjN , for j = 1, 2. Let us recall how α1N and α2N act on multivalued holomorphic functions on fKa,N )0 (C): (M C l l

β1 : (z, τ ) 7−→ (z + N, τ )

and β2 : (z, τ ) 7−→ (z − N τ, τ ) are automorphisms of (lR × lR \ N ZZ × N ZZ) × H2+ . In order to define α1N = (β1−1 )∗ and α2N = (β2−1 )∗ on the level of multivalued functions, we have to specify paths connecting (−r20 i+r10 , i) and βj−1 (−r20 i+r10 , i), −1 −1 and we choose these to be prN ((α1N )−1 ) and prN ((α2N )−1 ) respectively.

So αjN (f ) is the multivalued function, whose germ at (−r20 i+r10 , i) is the analytic −1 continuation of the germ of f at (−r20 i + r10 , i) via prN ((αjN )−1 ), pulled back

via βj−1 . We shall frequently write (z, τ ) 7−→ f (z − N, τ ) and (z, τ ) 7−→ f (z + N τ, τ ) for the functions α1N (f ) and α1N (f ), bearing in mind that the choice of α1N and α2N , which is suppressed in this notation really does matter. Similarly, we write (z, τ ) 7−→ f (z, τ − 1) and

z 1 (z, τ ) 7−→ f − , − τ τ for the functions T (f ) and S(f ), where again T and S are understood to consist 



of automorphisms T : (z, τ ) 7−→ (z, τ + 1) ,   z 1 S : (z, τ ) 7−→ ,− τ τ

11

of (lR × lR \ N ZZ × N ZZ) × H2+ as well as paths connecting (−r20 i + r10 , i) and its images under T −1 and S −1 respectively: −1 prN (T −1 ) : [0, 1] −→ (lR × lR \ N ZZ × N ZZ) × H2+ ,

t 7−→ (r10 , r20 , i − t) ,

0

N r

r

(r10 , r20 )  −1  r ) prN (S −1 ) 0 0 (r2 , −r1 ) r

−N r

So instead of regarding πe 1 as a group of homotopy classes of paths we consider

e 0 and the induced action on multivalued holomorphic functions. its action on X 2,a

This viewpoint allows us to slightly enlarge πe 1 so as to include elements like S, fKa,N )0 (C). which don’t have an interpretation as closed paths on (M l C l

Lemma 2.2: a) We have the following relations: T α1N T −1 = α1N , T α2N T −1 = α2N α1N , Sα1N S −1 = α2N , Sα2N S −1 = α2N (α1N )−1 (α2N )−1 , S 2 , (ST )3 ∈ hα1N , α2N i. In particular, there is an exact sequence fKa,N (C) e −→ SL (ZZ)/h−idi −→ 1 , 1 −→ π1 (M l i , pa,N (xe )) −→ Π 1 2 e is the group of automorphisms of X e 0 generated by α , α , T where Π 1 1 2 2,a

and S.

e × b) πe1 = Π 1 SL2 (ZZ)/h−idi Γ.

12

e 0 onto H+ induces a surjection of π e1 onto the set Proof: The projection of X 2 2,a

of automorphisms of H2+ given by the usual action of SL2 (ZZ)/h−idi. Its kernel fKa,N (C) is π1 (M l i , pa,N (xe)). It remains to show the four relations in a). Observe

that if (σ1 , γ1 ) and (σ2 , γ2 ) are pairs consisting of automorphisms σj of

(lR×lR\N ZZ×N ZZ)×H2+ and paths γj joining (−r20 i+r10 , i) and σj−1 (−r20 i+r10 , i), their composition is given by (σ1 σ2 , (σ2−1 γ1 ) ◦ γ2 ). Given this, it is straightforward to check the relations.

q.e.d.

The action of πe 1 on ForQl (j ∗ Log(ia , Ka,N )(1)) is induced by the following action

e : of Π 1

α1N maps 2πi·ek1 el2 to 2πi·ek1 el2 exp(N e1 ) , α2N maps 2πi·ek1 el2 to 2πi·ek1 el2 exp(N e2 ) , T

maps 2πi·ek1 el2 to 2πi·ek1 (e1 + e2 )l ,

S

maps 2πi·ek1 el2 to 2πi·(−e1 )l ek2 .

∗

g] µ N N On ForQl ([π a L,∞ (V2 )), α1 and α2 act trivially, and T and S act in the usual

manner.

So if the canonical base vectors are ε1 and ε2 , T (ε1 ) = ε1 , T (ε2 ) = ε1 + ε2 , S(ε1 ) = ε2 , S(ε2 ) = −ε1 . The stalk at pa,N (xe) of ForQl (pol) is the vector space

E := hε1 , ε2 iQl ⊕ h2πi·ek1 el2 | k, l ∈ lN0 iQl .

e as follows: We define the action of Π 1

α1N :

ε1 7−→ ε1 , e2 exp(N e1 ) exp(N e2 ) − 1 ∞ X N k+l−1 Bl ε2 − ·2πi·ek1 el2 , k! l! k,l=0

ε2 7−→ ε2 − 2πi· =

2πi·ek1 el2 7−→ 2πi·ek1 el2 exp(N e1 ) , for k, l ∈ lN0 , α2N :

e1 exp(N e1 ) exp(N e1 ) − 1 ∞ X N k−1 Bk ·2πi·ek1 , ε1 + (−1)k k! k=0

ε1 7−→ ε1 + 2πi· =

ε2 7−→ ε2 , 2πi·ek1 el2 7−→ 2πi·ek1 el2 exp(N e2 ) , for k, l ∈ lN0 , 13

T :

ε1 7−→ ε1 , ε2 7−→ ε1 + ε2 +2πi·((exp(N e1 ) − 1)(exp(N e2 ) − 1)(exp(N e1 + N e2 ) − 1))−1 · exp(N e1 )(e2 (exp(N e1 + N e2 ) − 1) − (e1 + e2 )(exp(N e2 ) − 1)) ε1 + ε2 −

=

k Bp Bl+k−p+1 N k+l−1 X (−1)p ·2πi·ek1 el2 , l! p! (k − p + 1)! p=0 k,l=0 ∞ X

2πi·ek1 el2 7−→ 2πi·ek1 (e1 + e2 )l , for k, l ∈ lN0 , S:

ε1 7−→ ε2 , e1 exp(N e1 ) exp(N e1 ) − 1 ∞ X N k−1 Bk −ε1 − (−1)k ·2πi·ek1 , k! k=0

ε2 7−→ −ε1 − 2πi· =

2πi·ek1 el2 7−→ 2πi·(−e1 )l ek2 , for k, l ∈ lN0 . e . It is a straightforward matter to check that this really defines an action of Π 1

However, we advise the reader to look at the proof of the next result first. For l ∈ lN0 , define S l to be Syml (µL,∞ (V2 )).

e . Its restriction to π e 1 is the Theorem 2.3: The above defines an action of Π 1

fKa,N )0 (C) restriction to (M l of the pro–local system ForQl (pol) underlying pol. C l

Proof: Set IF2 := hα1N , α2N i.

2 For V ∈ ModQ[IF l 2 ] , RΓ(IF , V) is represented by the complex

V −→ V ⊕ V v 7−→ (α1N − 1)v, (α2N − 1)v) . 2

2 If we let ResIF e (E) be the above object with the induced action of IF , then the Π 1

boundary homomorphism 2

∗

2

IF 1 2 IF ∗ g δ : H 0 (IF2 , ResΠ e ForQl ([πa ] µL,∞ (V2 ))) → H (IF , ResΠ e ForQl (j Log(1))) 1

1

belonging to the exact sequence 2

2

2

1

1

1

∗

IF IF ∗ g 0 −→ ResIF e ForQl ([πa ] µL,∞ (V2 )) −→ 0 e (E) → ResΠ e ForQl (j Log(1)) −→ ResΠ Π

14

maps the classes of the cocycles ε1 and ε2 to the classes of the cocycles e1 0, 2πi· exp(N e1 ) exp(N e1 ) − 1

!

e2 −2πi· exp(N e1 ), 0 exp(N e2 ) − 1

and

!

respectively. The reader may check, using the relations of 2.2, that the action of SL2 (ZZ)/h−idi 2

IF on H 1 (IF2 , ResΠ e 1 ] , is induced by e (V)), for V ∈ ModQ[ l Π 1

T : (v1 , v2 ) 7−→ (T v1 , T v2 − α2N T v1 ) ,

S : (v1 , v2 ) 7−→ ((α1N − 1)(α2N )−1 Sv1 − α1N (α2N )−1 Sv2 , Sv1 ) . A short calculation shows that δ is in fact SL2 (ZZ)/h−idi–equivariant. Since there are no nontrivial IF2 –equivariant homomorphisms ∗

2

2

IF IF ∗ g ResΠ e ForQl (j Log(1)) , e ForQl ([πa ] µL,∞ (V2 )) −→ ResΠ 1

1

it follows from the Hochschild–Serre spectral sequence that the action of IF2 on 2 e . ResIF (E) can be extended uniquely to an action of Π 1 e Π 1

It turns out to be determined already by the following relations: T α1N = α1N T , Sα1N = α2N S and the requirement that it be compatible with the actions on the local systems ∗

g] µ ForQl (j ∗ Log(1)) and ForQl ([π a L,∞ (V2 )).

We leave it to the reader to verify that our definition satisfies these relations. So we have a Q[ l πe 1 ]–module E sitting in an exact sequence ∗

g] µ 0 −→ ForQl (j ∗ Log(1)) −→ E −→ ForQl ([π a L,∞ (V2 )) −→ 0 .

(∗∗)

As in [W4], § 1, we have the diagram

M L (G2 , H2 )C0l  H

H

[ia ]

HH

id

-

HH

M Ka,N (P2,a , X2,a )C0l 

HH j H

[πa ] ?

M L (G2 , H2 )C0l

15

  

j



fKa,N (P , X )0
M 2,a 2,a C l

  g]  [π a



[ia ] being the inclusion of the zero section, and an exact triangle [ia ]∗ [ia ]∗ V(−1)[−2]

−→

V

shift by [1] -

(∗)

. ∗

j∗ j V for mixed, but also for topological smooth sheaves V. We have to look at [πa ]∗ (∗). As in [W4], Theorem 1.3, one proves that g] For (j ∗ Log(1)) = [π a ∗ Q l

Y

ForQl (S m )[0].

m≥1

It follows as in [W4], Theorem 1.5 that already on topological level, the group ∗

g] µ ∗ of one–extensions of ForQl ([π a L,∞ (V2 )) by ForQ l (j Log(1)) is one–dimensional:

the natural morphism

Ext1 e Ka,N (MCl

∗

)0 (C) l

g] µ ∗ (ForQl ([π a L,∞ (V2 )), ForQ l (j (Log(1)))

1 −→ Hom(M L )C0l (C) l (S ), l (ForQ

Q

m≥1

ForQl (S m ))

is an isomorphism. So E is a rational multiple of ForQl (pol). This rational multiple can be detected by looking at the elements of Ext1 e Ka,N (MCl

∗

)0 (C) l

f µ (ForQl ([π] l l (Q(1))) a L,∞ (V2 )), ForQ

coming from E and ForQl (pol) via the map induced by the projection of Log(1) onto Q(1). l Observe that any such extension is characterized by the associated boundary homomorphism g] For (Q(1)) , δ : ForQl (µL,∞ (V2 )) −→ H0 [π a ∗ Q l l

or even by its fibre at i. The right hand side equals H 1 (IF2 , ForQl (Q(1))), l and under our identification the map δ(E) is given by sending ε1 and ε2 to the classes 







of 0, N1 2πi and − N1 2πi, 0 respectively. Now observe that the isomorphism ∼

H 1 (IF2 , ForQl (j ∗ Log(1))) −→

Y

ForQl (S m )i

m≤−1

g] (∗) induces an given by the fibre at i of the boundary map belonging to [π a ∗

isomorphism

∼

κ : H 1 (IF2 , ForQl (Q(1))) l −→ ForQl (µL,∞ (V2 )) . 16

We have to show that δ(E) = κ−1 . We have H 1 (IF2 , ForQl (Q(1))) l = H 1 (ZZ2 , ForQl (Q(1))), l and κ coincides with the boundary map tr

H 1 (ZZ2 , ForQl (Q(1))) l −→ H 2 (ZZ2 , ForQl (µL,∞ (V2 )(1))) −→ ForQl (µL,∞ (V2 )) belonging to the class of the extension ForQl (Log/W−2 Log(1)) in Ext1

Ka,N

(MCl

)0 (C) l

(ForQl (Q(1)), l ForQl (µL,∞ (V2 )(1))) .

We leave it to the reader to check, e.g. via the description of the cup product given in [Br], V, §§ 1–3, that the trace isomorphism maps the element [(0, 2πi)] ∪ [(2πi, 0)] ∈ H 2 (ZZ2 , ForQl (Q(2))) l to 2πi ∈ ForQl (Q(1)) l and hence that κ maps

h

0, N1 2πi

i

and

h

− N1 2πi, 0

to ε1 and ε2 .

i

back

q.e.d.

Remarks: a) We recall Remark d) at the end of § 2 of [W4]: polylogarithmic extensions can be defined for any relative elliptic curve π : E −→ B, in the category of admissible variations of F –Hodge structure, for F a subfield of lR, in the category of lisse ´etale F –sheaves, for F/Q l l finite, and even in the category of mixed systems of smooth sheaves (see [W2], § 2), a fortiori in the category of variations of Hodge–de Rham structure, to be defined in the next paragraph. They satisfy norm compatibility ([W4], § 5) and rigidity ([W4], § 2). More precisely, for Ee := E \i(B), where i denotes the zero section, we have: Ext1Sh(Ee) (πe ∗ V2 , j ∗ Log(1)) −→ Ext1Sh(Ee) (πe ∗ V2 , F (1))

−→ Ext1Sh(Eetop. ) (πe ∗ V2 , F (1)) −→ HomSh(Btop. ) (V2 , V2 )

is injective, where we denote by Sh(Xtop. ) the category of “topological sheaves” on a topological space Xtop. or a scheme over an algebraically closed field (see

17

[W4], § 1). In particular, pol is uniquely determined by the extension 0 −→ F (1) −→ pol≥−2 −→ πe ∗ V2 −→ 0 , and even by the topological extension underlying it. b) For future reference, we compare this extension to the restriction of the extension 0 −→ π ∗ V2 −→ Log≥−1 −→ F (0) −→ 0 e Observe that to E.

pol≥−2 ∈ Ext1Sh(Ee) (πe ∗ V2 , F (1)) = Ext1Sh(Ee) (F (0), πe ∗ V2∨ (1)) The last equality uses the isomorphism

= Ext1Sh(Ee) (F (0), πe ∗ V2 ) .

V2 −→ V2∨ (1) given by sending v to the map w 7−→ v ∪ w. On the level of local systems, this isomorphism can be described, using the basis (ε1 , ε2 ) of Theorem 2.3: it maps ε1 to −2πi · ε∨2 , and ε2 to 2πi · ε∨1 . In the proof of Theorem 2.3, we showed: the classes of pol≥−2 and Log≥−1 in Ext1Sh(Eetop. ) (F (0), πe ∗ V2 ) coincide.

e Indeed, we may reduce We claim that this holds already in the category Sh(E).

to the universal case considered before, where we have Ext 1Sh(B) (F (0), V2 ) = 0

([BL], Lemma 1.6.1). This means that the map Ext1Sh(Ee) (F (0), πe ∗ V2 ) −→ Ext1Sh(Eetop. ) (F (0), πe ∗ V2 )

−→ HomSh(Btop. ) (F (0), V2 ⊗F V2∨ )

is injective, and hence, that the first map, induced by the forgetful functor is injective. Similarly, one shows that the class of Log≥−1 in Ext1Sh(E) (F (0), π ∗ V2 ) equals the class [∆], where we let [ ] denote the Abel–Jacobi map. For any elliptic curve π, it maps the Mordell–Weil group E(B) to Ext1Sh(B) (F (0), V2 ). ∆ is considered as a section of the base change of E over itself. Finally, the logarithmic sheaf is the projective limit of the Symn (Log≥−1 ), n ∈ lN. This can be seen e.g. via the universal property ([W2], Theorem 3.5) 18

of Log, applied to the relatively unipotent sheaves Symn (Log≥−1 ), and the fact that due to the commutativity of the fundamental group of an elliptic curve, the statement certainly holds on topological level. Altogether, we arrive at a description of all the successive one–extensions occurring in pol. The next paragraph will be concerned with a description of the full mixed structure on pol in the Hodge–de Rham context.

§ 3 The Hodge–de Rham version of pol ˆ (LieV2 )) = Log(ia , Ka,N ). To achieve We start by giving a description of µa,K,∞(U this, what we did in [W5], § 3 was to write down an isomorphism between the bifiltered holomorphic vector bundles underlying the variations of Hodge structure Log(ia , Ka,N ) and

Q

n∈lN0 [πa ]

∗

Symn µL,∞ (V2 ) respectively. This amounted

to the same as to give a trivialization of the bifiltered vector bundle underlying Log(ia , Ka,N ). In our situation, such an isomorphism exists only on the C ∞ –level. So let Fordiff.Ka,N (MCl ∞

)0 (C) l

(Log(ia , Ka,N )) and Fordiff.Ka,N (MCl

)0 (C) l

(

Q

n∈lN0 [πa ]

the C –vector bundles underlying Log(ia , Ka,N ) and and let ek,l := ek1 el2 ∈ Symk+l (V2 ).

Q

∗

Symn µL,∞ (V2 )) be

n∈lN0 [πa ]

∗

Symn µL,∞ (V2 ),

Lemma 3.1: There is an isomorphism Y

n∈lN0

Fordiff.Ka,N (MCl

∼

)0 (C) l

([πa ]∗ Symn µL,∞ (V2 )) −→ Fordiff.Ka,N (MCl

)0 (C) l

(Log(ia , Ka,N ))

given by sending ek,l to the multivalued section ek1 el2 exp(r1 e1 ) exp(r2 e2 ) . The proof makes use of the monodromy of the functions r1 and r2 , which we note explicitly: Lemma 3.2: Let



100



 

 p= a α β  ∈ Λ . 

bγδ



Then r1 ◦ p = αr1 + βr2 + a , 19

r2 ◦ p = γr1 + δr2 + b . Proof: This follows directly from 1.2.

q.e.d.

In terms of the basis (ek,l | k ∈ lN0 ), ek1 el2 = ek,l exp(−r1 e1 ) exp(−r2 e2 ) . −1 We view prN ForQl (Log(ia , Ka,N )) as the pro–local system over Q l sitting in−1 side prN Fordiff.Ka,N (MCl

)0 (C) l

(

valued function

Q

n∈lN0 [πa ]



1    −r1  LW N :=

  −r2   1 2  r  2! 1  r r  1 2   1 r2  2! 2  .

∗

Symn µL,∞ (V2 )) described by the pro–matrix 

0

0 0 0 0 . . .  0 0 0 0 . . . 

1 0

1 0 0 0

−r1 0 1 0 0 −r2 −r1 0 1 0 0 −r2 0 0 1 .. .. .. .. .. . . . . .

..

e0,0 e1,0

 . . .   . . .   . . .   . . .  

e0,1 e2,0 e1,1 e0,2 .. .

We chose the ordering of the basis of multivalued sections of the bundle −1 prN Fordiff.Ka,N (MCl

)0 (C) l

(

Y

[πa ]∗ Symn µL,∞ (V2 ))

n∈lN0

as indicated on the right of LW N. From Lemma 2.1, it follows how the Hodge filtration of looks like:

Q

n∈lN0 [πa ]

Hp,q = 0 if p or q > 0, and −1 p,q Hp,q is of rank one, a global generator of prN H being

hp,q : (z, τ ) 7−→ (τ e1 + e2 )−q (τ e1 + e2 )−p if p, q ≤ 0.

20

∗

Symn µL,∞ (V2 )

We also want to describe a set of holomorphic generators of F p : F p = 0 if p > 0, and −1 p F p is of infinite rank, a basis of global generators of prN F being 0

(fp0 ,q : (z, τ ) 7−→ (τ e1 + e2 )−q e2−p | p ≤ p0 ≤ 0, q ≤ 0) if p ≤ 0. We are now in a position to describe the weight and Hodge filtrations of Log(ia , Ka,N ). Denote by Hp,q (Log(ia , Ka,N )) the C ∞ –subbundle, on whose fibres the Deligne torus acts via multiplication by z1−p z2−q . As remarked in [W3], § 1, this yields the unique decomposition of the C ∞ –bundle underlying Log, which satisfies Hq,p = Hp,q mod

M

0

0

Hp ,q .

p0

0} . Note that because of our choice of xe, all the maps (z, τ ) 7−→ qτj qz , j ≥ 0 and (z, τ ) 7−→ qτj /qz , j ≥ 1 send xe to a point in this domain.

23

e for N = 1. The proofs We need to study the behaviour of the Λm,n under Π 1

of 3.5, 3.7 and 3.8, up to the calculation of the rational constants in 3.8, were

already performed in [BLpp], 4.5 and 4.9. Remember that γ ∈ πe 1 sends a multivalued function f to f ◦ γ −1 .

First, we need a combinatorial result: Lemma 3.4: Let l, m ∈ lN0 , and define Al,m := l!

l X

Bm+k+1 1 , k=0 m + k + 1 k!(l − k)! m X

Bm−i+l+1 (−1)i Bl,m := m! . i=0 m − i + l + 1 i!(m − i)! Then Al,m − Bl,m = (−1)

m

!

m! l! δl,0 − . (m + l + 1)!

Proof: The defining identity ∞ X t Bk k = t exp(t) − 1 k=0 k!

shows that Al,m =

Bl,m =

d dt

!l

d dt

!m 

d exp(t)· dt exp(−t)·

!m

d dt

!l

1 1 − exp(t) − 1 t

!!



t=0

,

!

1 1  . − t=0 exp(t) − 1 t

The product formula for differentiation implies Al+1,m = Al,m + Al,m+1 , Bl,m+1 = −Bl,m + Bl+1,m . So we have Al,m+1 − Bl,m+1 = −(Al,m − Bl,m ) + (Al+1,m − Bl+1,m ) , and we may use induction on m. For m = 0, the claim is checked directly.

24

q.e.d.

Lemma 3.5: Let m ∈ lN0 and n ∈ lN. a)

α11 (Λm,n )

=

n−1 X i=0

b)

α21 (Λm,n )

(−1)i Bm+1 Λm,n−i − δn,1 . i! (m + 1)!

n X 1 (−1)i m−1 l Λm−i,n +(−1) cCn−l = l c H2 i! i=0 l=0 (n − l)!(m + l + 1)! B n . + (−1)m+n n!(m + 1)! m X

Proof: We prove b), leaving a) to the reader. α21 (Λm,n ) −

m X (−1)i

i!

i=0

Λm−i,n



∞  ∞ X m   X 1 X m m−i m i m−i m+n+1 j Λn (qH2 /qCl ) = (−1) j Λ (q q ) + (−1) n H C l 2 i m! j=1 i=0 i j=0

−

m X i=0

+

Bm+k+1 (cCl + cH2 )n−k ckH2 Bm+1 Bn + (−1)n+1 (n − k)!k! m + 1 n! k=0 m + k + 1 m X i=0

1 m!

!

n X

− =





∞ ∞ X X m (−1)i  j m−i Λn (qH2 qCl ) + (−1)m−i+n+1 j m−i Λn (qH2 /qCl ) i j=1 j=0

n X cCn−k ckH2 Bm−i+k+1 m Bm−i+1 Bn l (−1)i + (−1)n+1 i m − i + 1 n! k=0 m − i + k + 1 (n − k)!k!

!

!!

(−1)m+n+1 Λn (1/qCl ) − (−1)m Λn (qCl ) +

n X

n Bm+k+1 1 X 1 l cCn−l l c H2 m + k + 1 k! (n − l)!(l − k)! l=k k=0

m l X cCn−l m Bm−i+l+1 l c H2 (−1)i − m−i+l+1 l=0 (n − l)!l! i=0 i n X

+(−1)

=

1 m!

!

n+1

m Bm−i+1 Bn Bm+1 Bn X m (−1)n+i+1 − m + 1 n! m − i + 1 n! i=0 i

!

!

(−1)m+n+1 Λn (1/qCl ) − (−1)m Λn (qCl ) cn−l cl Bn + (Al,m − Bl,m ) Cl H2 + (−1)n+1 (A0,m − B0,m ) . (n − l)!l! n! l=0 n X

!

So the claim follows from 3.4 and the equality (−1)m+n+1 Λn (1/qCl ) − (−1)m Λn (qCl ) = (−1)m+1 which is proven in 3.6.

1 n (cCl − (−1)n Bn ) n! q.e.d.

25

Lemma 3.6: For t ∈ lP1 (C)\{0, l 1, ∞} and n ∈ lN, we have the identity 1 Λn (t) + (−1) Λn (t ) = n! n

log t 2πi

−1

!n

n

− (−1) Bn

!

.

Here, the multivalued function t 7−→ Λ(t−1 ) takes the value Λ(t−1 ) at t ∈ {s ∈ Cl | |s| < 1, Re (s) > 0, Im (s) > 0} defined by joining t and t−1 with a path not meeting {s ∈ Cl | |s| ≥ 1 , Im (s) ≥ 0} . Proof: The claim follows from the identity Lik (t) + (−1)k Lik (t−1 ) = −

k X

(log(t))k−j (2πi)j Bj (k − j)!j! j=0

which in turn is proven by induction on k. The constant in the induction step is calculated by forming the limit t −→ 1.

q.e.d.

Lemma 3.7: Let m ∈ lN0 and n ∈ lN. Then (n − 1)dΛm,n = (m + 1)cH2 dΛm+1,n−1 + cCl dΛm,n−1 . Proof: This is a direct computation using the formula (k − 1) dΛk =

log dΛk−1 . 2πi

q.e.d.

This lemma will translate into Griffiths transversality for the pro–variation, that we intend to define. However, we already need it to compute the effect of S on the Λm,n : Lemma 3.8: Let m ∈ lN0 and n ∈ lN. n 1 X m+i 1 Bm+i+1 Bn−i Λm+i,n−i + (−1)n a) T (Λm,n ) = (−1) . m m! i=1 i! m + i + 1 (n − i)! i=0 n−1 X

i

!

1 cCm+n+1 l b) S(Λm,n ) = (−1) Λn−1,m+1 + (−1) (m + n + 1)! cnH2 Bm+1 Bn +(−1)n−1 . (m + 1)! n! m

m+n+1

Proof: We prove b), leaving a) to the reader. Let tm,n := S(Λm,n ) + (−1)m+1 Λn−1,m+1 + (−1)m+n 26

cCm+n+1 1 l . (m + n + 1)! cnH2

First we show by induction on (m + n) that tm,n is constant: let m = 0, n = 1. Λ0,1 : (z, τ ) 7−→

∞ X

Λ1 (qτj qz )

+

j=0

f := exp(2πiΛ0,1 ) : (z, τ ) 7−→

∞ X

1 1 1 Λ1 (qτj /qz ) − z + τ + . 2 12 4 j=1

−1 iqτ qz 2 (1 1 12

− qz )

∞ Y

(1 − qτj qz )(1 − qτj /qz )

j=1

coincides, up to a multiplicative constant, with the same noted function from [L], § 19. By [L], § 19, Theorem 1, we have πicC2l S(f )/f = ζ ·exp c H2

!

for some ζ ∈ Cl ∗tors. .

More precisely, [L], § 19, Theorem 1 shows that such a relation holds for some e , which a priori differs from S by some element of the commutator of S0 ∈ Π 1

hα11 , α21 i. By Lemma 3.5, any such element maps Λ0,1 to Λ0,1 +t for some constant t. So the above claim holds for S itself, too. This proves that t0,1 is a constant. So let (m + n) ≥ 2. Λm,n and tm,n are regular at cCl = 0. We show that dtm,n is invariant under α1 and α2 : by 2.2.a), we have α21 S = Sα11 , so

α21 (tm,n ) − tm,n = Sα11 (Λm,n ) + (−1)m+1 α21 (Λn−1,m+1 ) + (−1)m+n −S(Λm,n ) − (−1)

=

n−1 X i=1

m+1

n−1 X i=1

+(−1)

m+n+1

+(−1)n+1

i=1

cCm+n+1 l (m + n + 1)!cnH2

(−1)i Λn−i−1,m+1 i!

m+1 X l=0

n−1 X

Λn−1,m+1 − (−1)

m+n

(−1)i Bm+1 S(Λm,n−i ) − δn,1 i! m+1

+(−1)

=

m+1

(cCl + cH2 )m+n+1 (m + n + 1)!cnH2

cCm+1−l clH2 l (m + 1 − l)!(n + l)!

m+n+1 X cCm+n+1−l cl−n Bm+1 l H2 + (−1)m+n (m + 1)!n! l!(m + n + 1 − l)! l=1

Bm+1 Bm+1 (−1)i tm,n−i − δn,1 + (−1)n+1 . i! (m + 1)! (m + 1)!n! 27

(∗)

By our induction hypothesis, dtm,n is invariant under α21 . The claim for α11 is proven by using the relation α11 S = S(α11 )−1 (α21 )−1 α11 . We leave the calculation to the reader. Because of the regularity of dtm,n at cCl = 0, it is necessarily of the shape (z, τ ) 7−→ am,n (τ )dτ + bm,n (τ )dz . When calculating dtm,n , we may set cCl = 0. By 3.7, we have dΛm,n |cCl =0 =

(m + n − 1)! n−1 c dΛm+n−1,1 |cCl =0 . (n − 1)! m! H2

(∗∗)

Using the explicit formula for Λk,1 , k ≥ 1, we see that   

0,k |z=0 =  1 Bk+2  Ek+2 (τ ) , k k! k + 2  !   1 Bk+1 Ek+1 (τ ) , k ∂ Λk,1 (z, τ ) |z=0 = k! k + 1  ∂z  0,k

∂ Λk,1 (z, τ ) ∂τ

!

odd even

,

≥ 3 odd

.

even

Here, the Ek are the usual normalized Eisenstein series of respective weight k. When checking the above formulae, the reader will note that at one point it is necessary to change the order of summation in a certain double sum. For k = 1, the second formula does not hold because we only have conditional convergence for the double sum calculating ∂ Λ1,1 (z, τ ) ∂z

!

|z=0 .

For m + n ≥ 3, we are done since the modular properties of the Ek together with (∗∗) imply that dtm,n = 0. So let m + n = 2, i.e., (m, n) ∈ {(1, 1), (0, 2)}. By the first formula, we know at least that am,n = 0. (∗∗) gives the relation dΛ0,2 |cCl =0 = cH2 dΛ1,1 |cCl =0 . Because of dt0,2 = SdΛ0,2 |cCl =0 − dΛ1,1 |cCl =0 28

and dt1,1 = SdΛ1,1 |cCl =0 + dΛ0,2 |cCl =0 , we get dt0,2 = −

1 c H2

i.e., b0,2 = −

1 c H2

dt1,1 ,

b1,1 .

On the other hand, ∂ ∂ ∂ ∂ bm,n (z, τ ) = tm,n (z, τ ) = am,n (z, τ ) = 0 , ∂τ ∂τ ∂z ∂z so both b1,1 and b0,2 are constants. Now that we know that tm,n is constant we use (∗) to calculate its value: n−1 X i=1

(−1)i Bm+1 Bm+1 tm,n−i = δn,1 + (−1)n . i! (m + 1)! (m + 1)!n!

We use induction on n: for n = 2, we get Bm+1 Bm+1 B1 =− . 2(m + 1)! (m + 1)! 1!

−tm,1 =

Let n ≥ 3, and assume that we already showed that tm,n−i = (−1)n−i−1

Bm+1 Bn−i (m + 1)! (n − i)!

for i = 2, . . . , n − 1 .

By the above formula, we have Bm+1 −tm,n−1 = (m + 1)!

X (−1)n−1 Bn−i (−1)n n−1 − n! i! (n − i)! i=2

!

n X 1 Bn−i Bm+1 Bn−1 = (−1) − (m + 1)! i=1 i! (n − i)! (n − 1)! n

|

= (−1)n−1

{z

=0

!

}

Bm+1 Bn−1 . (m + 1)! (n − 1)!

Here, the last sum is zero because it is the coefficient of tn in (exp(t) − 1)·

t = t. exp(t) − 1 q.e.d. 29

Since we chose to describe our multivalued functions in terms of the partial covering prN |(lR×lR\N ZZ×N ZZ)×H+2 , it is necessary to introduce functions ΛN m,n for any N . Let ∼

[ϕN ] : (lR × lR \ N ZZ × N ZZ) × H2+ −→ (lR × lR \ ZZ × ZZ) × H2+ be multiplication by N −1 . Definition: For m ∈ lN0 and n ∈ lN, we define + ◦ ΛN l. m,n := Λm,n [ϕN ] : (lR × lR \ N ZZ × N ZZ) × H2 −→ C

The multivalued holomorphic functions ΛN m,n will turn up as entries of the period matrix Ωpol,N of the elliptic polylogarithm. As remarked before, the matrix PNW comparing the rational structures of the trivial extension and of pol cannot be expected to contain only holomorphic functions. As in [W5], § 3, we look for multivalued functions “trivializing” the action of e given in 2.3. Π 1

First we define the multivalued functions “of level 1”: Definition: Let k, l ∈ lN0 . 1 Bl+1 i) Q0,l r2l+1 − . 1 := (l + 1)! (l + 1)! For k ≥ 1, k X (−1)i Bk Bl+1 k,l Q1 := −Λl,k + r1k−i r2l+i+1 ciH2 + (−1)k+1 . k! (l + 1)! i=0 (k − i)!(l + i + 1)! ii) Qk,0 2 := −

Bk+1 1 r1k+1 + (−1)k+1 . (k + 1)! (k + 1)!

For l ≥ 1, Qk,l 2 := −Λl−1,k+1 −

k X i=0

(−1)i r k−i r l+i+1 ci+1 H2 . (k − i)!(l + i + 1)! 1 2

e In order to study the behaviour of the Qk,l 2 under Π1 for N = 1, we again need

a combinatorial result:

Lemma 3.9: Let p, q, r ∈ lN0 , and define Cp,q,r Then Cp,q,r

p X

i+r := (−1)i r i=0

!

p+q = . p 30

!

!

p+q+r+1 . p−i

Proof: If p = 0, then the claim is trivial. For r = 0, one shows that Cp+1,q−1,0 = −Cp,q,0 +

(p + q + 1)! . (p + 1)!q!

Using C0,p+q,0 = 1, the claim for r = 0 follows by induction on p. Finally, one uses induction on (p + r) and the formula Cp,q,r = Cp,q+1,r−1 − Cp−1,q+1,r . q.e.d. Lemma 3.10: Let k, l ∈ lN0 , and N = 1. a) α1N (Qk,l 1 ) = α2N (Qk,l 1 )

=

k X i=0

(−1)i k−i,l Q1 , i!

l X

(−1)i k,l−i 1 Bk Q1 + (−1)k+l+1 , i! l! k!

i=0

T (Qk,l 1 )

k X

!

l + i k−i,l+i Q1 , = (−1) l i=0 i

l l,k S(Qk,l 1 ) = (−1) Q2 .

b) α1N (Qk,l 2 ) = α2N (Qk,l 2 ) =

k X i=0

(−1)i k−i,l Bl + δk,0 , Q2 i! l!

l X

(−1)i k,l−i Q2 , i!

i=0

T (Qk,l 2 )

k X

k 1X l+i Bi Bk+l+1−i (Q1k−i,l+i +Q2k−i,l+i )+(−1)k = (−1)i , l l! i=0 i! (k + 1 − i)! i=0

!

l+1 l,k S(Qk,l Q1 + δk,0 2 ) = (−1)

Bl . l!

Proof: As the reader might have guessed, the proof is a long and tedious computation using the previous lemmata. As an example, we derive the formula for T (Qk,l 1 ), where k ≥ 1: T (Qk,l 1 ) = −T (Λl,k ) +

k X i=0

+(−1)k+1

(−1)i (r1 − r2 )k−i r2l+1+i (cH2 − 1)i (k − i)!(l + 1 + i)!

Bk Bl+1 k! (l + 1)! 31

k−1 X

k 1 Bl+i+1 Bk−i l+i 1X = (−1) (−Λl+i,k−i ) + (−1)k+1 l! i=0 i! l + i + 1 (k − i)! l i=0

!

i

i k−i X i (−1)q (−1)i X k−i−q l+1+i+q (−1)i−r crH2 r1 r2 + r=0 r i=0 (l + 1 + i)! q=0 q!(k − i − q)! k X

!

k−1 X

k l + i k−i,l+i 1 Bl+i+1 Bk−i 1X = (−1) Q1 + (−1)k+1 l l! i=0 i! l + i + 1 (k − i)! i=0

!

i

k−1 X

!

Bl+i+1 l+i Bk−i − (−1) (−1)k+1−i (k − i)! (l + i + 1)! l i=0 i

p p−r X j+r (−1)j (−1)p k−p l+p+1 X r r1 r2 c H2 + r (p − j − r)!(l + j + r + 1)! p=0 (k − p)! r=0 j=0 k X

−

k X

!

(−1)

q=0

+(−1) k X

q

X l + q k−q (−1)r r1k−q−r r2l+q+r+1 crH2 (k − q − r)!(l + q + r + 1)! l r=0 !

!

1 k+l r2k+l+1 l (k + l + 1)!

k

!

l + i k−i,l+i = (−1) Q1 l i=0 i

p−r p X X (−1)p 1 k−p l+p+1 r Cp−r,l,r + c H2 r1 r2 p=0 (k − p)! (l + p + 1)! r=0 j=0 k X

−

p X (−1)p (l + p − r)! r 1 r1k−pr2l+p+1 c H2 . r=0 l!(p − r)! p=0 (k − p)! (l + p + 1)! k X

Our claim follows from 3.9.

q.e.d.

Definition: Let k, l ∈ lN0 , j ∈ {1, 2}. ◦ [ϕN ]. QN,k,l := Qk,l j j

More explicitly, we have: = QN,0,l 1

1 Bl+1 l+1 r − , 2 N l+1 (l + 1)! (l + 1)!

QN,k,l = −ΛN 1 l,k +

1 N k+l+1

+(−1)k+1

k X i=0

(−1)i r k−i r l+i+1 ciH2 (k − i)!(l + i + 1)! 1 2

Bk Bl+1 , k! (l + 1)! 32

for k ≥ 1 ,

QN,k,0 =− 2

1 k+1 k+1 Bk+1 r + (−1) , 1 N k+1 (k + 1)! (k + 1)!

QN,k,l = −ΛN 2 l−1,k+1 −

1 N k+l+1

k X i=0

(−1)i r k−i r l+i+1 ci+1 H2 , (k − i)!(l + i + 1)! 1 2

for l ≥ 1 .

As follows from the shape of the formulae, 3.10 holds for the QN,k,l and arbitrary j N. By abuse of notation, we allow ourselves to write S n for the pullback of S n via g]. Recall the basis of multivalued sections (2πi·e | k, l ∈ lN ) of [π a k,l 0

Fordiff.Ka,N (MCl

)0 (C) l

(Log(ia , Ka,N )(1)).

Lemma 3.11: This basis can be completed to give an isomorphism Fordiff. e Ka,N (MCl

)0 (C) l

(S 1 ) ×

Y

n∈lN0

Fordiff. e Ka,N (MCl

by sending ε01

∞ X

to f1 := ε1 +

∼

)0 (C) l

(S n (1)) −→ Fordiff. e Ka,N (MCl

)0 (C) l

pol

·2πi·ek1 el2 N k+l−1 QN,k,l 1

k,l=0

and ε02

to f2 := ε2 +

∞ X

·2πi·ek1 el2 . N k+l−1 QN,k,l 2

k,l=0 ∗

g] µ Here, ε01 and ε02 denote the usual multivalued sections of S 1 = [π a L,∞ (V2 ).

Proof: We have to show that f1 and f2 are invariant under α1N and α2N and that T (f1 ) = f1 ,

S(f1 ) = f2 ,

T (f2 ) = f1 + f2 , S(f2 ) = −f1 . Let us show, for example, the invariance of f2 under α1N : α1N (f2 ) = α1N (ε2 ) +

∞ X

N k+l−1 α1N (QN,k,l )·α1N (2πi·ek1 el2 ) 2

k,l=0

= ε2 − 2πi·

+

∞ X

k,l=0

e2 exp(N e1 ) exp(N e2 ) − 1

N k+l−1

k X i=0

(−1)i N,k−i,l Bl Q2 + δk,0 ·2πi·ek1 el2 exp(N e1 ) i! l! !

33

= ε2 −

+

∞ X N k+l−1 Bl N k+l−1 Bl ·2πi·ek1 el2 + ·2πi·ek1 el2 k!l! k!l! k,l=0 k,l=0 ∞ X

∞ X

N k+l−1

k,l=0

= ε2 +

∞ X

N

k X

∞ X (−1)i N,k−i,l 1 j k+j l Q2 ·2πi· N e1 e2 i! i=0 j=0 j! k X

k+l−1

i=0

k,l=0

= ε2 +

∞ X

N

k−i (−1)i X 1 N,k−i−j,l Q ·2πi·ek1 el2 i! j=0 j! 2

k k−p X X

(−1)k−p−j QN,p,l ·2πi·ek1 el2 2 p=0 j=0 (k − p − j)!j!

k+l−1

k,l=0

|

= ε2 +

∞ X

{z

}

=δp,k

N k+l−1 QN,k,l ·2πi·ek1 el2 = f2 . 2

k,l=0

q.e.d. So ForQl pol(0, ia , Ka,N ) is the pro–local system over Q l sitting inside the C ∞ –pro–bundle Fordiff. e Ka,N (MCl

)0 (C) l

(S 1 ) ×

Y

n∈lN0

Fordiff. e Ka,N (MCl

)0 (C) l

(S n (1))

given by the inverse PNW of the pro–matrix valued function 

(PNW )−1 =

1

   0   1 N,0,0  Q1  N   QN,1,0 1    QN,0,1  1   N QN,2,0  1  .

0 1

0

0

0

1 QN,0,0 1 N 2 QN,1,0 r1 2 N,0,1 Q2 r2 N,2,0 1 2 r N Q2 2! 1

..

.. .

.. .

0

0

0

0

0

0

0

0

1

0

0

0

1

0

r1 .. .

0 .. .

1 .. .



f1

 . . .   . . .   . . .   . . .   . . .  

f2

. . .

2πi·1 2πi·e1 2πi·e2 2πi·e21 .. .

So if (ε01 , ε02 , 2πi·ek,l | k, l ∈ lN0 ) is the canonical basis of the C ∞ –pro–bundle, then PNW (ε01 ), PNW (ε02 ) and all PNW (2πi·ek,l ) are multivalued sections of the pro–local system ForQl (pol). For the computation of the entries of PNW , another combinatorial observation will turn out to be useful: 34

Lemma 3.12: Let p, q, r ∈ lN0 , and define Dp,q,r :=

r X i=0

p i

!

!

q . r−i

!

m := 0 for m < n. Here, we define n ! p+q Then we have Dp,q,r = . r Proof: Dp,q,r is the coefficient of xr in (x + y)p · (x + y)q = (x + y)p+q .

q.e.d.

Definition: i) For m, n ∈ lN0 , we define the (m, n)–th elliptic higher logarithm to be 

n 1 X 1 1 n−p Lim,n := cH 2 p m! p=1 (n − p)! (−2πi)



·

∞ X

(j − r2 )

m+n−p

j Lip (qH ql ) 2 C

+ (−1)

m+n−1

j=0

∞ X

(j + r2 )

m+n−p

j=1

Bm+n+1 (−r2 ) 1 n c + m + n + 1 n! H2

!



j Lip (qH /qCl ) 2

.

Here, Bk (X) denotes the k–th Bernoulli polynomial: Bk (X) =

Pk

q=0

!

k Bk−q X q . q

Also, we define Li−1,n to be zero. ii) For k, l ∈ lN0 , let R1k,l := Lil,k . iii) For k, l ∈ lN0 let R2k,l := Lil−1,k+1 + (−1)k

1 Bk+1 (r1 )Bl (−r2 ) . (k + 1)! l!

iv) For k, l ∈ lN0 and j ∈ {1, 2}, let RjN,k,l := Rjk,l ◦ [ϕN ] . Lemma 3.13: Denote by (ε01 , ε02 , 2πi·ek,l | k, l ∈ lN0 ) the canonical basis of the C ∞ –pro–bundle Fordiff. (S 1 ) × e Ka,N (C) M l PNW (εj ) = ε0j +

∞ X

Q

n∈lN0

Fordiff. (S n (1)). Then we have e Ka,N (C) M l

N k+l−1 RjN,k,l · 2πi·ek,l

k,l=0

35

for j = 1, 2 .

So PNW is the pro–matrix valued function 

1

   0   1 N,0,0  R1 N   RN,1,0 1    RN,0,1  1   N RN,2,0  1  .

0

0

0

0

0

1

0

0

0

0

0

0

0

1 N,0,0 R 1 N 2 N,1,0 R2 −r1 R2N,0,1 −r2 N R2N,2,0 2!1 r12

..

.. .

.. .

1

0

0

0

1

0

−r1 .. .

0 .. .

1 .. .

. . .



ε01

. . . 

ε02

. . . 

2πi·e0,0

 

 . . .   . . .   . . .  

2πi·e1,0 2πi·e0,1 2πi·e2,0 .. .

Proof: By 3.1 and 3.11, we have to show the equality −

k X l X

(−1)p+q k+l−p−q−1 p q N,k−p,l−q N,k,l N r1 r2 Q m = N k+l−1 Rm p! q! p=0 q=0

for m = 1, 2 .

Because of its structure, we may suppose N = 1. First, we carry out the computation for the terms Tp,q :=

k−p X i=0

(−1)i r1k−p−ir2l−q+i+1 ciH2 (k − p − i)!(l − q + i + 1)!

of Q1k−p,l−q : k X l X

(−1)p+q p q r r Tp,q p! q! 1 2 p=0 q=0

−

X (−1)p+q k−p (−1)i =− r1k−i r2l+i+1 ciH2 p! q! i=0 (k − p − i)!(l − q + i + 1)! p=0 q=0 k X (−1)i =− r1k−i r2l+i+1 ciH2 δk,i (−1)l Ci,l,0 (k − i)!(l + i + 1)! i=0 ! k + l (−r2 )k+l+1 k c . = 3.9 l (k + l + 1)! H2 k X l X

Similarly, let Sp,q := −Λl−q,k−p + (−1)k−p+1

Bk−p Bl−p+1 , (k − p)! (l − q + 1)!

where we define Λl−q,0 to be zero. Then −

k X l X

(−1)p+q p q r r Sp,q p! q! 1 2 p=0 q=0

(1)

∞ X (−1)p+q 1 j p q j l−q Λk−p (qH ql ) = r1 r2 2 C p! q! (l − q)! p=0 q=0 j=0 k−1 l XX

36

(2)

+(−1)k+l+1

∞ X 1 1 j r1p r2q j l−q Λk−p (qH /qCl ) 2 p! q! (l − q)! p=0 q=0 j=1

k−1 l XX

k−p X Bl−q+r+1 cCk−p−r crH2 (−1)p+q 1 p q l + r1 r2 . p! q! (l − q)! p=0 q=0 r=0 l − q + r + 1 (k − p − r)! r! k X l X

(3) (4)

For fixed j, term (2) contributes j k X X Lin (qH q l ) ((j − r2 )cH2 + r1 )k−p−n (−1)p p k−p 1 (−1)q q l−q 2 C r j r q! (l − q)! 2 p! 1 n=1 (−2πi)n (k − p − n)! p=0 q=0 l X

=

j k X X (−1)p Lin (qH q l ) k−n 1 1 2 C (j − r2 )l r1p ((j − r2 )cH2 + r1 )k−p−n n l! (−2πi) p! (k − p − n)! n=1 p=0

=

k 1X 1 1 j k−n (j − r2 )k+l−n cH Lin (qH ql ) . 2 2 C l! n=1 (k − n)! (−2πi)n

Similarly, term (3) contributes (−1)k+l−1

k X

1 1 j k−n (j + r2 )k+l−n cH Lin (qH /qCl ) 2 2 n (k − n)! (2πi) n=1

for fixed j. Finally, (4) equals k−p r+s X Bl−q+r+1 1 k−p−r X r1k−r−s (−r2 )q+s cH (−1)p 1 1 2 p! q! (l − q)! l − q + r + 1 r! s!(k − p − r − s)! p=0 q=0 r=0 s=0 k X l X

=

k X

r1k−R cR H2

p=0

R=0

=

=

3.12

ckH2

k−R X

k+l X

|

R l X X (−1)p Bl+R−q−t+1 (−r2 )q+t 1 p!(k − R − p)! q=0 q!(l − q)! t=0 l + R − q − t + 1 t!(R − t)!

{z

=δR,k

}

T X Bk+l+1−T l T 1 (−r2 ) k! l! q=0 q T =0 k + l + 1 − T

|

!

=Dl,k,T

k+l (−r2 )p k+l X Bk+l+1−p ckH2 . l p! (k + l + 1 − p)! p=0

!

{z

k T −q

! }

The same calculations, with k replaced by k + 1 and l by l − 1, show our claim for m = 2. We leave the details to the reader.

q.e.d.

We want to extend the mixed structure Log(ia , Ka,N )(1) described in 3.3 to the whole of ForQl (pol). The bundles (F 0 /F 0 ∩ W−2 )(pol) and (F −1 /(F 0 + F −1 ∩ W−2 ))(pol) must be of rank one. 37

Let ϕ0,−1 := cH2 ε01 + ε02 . It generates a C ∞ –vector bundle of rank one. Actually, ϕ0,−1 is holomorphic with respect to the structure given by (ε1 , ε2 , 2πi·ek1 el2 | k, l ∈ lN0 ): ϕ0,−1 = cH2 ε1 + ε2 +

∞ X

N k+l−1 (cH2 QN,k,l + QN,k,l )·2πi·ek1 el2 , 1 2

k,l=0

and N k+1 cH2 QN,k,l + QN,k,l = −cH2 ΛN 1 2 l,k − Λl−1,k+1 + (−1)

Bk Bl+1 cH , k! (l + 1)! 2

where we define ΛN m,0 := 0 and

1

Bn . n! In the above identity, we used the definition of the QN,k,l and the relation j ΛN −1,n :=

−

cn N n n! Cl

− (−1)n

k X 1 1 (−1)i r1k+1 + r1k−i (r2 cH2 )i+1 = − cCk+1 . l (k + 1)! (k − i)!(i + 1)! (k + 1)! i=0

We intend to let ϕ0,−1 be a section of F 0 (pol). In order to define F −1 (pol), we need to exhibit another multivalued holomorphic section. First we write down another formula for ε02 : ε02 = ε2 +

∞ X

N k+l−1 QN,k,l ·2πi·ek1 el2 2

∞ X

k l N k+l−1 ΛN l−1,k+1 ·2πi·e1 e2

k,l=0

= ε2 −

k,l=0

+

∞ X

N k−1

k=0

−

∞ ∞ X X

N k+l−1

k=0 l=1

= ε2 −

∞ X

1 (cCk+1 − r1k+1 )·2πi·ek1 l k+1 N (k + 1)! 1 N k+l+1

k X

(−1)i k l r1k−i r2l+i+1 ci+1 H2 ·2πi·e1 e2 i=0 (k − 1)!(l + i + 1)!

k l N k+l−1 ΛN l−1,k+1 ·2πi·e1 e2

k,l=0

−

∞ X k (−1)i 1 X k l r k−1 r l+i+1 ci+1 H2 ·2πi·e1 e2 . N 2 k,l=0 i=0 (k − i)!(l + i + 1)! 1 2

38

Now recall the basis of multivalued sections (2πi·fp0 ,q | p + 1 ≤ p0 ≤ 0, q ≤ 0) of F p (

Q

n∈lN0 [πa ]

∗

Symn µL,∞ (V2 )(1)).

Let ϕ−1,0 := ε02 −

∞ 1 X (−r2 )j+1 cH ·2πi·f0,−j . N 2 j=0 (j + 1)! 2 ∞ X

(−r2 )k ·2πi·f0,q−k , for q ≤ 0, generate a k! k=0 C ∞ –pro–vector bundle. Again, ϕ−1,0 is holomorphic: ϕ0,−1 , ϕ−1,0 and the 2πi·ψ0,q =

2πi·f0,−j = 2πi·(cH2 e1 + e2 )j exp(r1 e1 ) exp(r2 e2 ) =

j X i=0

∞ ∞ j X 1 X 1 p q i q+j−i r1 r2 cH2 ·2πi·ep+i , 1 e2 i p=0 p! q=0 q!

!

so ∞ X

(−r2 )j+1 cH2 ·2πi·f0,−j j=0 (j + 1)! j ∞ ∞ X 1 1 X 1 p q+j+1 i+1 (−1)j+1 X q+j−i r1 r2 cH2 ·2πi·ep+i = 1 e2 j=0 j + 1 i=0 i!(j − i)! p=0 p! q=0 q! ∞ X

=−

∞ X k X

(−1)i k l Cl,0,i r1k−i r2l+i+1 ci+1 H2 ·2πi·e1 e2 (k − i)!(l + i + 1)!

∞ X k X

(−1)i k l r k−i r l+i+1 ci+1 H2 ·2πi·e1 e2 , (k − i)!(l + i + 1)! 1 2

k,l=0 i=0

=−

k,l=0 i=0

where we used 3.9. So we get the formula ϕ−1,0 = ε2 +

∞ X

k l N k+l−1 (−ΛN l−1,k+1 )·2πi·e1 e2 .

k,l=0

Theorem 3.14: If we let ϕ0,−1 be a section of F 0 and ϕ−1,0 a section of F −1 , then these data define an admissible pro–variation of Hodge structure fKa,N )0 . It coincides with the restriction to (M fKa,N )0 of pol. on (M C l C l

Proof: We have defined the underlying local system, the Hodge filtration by holomorphic sub–vector bundles and the weight filtration by rational sub–local

systems. Since the weight and Hodge filtrations induce variations of Hodge structure on W−2 and on the quotient by W−2 , [GS], Observation 1.16 tells us 39

that we only need to check Griffiths transversality and admissibility. The formulae ϕ0,−1 = cH2 ε1 + ε2 +

∞ X

N

k+l−1

−cH2 ΛN l,k

−

ΛN l−1,k+1

+ (−1)

k+1 Bk

Bl+1 cH k! (l + 1)! 2

k,l=0

·2πi·ek1 el2 , ϕ−1,0 = ε2 +

X

k l N k+l−1 (−ΛN l−1,k+1 ) · 2πi·e1 e2 ,

k,l=0

2πi·ψp,q = 2πi(cH2 e1 + e2 )−q e−p l e1 ) 2 exp(cC =

∞ X

k=0

−p−q X

l=−p k+l≥−p−q

(−q)! (k + l + p + q)!(−l − p − q)!(l + p)! −l−p−q k+l+p+q cCl · 2πi·ek1 el2 ·cH 2

give the entries of the period matrix Ω of our object. So 

Ω=

c H2

0

    1     1 1 1  c H2 + − 2 cCl +  N 2N 2N    1 2 1 1  N cCl + cH2 +  −cH2 Λ0,1 − 2  2N 4 12    1  −ΛN cH 0,1 −  12 2    1 3 N  N cCl + cH2  −N cH2 Λ0,2 − 2  6N 24    N  −N cH ΛN − N ΛN − cH 2 1,1 0,2  24 2     −N ΛN 1,1   

1 1 1 cCl + 2 N 2N 1 1 − 2 cC2l + 2N 12 −

.. .

−ΛN 0,1 −

1 3 c 6N 2 Cl

−N ΛN 0,2 −N ΛN 1,1 .. .

···

   · · ·     · · ·     · · ·      · · ·     · · ·      · · ·     · · ·   

From 3.7, we conclude that N (n − 1)dΛN m,n = (m + 1)cH2 dΛm+1,n−1 +

40



1 cCl dΛN m,n−1 . N

.

!

By induction, we get dΛN m,n

X 1 n−1 (m + j)! = cjH2 cCn−j−1 dΛN l m+j,1 . n−j−1 m! j=0 N j!(n − j − 1)!

So we have 1 1 d ϕ0,−1 (z, τ ) − ϕ0,−1 (z, τ ) + ϕ−1,0 (z, τ ) dτ τ τ ∞ X

=

N

k+l−1

k,l=0 ∞ X ∞ X

=

N

k+l−1

k=1 l=0

+

N

∞ X ∞ X

k N l+j−1 (l + j − 1)! j k−j d N 1X τ z Λ (z, τ ) · 2πi·ek1 el2 l! j=1 (j − 1)!(k − j)! dτ l+j−1,1

∞ X ∞ X

k X 1 N l+j−1 (l + j − 1)! j k−j d N τ z Λ (z, τ ) · 2πi·ek1 el2 (l − 1)! j=0 j!(k − j)! dτ l+j−1,1

∞ X ∞ X

k N l+j−1 (l + j)! j k−j d N 1X τ z Λl+j−1,1 (z, τ ) · 2πi·ek1 el2 l! j=0 j!(k − j)! dτ

k=0 l=0

−

k=0 l=1

=−

k=0 l=0

=−

!

d k l −τ ΛN l,k (z, τ ) · 2πi·e1 e2 dτ

∞ X ∞ X

k+l−1

k=0 l=1

=−

!

d N d Λ (z, τ ) · 2πi·ek1 el2 −τ ΛN l,k (z, τ ) − dτ dτ l−1,k+1

−∞ X

N −q−1

q=−1

!

d (z, τ ) · 2πi·ek1 el2 − ΛN dτ l−1,k+1

d N Λ (z, τ ) · 2πi·ψ0,q (z, τ ) . dτ −q−1,1

(∗)

We also have d 1 1 2πi·ψp,q (z, τ ) = · 2πi·ψp,q−1(z, τ ) − · 2πi·ψp−1,q (z, τ ) dz τ τ for p, q ≤ 0. It follows that 0 is the only multivalued section g of the holomorphic bundle hol. ForM (j ∗ Log(ia , Ka,N )(1)) satisfying e Ka,N (C) l

a) g is invariant under T .

b) 5(g) | d ∈ F −1 . dz

Namely, if we let g=

X

gp,q · 2πi·ψp,q ,

p,q≤0

41

then we get X d d g(z, τ ) = gp,q (z, τ ) · 2πi·ψp,q (z, τ )+ dz p,q≤0 dz

1 + gp,q (z, τ ) · (2πi·ψp,q−1(z, τ ) − 2πi·ψp−1,q (z, τ )) . τ 

So condition b) means that d 1 gp,q (z, τ ) = (gp+1,q (z, τ ) − gp,q+1 (z, τ )) dz τ for p ≤ −1 and q ≤ 0. On the other hand, as a short computation shows, the invariance of g under T means that T gp,q (z, τ ) = (−τ )

−p

−q X

r=0

!

r−p 1 gp−r,q+r (z, τ ) r (1 − τ )r−p

for p, q ≤ 0. Now the commutation rule

d d = T dz dz together with the last two formulae yields T

−q X

!

r−p−1 (−1)r−p−1 gp−r+1,q+r = 0 r−1

r=1

for p ≤ −1 and q ≤ 0. By induction on q, we get gp,q = 0 for p ≤ −1 and q ≤ 0 . Since we have d 1 g−1,q (z, τ ) = (g0,q (z, τ ) − g−1,q+1 (z, τ )) dz τ for q ≤ 0, we finally get g = 0. It follows that there is at most one multivalued section ϕ of our vector bundle ∗

hol. g] µ mapping to cH2 ε1 + ε2 under the projection to ForM ([π a L,∞ (V2 )) sate Ka,N (C) l isfying i) ϕ is invariant under T .

ii) 5(ϕ) | d ∈ F −1 . dz

42

Furthermore, because F 0 of our variation is in any case mapped isomorphically ∗

g] (µ 0 to F 0 ([π a L,∞ (V2 ))), this ϕ would necessarily be a section of F .

By Lemma 3.11, the section ϕ0,−1 satisfies i). A calculation entirely analogous to (∗) yields the formula −∞ X d d N −q−1 ΛN ϕ0,−1 (z, τ ) = − (z, τ ) · 2πi·ψ0,q (z, τ ) . dz dz −q−1,1 q=−1

Hence ϕ0,−1 satisfies ii), and so our choice of F 0 is the only one, that allows Griffiths transversality to hold. From (∗), we deduce that ϕ−1,0 must then be a section of F −1 . This shows that ours is the only extension of the given data possibly satisfying Griffiths transversality. Since we know that pol exists and defines an admissible extension, the claim is proven.

q.e.d.

Theorem 3.14 tells us that the matrix Ω described in its proof is the period matrix Ωpol,N of pol. We have Ωpol,N = (PNW )−1 Ωtriv. PNF , where 

c  H2 Ωtriv. =

            

0

0

0

0

1

1

0

0

0

0

0

1

0

0

0

0

0

c H2

0

0 .. .

0 .. .

0 .. .

1 .. .

1 .. .

. . .



ε01

. . . 

ε02

. . . 

2πi·e0,0

 

 . . .   . . .  

2πi·e1,0 2πi·e0,1 .. .

is the period matrix of the trivial extension, 

1  PNF

=

0

 0 1   1 0 r c  N 2 2 H2  1 0 − r2 c  2N 2 2 H2  0 0  . .

..

..

0

0

0

0

0

0

1

0

0

−r2

1

0

0 .. .

0 .. .

1 .. .

compares the holomorphic structures, and 43



. . . . . .  

. . .  

 . . .   . . .  

2πi·f0,0 2πi·f0,−1 2πi·f−1,0 .. .



PNW

=

1

   0   1 N,0,0  R1 N   RN,1,0 1    RN,0,1  1  .

..

0

0

0

0

1

0

0

0

1 N,0,0 R N 2 R2N,1,0 R2N,0,1

1

0

0

.. .

−r1

1

0

−r2 .. .

0 .. .

1 .. .

. . .



ε01

. . . 

ε02

. . . 

2πi·e0,0

 

 . . .   . . .  

2πi·e1,0 2πi·e0,1 .. .

compares the rational structures. Remark: The following observation will be useful when evaluating pol at Levi sections: as follows from the definition of the sections ϕ0,−1 and ϕ−1,0 , the bifiltered C ∞ –pro–bundle underlying pol is the trivial, i.e., diagonally bifiltered pro–bundle Fordiff. (S 1 ) × e Ka,N (C) M l

Y

n∈lN0

Fordiff. (S n (1)). e Ka,N (C) M l

Next, we spell out what norm compatibility ([W4], Theorem 5.2) means in our situation: let N, M ∈ lN. The morphism [·1]Ka,N M ,Ka,N : M Ka,N M (P2,a , X2,a ) −→ M Ka,N (P2,a , X2,a ) is part of the following commutative diagram: [ϕM ] M Ka,N M −→ M Ka,N ∼

. [ϕ−1 M]

[·1] & M Ka,N

−1 Here, [ϕ−1 M ] = [ϕM ]Ka,N ,Ka,N is multiplication by M on the family of elliptic

curves M Ka,N over M L . We have the extension pol(0, i, Ka,N M ) |M Ka,N M \[·1]−1 (0) in C l

Ext1

Ka,N M

Sh(MCl

\[·1]−1 (0))

([πa ]∗ µL,∞ (V2 ) |M Ka,N M \[·1]−1 (0) , Log(1) |M Ka,M N \[·1]−1 (0) ) , C l

K

and its restriction to (MCl a,N M \ [·1]−1 (0))0 .

44

C l

Proposition 3.15: Under the norm map NKa,N M ,Ka,N (compare [W4], Theorem 5.2), this extension is mapped to the one described by PNF and the matrix–valued function, which sends (z, τ ) to 

1

   0   1 P N M,0,0  (y, τ )  N MX ∗ R1  N M,1,0  R1 (y, τ )   ∗  X N M,0,1  R1 (y, τ )   ∗  .

1 NM

X

∗ X ∗

..

Here, the sum K

P

∗

P

∗

0

0

0

0

1

0

0

0

1

0

0

−r1 (z, τ )

1

0

R2N M,0,0 (y, τ )

R2N M,1,0 (y, τ ) R2N M,0,1 (y, τ )

−r2 (z, τ )

0

1

.. .

.. .

.. .

.. .



. . . . . .  

. . .  

. . . .  

  . . .   

runs over the following set of representatives of points of K

f a,N M )0 (C) f a,N )0 (C) (M l mapping to pa,N (z, τ ) ∈ (M l under [·1]: C l C l

{(z + N bτ + N a, τ ) | 0 ≤ a, b ≤ M − 1} ⊂ X+ 2,a . For 0 ≤ a, b ≤ M − 1, i ∈ {1, 2} and k, l ∈ lN0 , the multivalued function (z, τ ) 7−→

RiN M,k,l (z

+ N bτ + N a, τ ) =

Rik,l

bτ + a z + ,τ NM M

!

is defined by letting the multivalued functions (z, τ ) 7−→ Lim,n

z bτ + a + ,τ NM M

!

for m, n ∈ lN0

denote the branches e 0 −→ C X l 2,a

taking the values given by the usual power series expression for the Lip , p ≥ 1 near xe . Note that because of our choice of xe, all the maps (z, τ ) 7−→ qτj q

z NM

+a + bτM

, j≥0

and (z, τ ) 7−→ qτj /q

z NM

send xe to the domain |s| < 1.

45

+a + bτM

, j≥1

Proof: The statement on the shape of the matrix comparing the rational structures is straightforward. So we have to show that the holomorphic structure on the C ∞ –pro–bundle Fordiff. e Ka,N (MCl

)0 (C) l

(S 1 ) ×

Y

n∈lN0

Fordiff. e Ka,N (MCl

)0 (C) l

(S n (1))

is still given by PNF . A priori, this structure is given by the 2πi·ψp,q , p, q ≤ 0 and the two sections ϕ0,−1 and ϕ0−1,0

:=

ε02

M −1 ∞ X X (−1)j+1 1 (r2 − N b)j+1 · 2πi·f0,−j − c H2 M 2 (N M ) j=0 (j + 1)! b=0

= ϕ−1,0 −

∞ j+1 −1 X 1 (−r2 )j+1−k M X 1 X (N b)k cH2 · 2πi·f0,−j N 2 M j=0 k=1 k! (j + 1 − k)! b=0

= ϕ−1,0 −

−1 X ∞ ∞ X (N b)k 1 MX (−r2 )q c · 2πi·f0,1−k−q H 2 N 2 M b=0 k=1 k! q! q=0

= ϕ−1,0 −

∞ −1 N k MX 1 X bk cH2 · 2πi·ψ0,1−k . N 2 M k=1 k! b=0

So ϕ0−1,0 and ϕ−1,0 define the same holomorphic structure.

q.e.d.

Corollary 3.16: (Distribution property.) For m, n ∈ lN0 and M ∈ lN, we have the following equality of multivalued funcfKa,1 )0 (C): tions on (M l C l

M m+n−1

X

Lim,n (Q) = Lim,n (P ) .

[M ]Q=P

More precisely, if we let Lim,n Tm,n :=

!

m+n Sm+n cn , where = Tm,n + (m + n + 1)! H2 m

n 1 X 1 1 n−p cH 2 p m! p=1 (n − p)! (−2πi)



·

∞ X

(j − r2 )

m+n−p

j ql ) Lip (qH 2 C

+ (−1)

m+n−1

∞ X

(j + r2 )

j=1

j=0

and Sk := Bk+1 (−r2 ) for k ≥ 0 , then the above formula holds for Tm,n and Sk separately: M m+n−1

X

Tm,n (Q) = Tm,n (P ) ,

[M ]Q=P

M k−1

X

Sk (Q) = Sk (P ) .

[M ]Q=P

46

m+n−p



j Lip (qH /qCl ) 2

Here the sum runs over the following set of representatives of points of K

K

f a,1 )0 (C) f a,1 )0 (C) (M l mapping to P = pa,1 (z, τ ) ∈ (M l under [M ]: C l C l 

1 (z + bτ + a), τ M





| 0 ≤ a, b ≤ M − 1 ⊂ X+ 2,a .

The multivalued functions on the left hand sides of the distribution equations are defined as in 3.15. Proof: Of course, the distribution property of the r–th Bernoulli polynomial M

r−1

M −1 X

X +b M

Br

b=0

!

= Br (X)

is well known and easy to prove by using the equation ∞ X tetX tr = B (X) . r et − 1 r=0 r!

However, at least for r ≥ 2, this will also follow directly from norm compatibility: by [W4], Theorem 5.2, the matrix appearing in 3.15, for N = 1, and P1W describe the same one–extension of variations of Hodge structure. By 3.15, the underlying bifiltered holomorphic subbundles coincide as subobjects of Fordiff. e Ka,1 (MCl

)0 (C) l

(S 1 ) ×

Y

n∈lN0

Fordiff. e Ka,1 (MCl

)0 (C) l

(S n (1)) .

The isomorphism between the two extensions induces the respective identities on the subobject j ∗ Log(ia , Ka,1 )(1) ⊂ object S 1 ⊂ Fordiff. e Ka,1 (MCl

)0 (C) l

Q

n∈lN0

Fordiff. e Ka,1 (MCl

)0 (C) l

(S n (1)) and the quotient

(S 1 ). It respects the Hodge filtration and hence must

map ϕ0,−1 to itself. Furthermore, it respects the rational structures and hence

maps εj to εj +

∞ X

qj,k,l · 2πi·ek1 el2

k,l=0

for j = 1, 2 and rational numbers qj,k,l . But since ϕ0,−1 = cH2 ε1 +ε2 +

∞ X

−cH2 Λl,k − Λl−1,k+1 + (−1)

k,l=0

!

Bl+1 cH ·2πi·ek1 el2 k! (l + 1)! 2

k+1 Bk

is mapped to itself, so is cH2 ε1 + ε2 and hence all qj,k,l are zero. So our isomorphism is the identity. The section, whose coordinates with respect to the basis (ε01 , ε02 , 2πi·ek,l | k, l ∈ lN0 ) are given by the first column of the matrix in 3.15, for N = 1, is therefore rational. 47

So for m, n ∈ lN0 , there are rational numbers qk,l such that 

M m+n−1

X

[M ]Q=P



Tm,n (Q) − Tm,n (P ) 

!

m + n  m+n−1 X M Sm+n (Q) − Sm+n (P ) cH2 (P )n + m [M ]Q=P =

∞ X

qk,l r1 (P )k r2 (P )l

k,l=0

K

f a,1 )0 (C) for all P ∈ (M l . C l

For the calculation of the qk,l , we may form the limit cH2 −→ i∞. Then the term with the Tm,n vanishes. If n > 0, then the term involving Sm+n converges if and only if it is identical to zero. Finally, a straightforward conputation shows our claim for S0 = B1 (−r2 ) = −r2 − 21 .

q.e.d.

So the functions R1k,l and R2k,l , for k, l ∈ lN0 , also satisfy distribution relations. For the classical higher logarithms Lik , we have the relations (−1)k−1 Lik (t−1 ) = Lik (t) +

k X

(log(t))k−r (2πi)r Br (k − r)! r! r=0

(compare the proof of 3.6). Here, the multivalued function t 7−→ Lik (t−1 ) takes the value Lik (t−1 ) at t ∈ {s ∈ Cl | |s| < 1 , Re (s) > 0 , Im (s) > 0} defined by joining t and t−1 with a path not meeting {s ∈ Cl | |s| ≥ 1 , Im (s) ≥ 0} . For the sake of completeness, we note the corresponding relation between Lim,n and the multivalued function P 7−→ Lim,n (−P ) , which takes the value Lim,n (−z, τ ) at (z, τ ) near (−r20 i + r10 , i) defined by joining (z, τ ) = (r1 , r2 , τ ) and (−z, τ ) = (−r1 , −r2 , τ ) with a small arc not meeting {(z, τ ) ∈ X+ 2 | Re z > 0 , Im z < 0}:

48

(−r1 , −r2 ) -• '

•0

• (r1 , r2 )

&

%

Lemma 3.17: For m, n ∈ lN0 , we have (−1)m+n−1 Lim,n (−P ) = Lim,n (P ) . Proof: It is straightforward to check, using the relation for Lip (t−1 ) and Lip (t), that (−1)m+n−1 Lim,n (−P ) 1 m+n n r c H2 m! n! 2 1 1 +(−1)m+n+1 (Bm+n+1 (r2 ) + (−1)m+n Bm+n+1 (−r2 ))cnH2 . m! n! m + n + 1

= Lim,n (P ) + (−1)m+n+1

Since

−te(−t)(−X) tetX − = −tetX , we have et − 1 e−t − 1 Bm+n+1 (X) + (−1)m+n Bm+n+1 (−X) = −(m + n + 1)X m+n . q.e.d.

Remark: Contrary to what we did in [W5], § 3, we described our multivalued functions in the parameters cH2 and cCl of the universal covering space X+ 2,a of K

(MCl a,N )0 . The reader should treat this parametrization with due caution as the map K

a,N 0 X+ ) 2,a −→ (MC l

c really depends on the choice of the element gf ∈ G2 (Z Z) representing the conK

nected component of MCl a,N .

49

It remains to study values at Levi sections, i.e., modular curves embedded in a scheme of d–torsion of M Ka,N . bi with coprime integers bi , fi , and define d to be the Let v ∈ V2 (Q), l write vi = fi smallest common multiple of d1 and d2 , where

di :=

fi · N . gcd(bi , N )

Recall the embedding ia,v : (G2 , H2 ) −→ (P2,a , X2,a ) , which on group level is given by 

ia,v : G2 −→ P2 : g 7−→ 

1



0

(1 − g)v

g

0  ,

c Lv,N = i−1 a,v (Ka,N ) = {g ∈ L | (1 − g)v ∈ N ·V2 (ZZ)},

and [ia,v ] : M Lv,N (G2 , H2 ) −→ M Ka,N (P2,a , X2,a ). A description of the map [ia,v ](C) l of complex manifolds was given in 1.5. We formulate the splitting principle ([W4], Proposition 6.1): ˆ (LieV2 ) splits canonically into a direct product Lemma 3.18: i∗a,v U ˆ (LieV2 ) = i∗a,v U

Y

Symn (V2 ) ,

n∈lN0

the base vector of Symk+l (V2 ) mapping to ek1 el2 under the natural projection ˆ (LieV2 )) −→ Symk+l (V2 ) i∗a,v W−(k+l) (U being given by ek1 el2 exp(v1 e1 ) exp(v2 e2 ) . Proof: This is a direct calculation using 2.1.

q.e.d.

As in [W4], § 6, we assume that v ∈ / Ka,N ∩ V2 (IAf ), i.e., that N does not divide bi or that fi is not equal to 1 for at least one i ∈ {1, 2}. So d > 1, and [ia,v ] fKa,N . factors through M

50

[ia,v ]C∗l pol is an element of Y

k∈lN0

Ext1

Lv,N

M HMQl (MCl

)

(S 1 , S k (1)) =

Y

Ext1

k∈lN0

=

Y

L

M HMQl (MCl v,N )

Ext1

L

M HMQl (MCl v,N )

k∈lN

(Q(0), l (S 1 )∨ ⊗Ql S k (1))

(Q(0), l S k−1 (1)) .

Here, we have used Proposition 3.19: a) For n ≥ 0, there is a canonical epimorphism of G2 –modules ϕn : V2∨ ⊗Ql Symn V2 −→ → Symn−1 V2 given by derivation: e∨j ⊗ f (e1 , e2 ) 7−→

∂ 1 f (e1 , e2 ) , j = 1, 2 . · n + 1 ∂ej

Here, we define Sym−1 V2 to be zero. The kernel of ϕn , which is a direct summand of V2∨ ⊗Ql Symn V2 , is isomorphic to (Symn+1 V2 )(−1), where Q(1) l denotes the determinant representation of G2 . b) For n ≥ 0, the Yoneda–Ext group Ext1

L

M HMQl (MCl v,N )

Proof: We leave a) to the reader.

(Q(0), l S n+1 ) is trivial.

For b), we use the Leray spectral sequence to see that our claim follows from the fact that there are no non–trivial morphisms of Hodge structures from Q(0) l to H 1 := H 1 (M Lv,N (C), l ForQl (S n+1 )). This in turn follows from the next result. q.e.d. Proposition 3.20: Let n ∈ lN0 , and m ∈ ZZ. Then there is an exact sequence of mixed Q–Hodge l structures δ 0 −→ H!1 −→ H 1 −→ H 0 (C(C), l Vn,m ) −→ Hc2 −→ 0 , where we define C to be the scheme of cusps of M Lv,N , W := ForQl (S n (m)), H!1 := im (Hc1 (M Lv,N (C), l W) −→ H 1 ), H 1 := H 1 (M Lv,N (C), l W), Hc2 := Hc2 (M Lv,N (C), l W). So Hc2 = 0 unless n = 0. H!1 is of Hodge type {(−m + 1, −n − m), (−n − m, −m + 1)}. Also, Vn,m is defined to be the local system on C(C) l of coinvariants under the local monodromy of W. It is uncanonically isomorphic to ForQl (Q(m l − 1)). 51

Proof: The sequence is induced by the exact cohomology sequence associated to p∗ of the exact triangle j! W

−→

j∗ W

[1] -

. i∗ i∗ j ∗ W

in the derived category of algebraic mixed Hodge modules, where p denotes the L

L

structural morphism of the smooth compactification (MCl v,N )− of MCl v,N , and i L

and j denote the complementary inclusions of CCl and MCl v,N respectively. By [S], 5.3.10, the Hodge structure on H!1 coincides with that of [Z], § 12, which shows the claim concerning the Hodge type. It remains to show that the “variation of Hodge structure” on C(C), l whose underlying perverse sheaf is (H 0 j∗ W) |C(C) l is none other than Q(m l − 1). Observe that up to a twist by 1, the exact sequence is dual to the exact sequence for the same n, and m replaced by −(n + m). So we may instead show that the Hodge structure on (H−1 j∗ W) |C(C) l + m). We may suppose m = 0. Since l is Q(n the invariants of the local monodromy groups are clearly one–dimensional, all we have to show is that the weight is −2n. But this follows from the definition of the weight filtration associated to the local monodromy ([SZ], §§ 2–3). It coincides with the weight filtration given by Saito’s formalism; compare the discussion on page 125 of [BZ].

q.e.d.

Via the Leray spectral sequence and 3.20, we may consider the k–th component of [ia,v ]C∗l pol, [ia,v ]C∗l polk ∈ Ext1

L

M HMQl (MCl v,N )

(Q(0), l S k−1(1))

as an element of HomM HQl (Q(0), l H 1 ) ⊂ H 0 (C(C), l Vk−1,1 ) if k ≥ 2. Here, we let H 1 := H 1 (M Lv,N (C), l ForQl (S k−1 (1))), and M HQl := M HMQl (Spec(C)) l denotes the category of graded–polarizable mixed Q–Hodge l structures. Remark: Observe that the injectivity of the maps Ext1

L

M HMQl (MCl v,N )

−→ H 1

can be interpreted as a rigidity principle for [ia,v ]C∗l pol itself: it is uniquely determined by the underlying extension of local systems. So if one is only interested in the extensions [ia,v ]C∗l pol, instead of calculating the whole of pol and then restricting it, one may first calculate the topological 52

extension underlying the [ia,v ]C∗l pol, and then use rigidity to make competent guesses about the mixed structure. In fact, this is the approach of [BL], § 2. It will turn out to be convenient to give a description of H 0 (C(C), l Vk−1,1 ) for k ≥ 1. Observe that by letting the standard cusp i∞ correspond to the class L

of 1, the set of cusps of a connected component (MCl v,N )0 can be interpreted as a quotient of the group SL2 (ZZ). At the standard cusp, a base vector of the coinvariants of Symk−1 (V2 ) for the local monodromy is given by h·ek−1 2 , where h is the ramification index of i∞, i.e., the smallest positive integer such that ±T h L

belongs to Γ(gf0 ), the fundamental group of (MCl v,N )0 (C). l † In order to have this definition extended SL2 (ZZ)–equivariantly, we define V (gf0 ) := {α : SL2 (ZZ) −→ Q l | α(g1 g) = α(g) ∀ g1 ∈ Γ(gf0 ) , α(gg2 ) = α(g) ∀ g2 ∈ B + , ∀ g ∈ SL2 (ZZ)} , where we define B to be the subgroup of upper triangular matrices of SL2 (ZZ), and B + ≤ B the subgroup of matrices, whose diagonal entries are 1. The map g 7−→ g(i∞) gives an identification of Γ(gf0 )\SL2 (ZZ)/B and the set C(gf0 )(C) l L

of cusps of (MCl v,N )0 . Let Vk−1,1 (gf0 ) denote the subspace of V (gf0 ) of functions c α satisfying α(−g) = (−1)k−1 α(g). Finally, let R ⊂ G2 (Z Z) be a set of represen-

tatives for the connected components of

c M Lv,N (C) l = SL2 (ZZ)\(H2+ × (G2 (Z Z)/Lv,N )) ,

and define Vk−1,1 (R) :=

M

Vk−1,1 (gf0 ) .

gf0 ∈R

Lemma 3.21: There is an isomorphism Vk−1,1 (R) −→ H 0 (C(C), l Vk−1,1 ) given by associating to a function α the element (hg(i∞) α(g)·ge2k−1)(gf 0 ,g(i∞)) ∈

M

H 0 (C(gf0 )(C), l Vk−1,1 ) = H 0 (C(C), l Vk−1,1 ) .

gf0 ∈R

Here, hg(i∞) is the ramification index of g(i∞). Proof: left to the reader. †

q.e.d.

The normalization by h will be justified later.

53

L

Now fix k ≥ 2. Let us recall the connection between Mk+1 (MCl v,N , C), l the vector L

space of modular forms of weight k + 1 on MCl v,N , and H 1 ⊗Ql C: l L

Mk+1 (MCl v,N , C) l is naturally contained in the space of global holomorphic sections of the (k + 1)–st tensor power ω k+1 of the invertible sheaf ω associated L

to the family of elliptic curves over MCl v,N (compare [D2], (2.1)). The Eichler– Shimura homomorphism L

f1 ⊗ C sh0 : Mk+1 (MCl v,N , C) l −→ H Q l l , f1 := H 1 (M Lv,N (C), where H l ForQl (S k−1 )∨ ), is essentially the boundary homomor-

phism of the de Rham–resolution of ForQl (S k−1 )∨ ⊗Ql Cl (compare [D2], (2.9)). In [V], Th´eor`eme 3.2.5, Eichler’s result on the shape of sh0 on cusp forms is given. The same formula holds for modular forms as well. Let t be the isomorphism ForQl (S k−1 )∨ −→ ForQl (S k−1 (1)) given by sending (e∨1 )p (e∨2 )k−1−p to 2πi·(−e1 )k−1−p ep2 , and fix R as in 3.21. Lemma 3.22: L

L

a) On each connected component (MCl v,N )0 of MCl v,N , L

l −→ H 1 ⊗Ql Cl H 1 (t) ◦ sh0 : Mk+1 ((MCl v,N )0 , C) is given as follows: for a modular form f , choose a holomorphic function Pf : H2+ −→ Cl satisfying dk Pf (τ ) = f (τ ) . (dτ )k αβ For any element g = γδ Lv,N 0 (MCl ) (C), l the function

!

of the fundamental group Γ(gf0 ) ≤ G2 (Q) l of

τ 7−→ (−γτ + α)k−1 Pf (g −1 τ ) − Pf (τ ) is a polynomial of degree at most k − 1 in τ , which we write in the form αg (X) =

k−1 X

ag,p (−1)p X k−1−p .

p=0

g 7−→

k−1 X

ag,p · 2πi·ep1 e2k−1−p

p=0

defines a ForQl (S k−1 (1)) ⊗Ql C–valued l one–cocycle, whose class in H 1 ⊗Ql Cl coincides with H 1 (t) ◦ sh0 (f ). 54

b) The composition of the map H 1 (t) ◦ sh0 with the map δ⊗Ql Cl of Proposition 3.20 and the isomorphism of 3.21 L

(δ ⊗Ql C) l ◦ H 1 (t) ◦ sh0 : Mk+1 (MCl v,N , C) l −→ Vk−1,1 (R) ⊗Ql Cl is given by associating to a modular form f the map αf , which sends g ∈ SL2 (ZZ) to



−1 (k−1)!

times the zeroeth term in the qH2 –expansion of f

[g]k+1

at i∞.

Proof: a) follows from [V], Th´eor`eme 3.2.5 and a calculation, which we leave to the reader. For b), we restrict our attention to the standard cusp i∞ and the case T ∈ Γ0 . The map δ ⊗Ql Cl : H 1 ⊗Ql Cl −→ C·e l 2k−1 factors through H 1 of a small punctured disc around the cusp. It is easily seen to send the class of a cocycle g 7−→

Pk−1 p=0

ag,p · 2πi·ep1 e2k−1−p to aT,0 · e2k−1 .

On the other hand, it is straightforward to check that the aT,0 belonging to a modular form f is equal to

−1 f (i∞). (k−1)!

q.e.d.

It follows from 3.22.b) and the injectivity of sh0 on cusp forms that sh0 is itself injective. So we may identify a modular form with the class in H 1 ⊗Ql Cl associated to it via H 1 (t) ◦ sh0 . Observe that thanks to our normalization of the isomorphism in 3.21, the map in 3.22.b) is nicely behaved under pullback of morphisms of modular curves. In order to compute the modular form belonging to [ia,v ]∗ polk , we need the following combinatorial result: Lemma 3.23: Let k ∈ lN and l, p ∈ lN0 . For 1 ≤ r ≤ k, define Er,l,p ∈ Q l as follows: Er,l,p := 0 for l ≥ r or p > k − r , 1 for p ≤ k − 1 , E1,0,p := p!(k − 1 − p)! 1 Er+1,0,p := + (k − r)Er,0,p for p ≤ k − r − 1 , p!(k − r − 1 − p)! Er+1,l,p := (k − r + l)Er,l,p + Er,l−1,p for l ≥ 1 . Then we have

k−1 X

(−1)k−1−p Er,l,p = (−1)k−1 δr,k δl,0 .

p=0

Proof: left to the reader.

q.e.d. 55

Definition: For k ≥ 2, define Hk+1 : (lR × lR \ ZZ × ZZ) × H2+ −→ Cl to be the function Hk+1





∞ ∞ X q j qCl q j /qCl  Bk+1 (−r2 ) X − (j − r2 )k H2j := + (−1)k−1 (j + r2 )k H2j k+1 1 − qH2 qCl 1 − qH2 /qCl j=0 j=1

We are finally in a position to calculate [ia,v ]C∗l pol. c Let R ⊂ G2 (Z Z) be a set of representatives for the connected components of

c M Lv,N (C). l For any gf ∈ R, let vgf ∈ V2 (Q) l be such that vgf − gf v ∈ N ·V2 (Z Z).

Theorem 3.24: For k ≥ 2, [ia,v ]C∗l polk ∈ Ext1

L

M HMQl (MCl v,N )

(Q(0), l S k−1(1))

corresponds to the modular form of weight k + 1 

N

k−1

Hk+1



 vgf , c H2 . N gf ∈R

Remark: We recall the remark made after Lemma 3.17: the parametrization M

H2+ −→ M Lv,N (C) l

gf ∈R

depends on the choice of R. For example, if we replace the representative gf by ggf , with g ∈ SL2 (ZZ), we not only replace vgf by gvgf but also cH2 by gcH2 , which in turn affects the shape of the isomorphism of 3.22.a). Proof of Theorem 3.24: By 1.5 and 3.18, what we have to do in order to L

calculate the restriction of [ia,v ]C∗l polk to the component (MCl v,N )gf indexed by gf ∈ R is to specialize PNF and PNW to rj = (vgf )j , j = 1, 2, and then apply the morphism induced by the one of 3.19.a). As follows from the remark after Theorem 3.14 and the shape of PNF , the bifiltered holomorphic pro–bundle underlying [ia,v ]C∗l pol is the trivial bifiltered pro–bundle Forhol. (S 1 ) × L M v,N (C) l

So [ia,v ]C∗l pol

“value” of †

L

g

Y

n∈lN0

Forhol. (S n (1)) . L M v,N (C) l

is in fact fully described by the first two columns of the

(MCl v,N ) f PNW at rj =

(vgf )j .† By 3.13 and 3.19.a), the cocycle with values in

This illustrates the injectivity of the map from Ext1

L

M HMQl (MCl v,N )

56

(Q(0), l S k−1 (1)) to H 1 .



(Symk−1 V2 )(1) corresponding to [ia,v ]C∗l pol

L

(MCl v,N )

gf

is defined as follows:

Let F : H2+ −→ Forhol. (S k−1 (1)) be given by H+ 2

F := N k−1

k−1 X

1  (p + 1)R1N,p+1,k−1−p(vgf , cH2 ) k + 1 p=0 

+(k − p)R2N,p,k−p(vgf , cH2 ) · 2πi·ep1 e2k−1−p , which is N k−1

k−1 X

Lik−1−p,p+1



p=0

  vgf , , cH2 + Cp · 2πi·ep1 ek−1−p 2 N

where Cp ∈ Q. l The cocycle is given by mapping g ∈ Γ(gf ) to the constant function τ 7−→ g(F (g −1τ )) − F (τ ) . We leave it to the reader to check that the function P := N k−1

k−1 X

Lik−1−p,p+1



p=0

  vgf k−1−p , cH2 + Cp (−1)p cH 2 N !

αβ has the following property: for all g = ∈ Γ(gf ), the function γδ τ 7−→ (−γτ + α)k−1 P (g −1 τ ) − P (τ ) is a polynomial of degree at most k − 1 in τ . It follows that the function dk P (τ ) τ− 7 → (dτ )k is a modular form of weight k + 1 for Γ(gf ), whose image under H 1 (t) ◦ sh0 coincides, by 3.22.a), with the class of the cocycle above; for the holomorphicity at the cusps look at the qH2 –expansion and use the proof of 3.25. So we need to calculate the k–th derivative of N

k−1

k−1 X p=0

Letting s :=

Lik−1−p,p+1



 vgf k−1−p . , cH2 (−1)p cH 2 N

vgf , we have N

dk k Bk+1 (−s2 ) dk k−1−p Li (s, τ )τ = + H0,p (τ ) , k−1−p,p+1 k−1−p (dτ )k k+1 (dτ )k !

57

where we define Hr,p , for 0 ≤ r ≤ k, to be zero if r > k − 1 − p, and Hr,p :=

p+1 X 1 1 1 ck−r−q (k − 1 − p − r)! q=1 (p + 1 − q)! (−2πi)q H2



·

∞ X



(j − s2 )k−q Liq e2πi((j−s2 )cH2 +s1 )

j=0

+(−1)k

∞ X

(j + s2 )k−q Liq

j=1





  e2πi((j+s2 )cH2 −s1 ) 

if r ≤ k − 1 − p. By induction, one obtains for 1 ≤ r ≤ k: r−1 l X dr k−r+l d H (τ ) = H (τ ) − E τ H(τ ) , 0,p r,p r,l,p (dτ )r (dτ )l l=0

where H :=

∞ X

(j − s2 )k

j=0

+(−1)

k−1

exp(2πi((j − s2 )cH2 + s1 )) 1 − exp(2πi((j − s2 )cH2 + s1 ))

∞ X

(j + s2 )k

j=1

exp(2πi((j + s2 )cH2 − s1 )) . 1 − exp(2πi((j + s2 )cH2 − s1 ))

Our claim follows from 3.23.

q.e.d.

Theorem 3.24, together with the invariance properties of pol, allows to compute the modular behaviour of the forms w Hk+1 := Hk+1 (w, cH2 ) , w ∈ V2 (Q)\V l 2 (ZZ) .

! αβ for the function For g = ∈ G2 (Q) l + and F : H2+ −→ C, l write F [g]k+1 γδ

τ 7−→ (γτ + δ)−(k+1) F (gτ ) . Corollary 3.25: For g ∈ SL2 (ZZ) and w ∈ V2 (Q)\V l 2 (ZZ), we have

w Hk+1

−1

[g]k+1

g w = Hk+1 .

Proof: Let p := ia (g) ∈ P (Q). l There is a commutative diagram (P2,a , X2,a ) x  [ia,w ] y[πa ]

(G2,a , X2,a )

[int(p−1 )] →

(P2,a , X2,a ) x 

[ia,g−1 w ] y[πa ] [int(g )] → (G2,a , X2,a ) −1

58

∼

and a canonical isomorphism [int(p−1 )]∗ pol −→ pol, which identifies [ia,w ]∗ polk with [int(g −1 )]∗ [ia,g−1 w ]∗ polk . But the map [int(g −1 )]C∗l on Ext1 corresponds to the

map F 7−→ F

[g −1 ]k+1

on modular forms under the monomorphism of 3.22.a). q.e.d.

Of course, it would pose no problem to write down, via the usual techniques, w the “Eisenstein type” formula for Hk+1 and then read off 3.25 from it.

3.25 enables us to compute the “boundary” of [ia,v ]C∗l polk , i.e., its image under the morphism δ : H 1 = H 1 (M Lv,N (C), l ForQl (S k−1 (1))) −→ Vk−1,1 (R) : Corollary 3.26: For k ≥ 2 and gf ∈ R, the component gf of δ([ia,v ]C∗l polk ), which is an element of Vk−1,1 (gf ), maps g ∈ SL2 (ZZ) to vg k − Bk+1 h(−g −1 f )2 i . (k + 1)! N 



c Here, vgf ∈ V2 (Q) l satisfies vgf − gf v ∈ N ·V2 (Z Z), and the map h i : lR → [0, 1[ is

unique with respect to the property of making the diagram -

lR @ R @ R

[0, 1[



lR/ZZ commutative. Proof: This follows from 3.22.b), 3.25 and the definition of Hk+1 .

q.e.d.

For gf = 1 and g = 1, this result differs, due to another normalization of the 1 . morphism δ, from [BL], Proposition 2.2.3 by the factor (k − 1)! c For n ≥ 3 and L = Ln := ker(G2 (Z Z) −→ G2 (ZZ/nZZ)) and gf = 1, the map 1 Q l V2 (ZZ)/V2 (ZZ) −→ Vk−1,1 (1) , n v 7−→ δ ([ia,v ]C∗l polk ) , 

or rather, its restriction to Q l



h

1 V (ZZ)/V2 (ZZ) n 2

i0

differs from the boundary of the

Eisenstein symbol in the modular case by the factor −n1−k . Here,  is the sign occurring at the end of § 4 of [SchSch]. 59

By [W4], Corollary 2.2, the class of polk is fixed under complex conjugation ι. L

Hence the same is true for its pullback to MCl v,N : [ia,v ]C∗l polk ∈ Ext1

L

M HMQl (MCl v,N )

Ext1

(Q(0), l S k−1(1)) lies in the subspace L

M HMQl (MCl v,N )

(Q(0), l S k−1(1))+

of elements fixed under complex conjugation. Contrary to the case of the classical polylogarithm (see Remark b) following [W5], Theorem 3.10), the values L

of polk at the Levi section [ia,v ]Cl (MCl v,N ) won’t generate the whole of the space L

Ext1

L M HMQl (MCl v,N )

(Q(0), l S k−1(1))+ unless all the connected components of MCl v,N

are defined over Q l or an imaginary quadratic number field. To obtain the correct analogue of the statement in the classical case, we define what can be seen as the Hodge–de Rham component of the category of mixed systems of smooth sheaves ([W2], § 2): Definition: Let k be a number field, X/k smooth, separated and of finite type, F ⊂ lR a field. HDRFs (X), the category of variations of mixed F –Hodge–de Rham structure over lR on X consists of families (VDR , V∞,σ , IDR,σ , I∞,σ | σ : k ,→ C) l , where a) VDR is a vector bundle on X, equipped with a flat connection ∇, which is regular at infinity in the sense of [D1], II, remark following D´efinition 4.5. Further parts of the data are an ascending weight filtration W· by flat subbundles and a descending Hodge filtration F · by subbundles. b) V∞,σ is a variation of mixed F –Hodge structure (F –M HS) on Xσ (C), l which is admissible in the sense of [Ka]. c) IDR,σ is a horizontal isomorphism FO (V∞,σ ) −→ VDR ⊗k,σ Cl of bifiltered vector bundles on Xσ (C). l Here, FO is a suitable forgetful functor. d) For any σ : k ,→ C, l the complex conjugation ι defines a diffeomorphism cσ : Xσ (C) l −→ Xι ◦ σ (C) l . 60

For a variation of F –M HS W on Xι ◦ σ (C), l we define a variation c∗σ (W) on Xσ (C) l as follows: the local system and the weight filtration are the pullbacks via cσ of the local system and the weight filtration on W, and the Hodge filtration is the pullback of the conjugate of the Hodge filtration on W. c∗σ preserves admissibility. I∞,σ is an isomorphism of variations of F –M HS V∞,σ −→ c∗σ (V∞,ι ◦ σ ) −1 such that c∗ι ◦ σ (I∞,σ ) = I∞,ι . ◦σ

Furthermore, we require the following: For each σ, let c∞,σ be the antilinear involution of Fdiff. (V∞,σ ), the C ∞ –bundle underlying V∞,σ , given by complex conjugation of coefficients. Likewise, let cDR,σ be the antilinear isomorphism Fdiff. (V∞,σ ) −→ c−1 σ (Fdiff. (V∞,ι ◦ σ )) given by complex conjugation of coefficients on the right hand side of the isomorphism in c). Our requirement is the validity of the formula ◦ Fdiff. (I∞,σ ) = cDR,σ ◦ c∞,σ = c−1 σ (c∞,ι ◦ σ ) cDR,σ .

It is straightforward to define Tate twists F (n) for n ∈ ZZ in the category of these data. The last condition we impose is the existence of a system of polarizations: there are compatible morphisms W GrW n VDR ⊗OX Grn VDR −→ FDR (−n) , n ∈ ZZ

of flat vector bundles on X, and polarizations W GrW l , n ∈ ZZ n V∞,σ ⊗F Grn V∞,σ −→ F (−n) , σ : k ,→ C

of variations of F –M HS such that the IDR,σ and I∞,σ and the corresponding morphisms of F (−n) form commutative diagrams. Remarks: a) As suggested by the notation, the category HDRFs (X) depends only on the scheme X, not on the base field k. 61

b) If π : X −→ Y is a finite Galois covering with Galois group G, then π ∗ induces an equivalence of categories HDRFs (Y ) −→ HDRFs (X)G , the right hand side denoting the category of descent data in HDRFs (X) with respect to π. Examples: i) If X is finite over Q l and m ≥ 1, then by [W5], Theorem 3.6, there is an isomorphism 

∼

Ext1HDRsF (X) (F (0), F (m)) −→  the superscript

+

M

+

x∈X(C) l

m  C/(2πi) l F ,

denoting the fixed part of the involution given by complex

conjugation on Cl as well as on X(C). l The reason for this is that on a point, any extension of F (0) by F (m) splits canonically on the level of underlying bifiltered “vector bundles”, and so the only interesting extension datum is the F –structure. ii) In general, if V ∈ HDRFs (X) is of weights smaller than zero, then there is an injection Ext1HDRs (X) (F (0), V) F

−→

M

Ext1M HMF (Xσ ) (F (0), V∞,σ )

!+

.

σ:k,→C l

For X = M Lv,N , note that by [W3], § 6, the variation S k−1 (1) ⊗Ql F is the Hodge component of a variation of Hodge–de Rham structure, which we also denote by S k−1 (1) ⊗Ql F . Here, the situation is in a sense complementary to that in i) in that the extensions are already completely determined by the underlying extension of flat vector bundles on M Lv,N . Lemma 3.27: Let VB(M Lv,N ) denote the category of flat vector bundles on M Lv,N , whose connection is regular at the cusps. For any k ≥ 2, the map Ext1HDRs (M Lv,N ) (F (0), S k−1(1) ⊗Ql F ) −→ Ext1VB(M Lv,N ) (O, ForO (S k−1 (1))) F

is injective. Proof: This follows from the injectivity of the map in ii) of the example and the fact (Proposition 3.20) that Ext1

L

M HMF (MCl v,N )

62

is contained in H 1 .

q.e.d.

By [W4], Corollary 2.2, polk is the Hodge component of a variation of Hodge–de Rham structure, also denoted by polk . Under mild restrictions on L, the space Ext1HDRsF (M L ) (F (0), S k−1 (1) ⊗Ql F ) is generated by the values of polk at M L . Before showing this, we prove the following technical result: Lemma 3.28: For n ≥ 2 and k ≥ 2, define a G2 (ZZ/nZZ)–equivariant map ωnk−1 : Q[(Z l Z/nZZ)2 ] −→ {α : SL2 (ZZ/nZZ) → Q l | α(−g) = (−1)k−1 α(g) , α(gg1 ) = α(g) , ∀ g1 ∈ B + , ∀ g ∈ SL2 (ZZ/nZZ)} by letting ωnk−1 (x) be the map 

g 7−→ Bk+1 h(−g

−1 x

n



)2 i .

i) ωnk−1 is surjective. ii) Let Pn := {x ∈ (ZZ/nZZ)2 | gcd(x1 , x2 ) = 1} and Nn := (ZZ/nZZ)2 \Pn . For any x ∈ (ZZ/nZZ)2 , let dx | n be maximal with respect to the property ∃ y ∈ (ZZ/nZZ)2 : x = dx y . Choose sets V 2 and N n of representatives in (ZZ/nZZ)2 of (ZZ/nZZ)2/ ± id and Nn / ± id. Then a basis of ker(ωnk−1 ) is given as follows: 

B := ((x) + (−1)k (−x) | x ∈ V 2 , x 6= −x) ∪ (x) − dxk−1

X

dx y=x



(y) | x ∈ N n  .

iii) Let P n be a set of representatives in (ZZ/nZZ)2 of Pn / ± id. Then the restriction of ωnk−1 to Q[P l n ] is bijective. Proof: i) is proven in [SchSch], 7.5. The dimension of the vector space on the right is ] SL2 (ZZ/nZZ)

.

∗∗ 0∗

!!

= ](Pn / ± id) .

Since Bk+1 (X) = (−1)k+1 Bk+1 (1 − X) and because of the distribution property of Bk+1 (X), the elements of B certainly belong to ker(ωnk+1 ). Since ]B = n2 − ](Pn / ± id) , all we have to show is the following: Q[N l n ] is generated by the image of B under the projection pr : Q[(Z l Z/nZZ)2 ] −→ → Q[(Z l Z/nZZ)2 ]/Q[P l n ] = Q[N l n] . 63

In the proof, we may replace B by 

B0 := (x) − dxk−1

X

dx y=x



(y) | x ∈ Nn  .

Our aim is to use the relations in B0 to generate, for any x ∈ Nn , a relation only involving (x) and a divisor with support in Pn . This is done using induction on the number of prime divisors of dx . n are coprime. This reduces us to It is easy to see that if dx y = x, then dy and dx   n those x satisfying dx , = 1. For any y satisfying dx y = x, we have dy | dx . dx There is exactly one y0 ∈ (ZZ/nZZ)2 satisfying dx y0 = x and dy0 = dx . We denote it by ϕ(x). So the relation (x) − dxk−1 (ϕ(x)) lies in the vector space generated by pr(B0 ). But this shows our claim: since dxk−1 > 1, whenever we have an equality ϕr (x) = x for some r ≥ 1, the divisor (x) belongs to the span of pr(B0 ). q.e.d. c Theorem 3.29: Let L ≤ G2 (Z Z) be neat, open and compact. Also, assume that

c L contains Ln := ker(G2 (Z Z) −→ G2 (ZZ/nZZ)) for some n ≥ 3, and fix k ≥ 2.

If the invariants in (ZZ/nZZ)2 under L coincide with those under Γ = SL2 (ZZ)∩L, then a basis of Ext1HDRsF (M L ) (F (0), S k−1 (1) ⊗Ql F ) is provided by 

1 [ia,v ] pol(ia , Ka,1 )k | v ∈ P n , Γv ≡ v mod V2 (ZZ) , n 

∗

where P n is a set of representatives in V2 (ZZ) of {x ∈ (ZZ/nZZ)2 | gcd(x1 , x2 ) = 1}/ ± id . Proof: By [P], Proposition 3.9, the connected components of MCLl are permuted transitively by the automorphism group of C. l Let (MCLl )0 be such a component. If follows from 3.27 that the map induced by the forgetful functor Ext1HDRsF (M L ) (F (0), S k−1 (1) ⊗Ql F ) → Ext1M HMF ((M L )0 ) (F (0), S k−1 (1) ⊗Ql F ) C l

is injective. Hence our claim follows if we show that the [ia,v ]∗ polk as in the claim yield a basis of Vk−1,1 (1) ⊗Ql F under δ ⊗Ql F . We may assume F = Q. l For L = Ln , our claim follows from Corollary 3.26 and Lemma 3.28. For arbitrary L, Ext1HDRsF (M L ) coincides with the L/Ln –invariants of the space Ext1HDRsF (M Ln ) , while the correct space of divisors under the G2 (ZZ/nZZ)–equivari-ant map is Q[(Z l Z/nZZ)2 ]Γmod n . Observe that only L–invariant elements of 64

(ZZ/nZZ)2 give rise to sections ia,v of Shimura data – hence the restriction on L. q.e.d. It follows from the proof that the map Ext1M HMQl (M L ) (Q(0), l S k−1 (1)) −→ H 0 (C(C), l Vk−1,1 ) C l

is in fact an isomorphism. So we recover the well–known fact that the sequence of Q–M l HS in 3.20 is actually split. Also, we see that by specializing the function Hk+1 to all rational values of r1 and r2 , we get a set of generators for the space of Eisenstein series. Observe that the fact that the restrictions of Hk+1 actually describe extensions of variations of Hodge–de Rham structure can be translated, with little effort, into saying that the Eisenstein series thus obtained are actually modular forms over the field of definition of the connected component of the modular curve one is considering. Of course, this fact can also be read off from the qH2 –expansions. It remains to examine the case k = 1. Fix N and v as before. Let us recall the following Theorem 3.30: ([W5], Theorem 3.7.) Let X/Cl be a smooth variety. a) The map



defines an isomorphism

g 7−→ 

1

0

1 − 2πi ·log g 1

 

∼

1 ∗ Γ(X(C), l OX( C) l ) ⊗ZZ F −→ Ext (F (0), F (1)) ,

the Ext–group being the one in the category of graded–polarizable variations of F –Hodge structure on X(C). l b) Under the isomorphism in a), the extension defines an admissible variation if and only if g is algebraic. Recall that by the normalization of [J], Lemma 9.2, the matrix 

Pz−1 := 

1

0

1 ·z 1 − 2πi

 

describes the following one–extension of Hodge structures: if e0 and e1 are the base vectors 1 ∈ Q l ⊂ Cl and 2πi ∈ Q(1) l ⊂ C, l then the two–dimensional Hodge 65

structure on Cl 2 corresponding to the above matrix is specified by F 0 := he0 iCl , W−2,Cl := he1 iCl , the rational structure being given by Pz−1 e0 = e0 −

66

1 ·ze1 and Pz−1 e1 = e1 . 2πi

Definition: Define the modified Siegel function Si : (lR × lR \ ZZ × ZZ) × H2+ −→ Cl to be the function Si := −q

B2 (−r2 ) 2 H2

exp (πiB1 (r1 )B1 (−r2 )) (1 − qCl )

∞ Y

j j (1 − qH q l )(1 − qH /qCl ) . 2 C 2

j=1

Recall that B1 (X) = X −

1 2

and B2 (X) = X 2 − X + 61 .

This function differs from the Siegel function defined in [K], (2.12) by the factor 



exp πi( 14 + 12 r1 + 21 r2 ) . c As before, let R ⊂ G2 (Z Z) be a set of representatives for the connected compo-

nents of M Lv,N (C). l As usual, for any gf ∈ R, let vgf ∈ V2 (Q) l be chosen such that c vgf − gf v ∈ N ·V2 (Z Z).

Theorem 3.31: Under the isomorphism of 3.30, [ia,v ]C∗l pol1 ∈ Ext1

L

M HMQl (MCl v,N )

(Q(0), l Q(1)) l

corresponds to the non–vanishing algebraic function 

Si

−1



 vgf , c H2 . N gf ∈R

Proof: By the procedure described in the proof of 3.24, we need to solve the equation      1 1 N,0,1 vgf N,1,0 vgf , c H2 + R 2 , c H2 =− ·log g . R1 2 N N 2πi

q.e.d. Recall that for g ∈ G2 (Q) l + and F : H2+ −→ C, l the function gF sends τ ∈ H2+ to f (g −1 τ ). + w For w ∈ V2 (Q)\V l 2 (ZZ), define Si to be the function Si(w, cH2 ) on H2 .

Corollary 3.32: a) For g ∈ SL2 (ZZ) and w ∈ V2 (Q)\V l 2 (ZZ), there exists a root of unity ξg,w such that g Siw = ξg,w · Sigw . b) If m·w ∈ V2 (ZZ), then (ξg,w )12m = 1 for all g ∈ SL2 (ZZ) .

67

Proof: a) is proven in a way analogous to 3.25. For b), note that by the proof of the last result, the function πi·(R11,0 + R20,1 )(w, cH2 ) is a logarithm of Si. In the proof of 3.13, we showed the equality 0,0 0,1 0,0 R11,0 + R20,1 = −Q1,0 1 + r 1 Q1 − Q 2 + r 2 Q2 .

Hence our claim follows from Lemma 3.10 and a short calculation, which we leave to the reader. Note that under the inclusion [ia,w ] : H2+ −→ X+ 2,a , the usual automorphisms S and T of H2+ will in general be mapped to automorphisms, that differ from S and T in 3.10 by elements of hα1 , α2 i.

q.e.d.

For a different proof of this result, see [KL], II, § 1. We need to prove an analogue of 3.22.b): Lemma 3.33: The composition of the isomorphism L

Γ(MCl v,N , O ∗

L

MCl v,N

) ⊗ZZ Q l −→ Ext1

L

M HMQl (MCl v,N )

(Q(0), l Q(1)) l

of Theorem 3.30 with the map Ext1

L

M HMQl (MCl v,N )

(Q(0), l Q(1)) l −→ H 1 := H 1 (M Lv,N (C), l ForQl (Q(1))) l

and the boundary morphism H 1 −→ V0,1 (R) of 3.20 and 3.21 is given by associating to a non–vanishing meromorphic function f the map αf , which sends g ∈ SL2 (ZZ) to the order of qH2 in the qH2 –expansion of g −1 f at i∞. Proof: left to the reader.

q.e.d.

Corollary 3.34: For gf ∈ R, the component gf of δ([ia,v ]C∗l pol1 ), which is an element of V0,1 (gf ), maps g ∈ SL2 (ZZ) to vg 1 − B2 h −g −1 f 2 N  

Proof: This follows from 3.31–3.33.

 

i .

2

q.e.d. 68

For a number field k and a scheme X/k, which is smooth, separated and of finite type, define a map ∗ Γ(X, OX ) ⊗ZZ F −→ Ext1HDRsF (X) (F (0), F (1))

as follows:

1 The underlying bifiltered vector bundle is trivial with basis e0 , ·e1 , and 2πi 1 ∗ · e1 iOX . For g ∈ Γ(X, OX ) ⊗ZZ F , the flat regular F 0 := he0 iOX , W−2 = h 2πi dg 1 1 ·e1 and maps e0 to · ·e1 . For any embedding connection is trivial on 2πi g 2πi σ of k into C, l the rational structure is given by   1 e0 − log gσ · ·e1 , e1 . 2πi 



Proposition 3.35: For a number field k and a smooth variety X/k, let Y /k be the finite reduced scheme of geometrically connected components of X. a) There is a commutative diagram of exact sequences -

1

Γ(Y, OY∗ ) ⊗ZZ F

-

∗ Γ(X, OX ) ⊗ZZ F



1

-

-

Div 0



?

Ext1HDRs (Y ) (F (0), F (1)) F

?

- Ext1

HDRsF (X) (F (0), F (1))

δ - Div 0

Here, Div 0 is the group of divisors of degree zero concentrated on Z \X, for a fixed smooth compactification Z of X. The map δ : Ext1HDRsF (X) (F (0), F (1)) −→ Div 0 is the composition of the inclusion of Ext1HDRsF (X) into M

Ext1M HMF (Xσ ) =

3.30

σ:k,→C l

M

∗ ) ⊗ZZ F Γ(Xσ , OX σ

σ:k,→C l

and the usual divisor morphism into DivC0l := turns out to land in Div 0 .

L

σ:k,→C l

Div 0 (Zσ\Xσ ), which

b) Ext1HDRsF (X) (F (0), F (1)) is the sum of the spaces Ext1HDRsF (Y ) (F (0), F (1)) ∗ and Γ(X, OX ) ⊗ZZ F . Hence the quotient of Ext1HDRsF (X) by Ext1HDRsF (Y )

canonically equals ∗ Γ(X, OX )/Γ(Y, OY∗ ) ⊗ZZ F .

69

Proof: We only prove b) in the case where Y = Spec(k). For any embedding σ0 : k ,→ C, l we get a non–vanishing function gσ0 on Xσ0 describing the rational structure, and hence also the connection. So for any automorphism τ of Cl over k, we get the equality d log gσ0 = d log gτσ0 . It follows that there exists z ∈ Cl ∗ such that zgσ0 is defined over k: divide gσ0 by its value at a k–rational point, and apply Hilbert 90!

q.e.d.

Corollary 3.34 actually calculates the image of [ia,v ]∗ pol1 in Div 0 . Here, as for k ≥ 2, the variation pol1 is the Hodge component of a variation of Hodge–de Rham structure, also denoted by pol1 . c Theorem 3.36: Let L ≤ G2 (Z Z) be neat, open and compact. Also, assume that

c L contains Ln = ker(G2 (Z Z) −→ G2 (ZZ/nZZ)) for some n ≥ 3. If the invariants

in (ZZ/nZZ)2 under L coincide with those under Γ = SL2 (ZZ) ∩ L, then 1 ([ia,v ]∗ pol(ia , Ka,1 )1 | v ∈ Pn , Γv ≡ v mod V2 (ZZ)) , n

where Pn is as in 3.29, generates Ext1HDRsF (M L ) (F (0), F (1))/Ext1HDRs (M ϕ(L) ) (F (0), F (1)) . F

Here, M ϕ(L) denotes the Shimura variety of connected components of M L . Proof: The proof runs along similar lines as the one of 3.29, with [SchSch], 7.5 replaced by [KL], I, Theorem 3.1, and Lemma 3.27 replaced by Proposition 3.35.b).

q.e.d.

Again, the proof shows even that the map δ Ext1HDRs (M L ) (Q(0), l Q(1)) l −→ Div 0 −→ Q[C l 0 (C)] l 0, Q l

for C 0 the scheme of cusps of a fixed connected component of MCLl , is surjective. This implies both the Manin–Drinfeld principle and the fact that every cusp is rational over the field of definition of the connected component, in which it is contained. The fact that the Siw describe extensions of variations of Hodge–de Rham structure implies that suitable powers are rational over the fields of definition of the connected components of the modular curves they live on: Proposition 3.35.a) says that there is a complex number zw ∈ Cl ∗ such that a power of zw ·Siw is rational. By specializing to CM –points and using [KL], XI, § 1, one sees that one may choose zw = 1. 70

Let us conclude by remarking that due to Corollary 3.16, the modified Siegel functions satisfy distribution relations of the shape Siw =

Y

Siv ,

v

where v runs over a suitably chosen set of representatives in V2 (Q) l of the set [M ]−1 (w mod V2 (ZZ)). Compare [K], § 2, where the distribution laws for the classical Siegel functions under arbitrary isogenies are calculated.

§ 4 Remarks on Beilinson’s conjectures for CM –elliptic curves In this final paragraph, we study the specializations to CM –points of the extensions of § 3. In particular, their real versions will turn out to be related to the elements in the Deligne cohomology of CM –elliptic curves constructed and studied in [De1] and [De2]. l Fix τ0 ∈ H2+ , whose minimal polynomial over lR has coefficients in Q. So Re τ0 and |τ0 |2 are rational. Define the torus T τ0 ≤ G2 by T τ0 (R) :=

  x − yRe τ 0   −y

y|τ0 |2 x + yRe τ0

   x, y ∈ R  

,

τ0 and write P2,a for the semidirect product V2 × T τ0 .

It is readily checked that hτ0 factors through TlRτ0 . Therefore, we have a cartesian diagram of Shimura data τ0 0 (P2,a , Xτ2,a ) −→ (P2,a , X2,a )

  yπa

  τ0 yπa

(T τ0 , {τ0 }) −→ (G2 , H2 ) . 0 Here, Xτ2,a is diffeomorphic to lR × lR, with (r1 , r2 ) corresponding to the mor-

l with (r1 , r2 ) phism kτr01 ,r2 of Lemma 1.2. Its complex structure is given by C, corresponding to −r2 τ0 + r1 . τ0 0 | m,Q(τ The reflex field E(P2,a , Xτ2,a ) equals Q(τ l 0 ), and T τ0 is equal to ResQ(τ l 0 )/Q lG l 0):

the latter provides a model of the Deligne torus S over Q, l and hτ0 descends to a Q–rational l isomorphism τ0 | m,Q(τ ResQ(τ , l 0 )/Q lG l 0 ) −→ T

71

also denoted by hτ0 . On Q(τ l 0 )–valued points, it is given by sending (z1 , z2 ) to 



i  τ0 z1 − τ0 z2 −|τ0 |2 (z1 − z2 )  . 2Im τ0 z1 − z 2 −τ0 z1 + τ0 z2 Note that hτ0 equals the map noted NormQ(τ l 0 )/Q l ◦ ResQ(τ l 0 )/Q l (µτ0 ) in [P], 11.4. ab From the description of the reciprocity law given there, it follows that GQ(τ l 0)

acts transitively on any Shimura variety formed from (T τ0 , {τ0 }). In other words, these Shimura varieties are just finite abelian extensions of Q(τ l 0 ). Lemma 4.1: i) The irreducible representations of T τ0 are τ0 τ0 Wp,q = Wq,p , p, q ∈ ZZ. τ0 Here, W−n,−n := Q(n) l is the n–th tensor power of the determinant repre-

sentation. We let 0 hτ−n,−n

n



2Im τ0  :=  q ·(2πi)n . |dQ(τ l 0) |

τ0 is the two–dimensional representation of Hodge type For p 6= q , Wp,q

{(p, q), (q, p)}: τ0 τ0 0 0 l 0 )) maps iQ(τ , hτq,p ⊗Ql Q(τ l 0 ) = hhτp,q Wp,q l 0 ) , and (z1 , z2 ) ∈ T (Q(τ 0 0 to z1−p z2−q hτp,q hτp,q 0 0 . and hτq,p to z1−q z2−p hτq,p

0 0 Conjugation in Gal(Q(τ l 0 )/Q) l acts by interchanging hτp,q and hτq,p .

ii) For n ∈ lN0 , there is an isomorphism of T τ0 –representations ∼

Symn V2 −→

M

τ0 . Wp,q

0≥p≥q p+q=−n

After tensoring with Q(τ l 0 ), it is given by mapping hp,q (τ0 ) = (τ0 e1 + e2 )−q (τ0 e1 + e2 )−p ∈ Symn V2 ⊗Ql Q(τ l 0) 0 . to hτp,q

Proof: left to the reader.

q.e.d. 72

Remark: Although we won’t ever use the basis in 4.1.i), we feel the need to justify our choice of normalization: given g ∈ G2 (Q) l + , there is an isomorphism ∼

int(g −1 )∗ V2 −→ V2 of representations of G given by sending a vector v to gv. The −(p+q)–th power of this isomorphism induces an isomorphism of representations of T τ0 , that fits into a commutative diagram −1 (τ

g int(g −1 )∗ Wp,q

∼

0)

-

τ0 Wp,q





?

∼-

int(g −1 )∗ Sym−(p+q) V2

?

Sym−(p+q) V2

where the vertical arrows are the maps of 4.1.ii) and the upper horizontal map comes from the isomorphism ∼

int(g −1 ) : T τ0 −→ g −1 T τ0 g = T g −1 (τ

Explicitly, it is given by sending hgp,q

0)

−1 (τ

0)

.

to

0 (−γτ0 + α)q (−γτ0 + α)p hτp,q

if

!

αβ g= . γδ

0 This explains our normalization of hτ−n,−n : we have

Im g −1 (τ0 ) = | − γτ0 + α|−2 Im τ0 . 0 We chose to include the factor two because then we have hτ−n,−n = f n ·(2πi)n if

ZZ ⊕ ZZτ0 is the order ZZ + f ·oQ(τ l 0 ) in oQ(τ l 0). Let Lτ0 ≤ T τ0 (IAf ) = IA∗f,Q(τ l 0 ) be neat, open and compact. Fix N ∈ lN, and define τ0 c Ka,N := (N ·V2 (Z Z))× Lτ0 . We have τ

τ

τ0 l = Q(τ l 0 )∗ \ IA∗f,Q(τ l = M L 0(T τ0 , {τ0 })(C) M L 0(C) l 0 ) /L . τ

l is the set of Q(τ l 0 )–linear embeddings of the field OMLτ0 into C. l Of M L 0(C) course, “Q(τ l 0 )–linear” refers to the canonical embedding σ0 of Q(τ l 0 ) into C. l

73

For v ∈ V2 (Q), l we have the Levi section τ0 0 0 iτa,v : (T τ0 , {τ0 }) −→ (P2,a , Xτ2,a ).

It is the base change of ia,v to (T τ0 , {τ0 }) and hence given by the same formulae: 

0 iτa,v (t) = 

1

0

(1 − t)v

0

 

t

,

0 iτa,v (τ0 ) = v = −v2 τ + v1 . 0 Also, we define iτa0 := iτa,0 .

τ0 0 0 Letting Lτv,N := (iτa,v )−1 (Ka,N ), we have the embedding τ0

τ0

τ0 0 0 [iτa,v ] : M Lv,N (T τ0 , {τ0 }) −→ M Ka,N (P2,a , Xτ2,a ).

c Fix v ∈ V2 (Q)\N l ·V2(Z Z). We attempt to determine the elements in

Ext1

s (M HDRQ l

τ L 0 v,N

)

(Q(0), l S k−1 (1)) τ0

τ0 ) at [iτa,v ](M Lv,N ). As in the previous given by the “value” of pol = pol(0, iτa0 , Ka,N

paragraph, we denote by S n the canonical construction of the representation Symn V2 . Again, an explicit description of the above Ext–group will be useful: Theorem 4.2: Let F ⊂ lR be a field, and denote by M HF the category of graded–polarizable mixed F –Hodge structures. For any H ∈ M HF , there is a canonical isomorphism ∼

W0 HCl /(W0 HF + W0 F 0 HCl ) −→ Ext1M HF (F (0), H) . It sends the class of h ∈ W0 HCl to the extension described by the matrix  

1

0

−h idH

 

.

This means that we equip Cl ⊕ HCl with the trivial weight and Hodge filtrations, and the rational structure extending the rational structure of HCl by the vector 1 − h ∈ Cl ⊕ HCl . Proof: [J], Lemma 9.2.

q.e.d. 74

As an immediate consequence, we get a result on the shape of the group of one–extensions in HDRFs (X), for X/Q l finite: Corollary 4.3: Let X be finite over Q, l and H ∈ HDRFs (X). Then the canonical map 

Ext1HDRsF (X) (F (0), H) −→  ∼

M

x∈X(C) l



−→  4.2

M

x∈X(C) l

+

Ext1M HF (F (0), H)

+

W0 HCl /(W0 HF + W0 F 0 HCl )

is an isomorphism. As before, the superscript

+

denotes the fixed part of the

involution given by complex conjugation on W0 HCl as well as on X(C). l Proof: Note that because this is true on the level of C–vector l spaces, any extension of F (0) by H splits on the level of underlying bifiltered “vector bundles”, though in general not canonically. Two such splittings differ by an element of W0 F 0 ForO (H). It is this observation, and the fact that elements of W0 F 0 HCl only define trivial extensions of Hodge structures (by. 4.2), which show the injectivity of the above map. We leave the details to the reader.

q.e.d.

Of course, this result also contains the case of schemes X, which are finite over arbitrary number fields: we just have to replace X(C) l by (Ql X)(C), l where Ql X denotes the restriction X −→ Spec(K) −→ Spec(Q) l of X to Q. l In particular, if K is imaginary quadratic, then the choice of an embedding of K into Cl identifies X(C) l with the orbits of complex conjugation on (Ql X)(C). l Corollary 4.4: Let K be imaginary quadratic and X a finite scheme over K. Also, let H ∈ HDRFs (X), and choose a fixed embedding of K into C. l Then the canonical map Ext1HDRsF (X) (F (0), H) −→ ∼

−→ 4.2

M

Ext1M HF (F (0), H)

M

W0 HCl /(W0 HF + W0 F 0 HCl )

x∈X(C) l

x∈X(C) l

is an isomorphism.

75

Remember that we have a bijection τ0

M Lv,N (C) l −→ GQ(τ l 0 ) (O

M

τ L 0 v,N

,→ C) l ,

the right hand side denoting the set of Q(τ l 0 )–linear embeddings of the field O

M

τ L 0 v,N

into C. l On the other hand, we have a commutative diagram

M

τ

0 Lv,N

H



0 [iτa,v ]-

HH

[·1]

HH j H

τ0

M Ka,N [πaτ0 ]

?

τ

ML 0 τ0

So [iτa,v ] is an O

τ L 0 M v,N

–valued torsion point of M Ka,N , considered as an elliptic

curve over the field OM Lτ0 . Denote this point by P . Let us fix some notation: we write K := Q(τ l 0 ), F := OM Lτ0 , and τ0

E := M Ka,N . We have O

M

τ L 0 v,N

= F (P ).

Theorem 4.5: Under the isomorphism of 4.4, 0 ∗ [iτa,v ] polk ∈ Ext1HDRQsl (Spec(F (P ))) (Q(0), l S k−1 (1))

is mapped to the class of 

−N k−1

k−1 X

Lik−1−p,p+1 ·

2πi·ep1 e2k−1−p

p=0





(σP )

.

σ∈GK (F (P ),→C) l

Proof: This follows directly from the computation carried out in the proof of 3.24.

q.e.d.

τ0 the canonical construction of the By slightly abusing notation, we denote by Wp,q τ0 τ0 can be identified with the of T τ0 . Recall from 4.1.ii) that Wp,q representation Wp,q

sub–Hodge–de Rahm structure of (iτa0 )∗ Sym−(p+q) V2 of type (p, q). On the other hand, by the remark following [W3], Lemma 2.5, the canonical construction S 1 of (iτa0 )∗ V2 on Spec(F ) is canonically isomorphic to the Hodge–de Rham 76

τ0 structure H1 (E, Q) l on Spec(F ). So a basis of the Q–Hodge l structure on (Wp,q )σ

can be described as follows: the restriction of σ defines an embedding of F into C. l Write σE for the elliptic curve E ×Spec(F ),σ Spec(C) l over C, l and choose an isomorphism ∼

ησ : σE(C) l −→ C/(Z l Zτσ ⊕ ZZ) for some τσ ∈ H2+ (not uniquely defined by σ). We leave it to the reader to check that τσ actually lies in K. Then hp,q (τσ ) = (τσ e1 + e2 )−q (τσ e1 + e2 )−p is of type (p, q) in S −(p+q) ⊗Ql K. 0 ∗ 0 ∗ In order to compute the image of [iτa,v ] polp,k−1−p of [iτa,v ] polk in

τ0 Ext1HDRs (Spec(F (P ))) (Q(0), l W−p,p−(k−1) (1)) Q l

τ0 under the projection from S k−1 (1) to W−p,p−(k−1) (1), we need to express the

class in 4.5 in terms of the h−p,p−(k−1) (τσ ) rather than the ep1 ek−1−p . 2 One last combinatorial result will be used: Lemma 4.6: Let p, q, r ∈ lN0 , and define Fp,q,r Then Fp,q,r = (−1)

p

p X

p := (−1) p−i i=0 i

!

!

q+r+i . r

!

!

q+r m . Here, is understood to equal zero if m < n or n r−p

n < 0. Proof: If p = 0, then the claim is trivial. For p > 0, we have p−1 X p q+r Fp,q,r = − (−1)i p−i−1 r i=0

!

q+r+i+1 = r

p−2 X p−1 q+r − (−1)i = p−i−2 r i=0

!

q+r+i+1 − r

!

q+r+i+1 = r

!

!

p−1 X

p−1 − (−1) p−i−1 i=0 i

!

!

!

= Fp−1,q,r − Fp−1,q+1,r . q.e.d. 77

Theorem 4.7: Let 0 ≤ p ≤ 0 ∗ [iτa,v ] pol in

k−1 0 ∗ , and denote by [iτa,v ] polp,k−1−p the image of 2

τ0 Ext1HDRs (Spec(F (P ))) (Q(0), l W−p,p−(k−1) (1)) . Q l

Then under the isomorphism of 4.4, this extension is represented by τ0 (W−p,p−(k−1) ⊗Ql C) l ,

M

(cσ )σ ∈

σ∈GK (F (P ),→C) l

with

cσ = (c−p · 2πi·h−p,p−(k−1))(σP ) +(cp−(k−1) · 2πi·hp−(k−1),−p)(σP ) , |

{z

if p6= k−1 2

where c−r denotes the multivalued function c−r = −N k−1

(cH2



· (−1)r

}

1 − cH2 )k−1

k X

1 s − 1 (cH2 − cH2 )k−s X s k,s (k − s)! r s=r+1 (−2πi) !

1 Bk+1 (−r2 ) (cH2 − cH2 )k + 2 (k + 1)!

!

.

Here, we define X

k,s

:=

∞ X

j (j − r2 )k−s Lis (qH q l ) + (−1)k−1 2 C

∞ X

j (j + r2 )k−s Lis (qH /qCl ) . 2

j=1

j=0

0 As before, P is the torsion point on E given by [iτa,v ].

Proof: We have the equalities e1 = e2 =

c H2 c H2

1 (h0,−1 − h−1,0 ) , − c H2 1 (−cH2 h0,−1 + cH2 h−1,0 ) . − c H2

n Hence em 1 e2 equals

(cH2 =

(cH2

n m X X 1 a m h (−1)n−b (−1) −a,a−m a − cH2 )m+n a=0 b=0 ! min(r,n) m+n X X 1 m n−r (−1) r−b − cH2 )m+n r=0 b=max(0,r−m)

!

!

n b c cH n−b h−b,b−n b H2 2 ! n b cH2 cH2 n−b h−r,r−(m+n) . b

So we get k−1 X

Lik−1−p,p+1 · 2πi·ep1 e2k−1−p

p=0

=

(cH2

k−1 X 1 (−1)k−1−r dr · 2πi·h−r,r−(k−1) k−1 − c H2 ) r=0

78

with dr =

k−1 X

min(r,k−1−p)

(−1)

p=0

p

X

b=max(0,r−p)

p r−b

!

!

k−1−p b cH2 cH2 k−1−p−b Lik−1−p,p+1 . b

After inserting the definition of Lik−1−p,p+1 and rearranging sums, we get dr =

r k X 1 Bk+1 (−r2 ) X (−1)t−r−1 1 ctH2 cH2 k−t k + 1 t=r+1 (k − t)!(t − r − 1)! i=0 i + t − r (r − i)!i!

+

k X

k X X 1 (−1)t−r−1 −s · c ctH2 cH2 k−t s H2 k,s (−2πi) (k − t)!(t − r − 1)! s=1 t=max(r+1,s) r X (s − 1)! t−r−1+i · (−1)i r! s−1 i=max(0,s−t+r)

The last sum in the first term equals equal to

(t − r − 1)! . t!

!

(t − r − 1)! · Cr,0,t−r−1 , which by 3.9 is t!

For s ≤ t − r, the last sum in the second term equals Fr,t−r−s,s−1 . For s ≥ t − r, it equals (−1)s−t+r

r! (t − r − 1)! · Ft−s,s−t+r,t−r−1 . (s − 1)! (t − s)!

In any case, Lemma 4.6 tells us that this sum is equal to (−1)r

(t − r − 1)! (s − r − 1)! (t − s)!

if s ≥ r + 1, and zero otherwise. Therefore, we have k k t Bk+1 (−r2 ) X c cH k−t dr = (−1)t−r−1 t H2 2 (k + 1)! t=r+1

!

+(−1)

k−1

k X

1 s − 1 (cH2 − cH2 )k−s X , s k,s r (k − s)! s=r+1 (−2πi) !

and hence k−1 X

(Lik−1−p,p+1 · 2πi·ep1 e2k−1−p )(σP )

p=0

=

k−1 X 1 2πi·h−r,r−(k−1) (τσ ) k−1 (τσ −τ σ ) r=0 ! k X s − 1 (τσ − τ σ )k−s X 1 r (η (σP )) · (−1) s k,s σ (k − s)! r s=r+1 (−2πi)

+

!

r . r−i

k X

!



k t k−t  Bk+1 (−r2 (ησ (σP ))) (−1)k−t τ τ . (k + 1)! t σ σ t=r+1 79

We still need to adjust the second term in this expression: for this, note that for r2 ∈ Q, l the expression k−1 X 1 Bk+1 (−r2 ) 2πi·h−r,r−(k−1)(τσ ) k−1 (τσ − τ σ ) (k + 1)! r=0   ! ! r k X 1 X k−t k k−t k t k−t t k−t  · − (−1) (−1) τ τ τ τ 2 t=0 t σ σ t σ σ t=r+1

defines a Q–rational l element of Symk−1 (V2 ), hence by 4.2 doesn’t change the extension class.

q.e.d.

s Of course, if we change coefficients to lR, then the classes in HDRlR (Spec(F (P ))) 0 ∗ of the [iτa,v ] polp,k−1−p are still described by the elements in Theorem 4.7. In order

to compare these classes to the elements in the Deligne cohomology of E and Spec(F (P )) constructed in [De1], it turns out to be necessary to calculate their images under M σ

Re

τ0 τ0 (W−p,p−(k−1) ⊗Ql C)/(W l l lR) −→ −p,p−(k−1) (1) ⊗Q ∼

M σ

τ0 (W−p,p−(k−1) ⊗Ql lR) ,

i.e., the unique representatives fixed under complex conjugation. So we need to l compute 12 (cσ + cσ ) for each σ ∈ GK (F (P ) ,→ C). Definition: a) A(τ ) :=

Im (τ ) . π

b) The Pontryagin pairing ( , )τ : (C/(Z l Z + ZZτ )) × (ZZ + ZZτ ) −→ Cl ∗ is defined by (z, γ)τ := exp(A(τ )−1 · (zγ − zγ)) . Explicitly, if z = −r2 τ + r1 and γ = mτ + n, then (z, γ)τ = exp (2πi(−nr2 − mr1 )) . c) The Kronecker double series Ga,b : (lR × lR \ ZZ × ZZ) × H2+ −→ C, l for a, b ∈ lN is defined by X

Ga,b (z, τ ) :=

γ∈(ZZ+ZZτ )\0

(z, γ)τ . γaγb

For a = b = 1, where the right hand side does not converge absolutely, the sum is defined as

(z, γ)τ . s→1 |γ|2s γ∈(ZZ+ZZτ )\0 lim

X

80

Theorem 4.8: Under the isomorphism of 4.4, τ0 0 ∗ [iτa,v ] polp,k−1−p ∈ Ext1HDRs (Spec(F (P ))) (lR(0), W−p,p−(k−1) (1) ⊗Ql lR) lR

is mapped to (cσ,lR )σ ∈

τ0 (W−p,p−(k−1) ⊗Ql lR) ,

M

σ∈GK (F (P ),→C) l

with cσ,lR = (c−p,lR · 2πi·h−p,p−(k−1))(σP ) +(cp−(k−1),lR · 2πi·hp−(k−1),−p)(σP ) . |

{z

if p6= k−1 2

}

Here, c−r,lR denotes the multivalued function cH − c H 1 c−r,lR = N k−1 2 k+12 Gk−r,r+1 , 2 (2πi) 0 and P ∈ E corresponds to [iτa,v ].

Proof: By 4.7, we get the following expression for c−r,lR = 12 (c−r − cr−(k−1) ): c−r,lR = −

1 1 k + 1 k−1 N 2 k (cH2 − cH2 )k−1



k X

s − 1 (cH2 − cH2 )k−s X 1 · (−1) s k,s r (k − s)! s=r+1 (−2πi) r

+(−1)

k−r+1

!

k X

s−1 (cH2 − cH2 )k−s X 1 s k−r−1 k,s (k − s)! s=k−r (−2πi) !

Bk+1 (−r2 ) (cH2 − cH2 )k + (k + 1)!

!

.

For k = 1, our claim amounts to Kronecker’s second limit formula ([L], § 20). 1 So let k ≥ 2. The term in the large brackets equals ·Dr+1,k−r , where (−2πi)k Da,b is the function defined and studied in [Za]. By [Za], Theorem 1, we have the equality Da,b = (−1)a+b

(cH2 − cH2 )a+b−1 Gb,a .† 2πi

This proves our claim.

q.e.d.

Remark: It is not only the values at special points of Ga,b that matter: in [BL], 3.3, the real version of the elliptic polylogarithm of § 3 is described in a language different from but equivalent to ours. It involves Kronecker double series as a whole. †

The reader may find it necessary to analyse the proof in order to see that this is the

correct formulation of the result.

81

The isomorphism ∼

H1 (E, Q) l = H 1 (E, Q) l ∨ −→ H 1 (E, Q)(1) l given by Poincar´e duality is the canonical construction of ∼

V2 −→ V2∨ (1) , e1 7−→ −2πi·e∨2 , e2 7−→ 2πi·e∨1 . If we let ω := e∨1 − cH2 e∨2 denote the generator of the sub–C ∞ bundle H1,0 of S1∨ , then this isomorphism identifies h0,−1 and 2πi·ω , h−1,0 and 2πi·ω . Under the induced isomorphism ∼

∨ Sk−1 (1) −→ Sk−1 (k) ,

the section 2πi·hp,q is sent to (2πi)k ·ω −q ω −p . 0 ∗ ] polk is mapped to the element Corollary 4.9: Under this isomorphism, [iτa,v

∨ (Sk−1 (k − 1) ⊗Ql lR)

M

(dσ,lR )σ ∈

σ∈GK (F (P ),→C) l ∨ = Ext1HDRs (Spec(F (P ))) (lR(0), Sk−1 (k) ⊗Ql lR) , lR

with dσ,lR =

k−1 X

(dp,lR · ω k−1−pω p )(σP ).

p=0

Here, dp,lR denotes the multivalued function dp,lR = N k−1

A(cH2 ) Gk−p,p+1 , 2

0 ]. and P ∈ E corresponds to [iτa,v

Proof: straightforward.

q.e.d.

82

0 ∗ Remarks: a) It follows that the extension [iτa,v ] polk is of motivic origin. For

k = 1, this is clear since the extension in 3.31 came from an algebraic function. So let k ≥ 2. Let Z ≤ E(F ab ) be a finite group generated by a point P , and write F (Z) := F (P ) and EF (Z) := E ⊗F F (Z). By [De1], §§ 9 and 10, the map k (]Z)1−k · rD ◦ EM : Q[Z(F l (Z))]0 −→ HM (EFk−1 l sgn (Z) , Q(k))

−→ HDk (EFk−1 l lR, lR(k))sgn , (Z) ⊗Q the latter space being identified with HBk−1 (EFk−1 l lR(k − 1))+ l C, sgn = (Z) ⊗Q

M

∨ (Sk−1 (k − 1) ⊗Ql lR) ,

σ∈GK (F (Z),→C) l

maps a divisor β =

P

Q∈Z

βQ · (Q) to the element



−N 1−k · 

X

Q∈Z

βQ



k−1 X

(dp,lR · ω k−1−pω p )(σQ) . p=0 σ

So if we take Galois invariants and the direct limit, we get a map rD ◦ EeM : Q[E l tors. ]0 −→

M

∨ (Sk−1 (k − 1) ⊗Ql lR) ,

σ∈GK (F (Z),→C) l

which up to the factor −N 1−k coincides with the map “evaluation at torsion points of polk ”. Note that because of k ≥ 2, the distribution relation is inhomogeneous (see the l tors. ]0 ) coincides with the vector space generproof of 3.28), hence rD ◦ EeM (Q[E ated by the values at torsion points of polk .

Actually, [De1], Theorem 10.9 holds without any hypothesis on the elliptic curve. On the other hand, we could specialize the polylogarithmic extension of § 3 to a non–CM elliptic curve E/F . Its value at a torsion point P would still be given by the formula in 4.9, of course with “ by “

L

σ∈GQl (F (P ),→C) l

+

L

σ∈GK (F (P ),→C) l ”

replaced

”. This shows that these values are of motivic origin in

any case. This again supports the hope that polk itself should be of motivic origin. b) As in the case of the classical polylogarithm ([W5], § 3) and as in § 3, the values at Levi sections of the polylogarithm on CM –elliptic curves E/F associτ0 0 ated to (P2,a , Xτ2,a ) generate a group of extensions in the category of variations

of lR–Hodge–de Rham structures. They define a Q–structure l on it that has 83

the relation to the value at k of the L–function of the motive Sym k−1 h1 (E) predicted by Beilinson’s conjecture. In [De1], this is shown for the part correτ0 ∨ sponding to the direct summand W−l,−l−1 (2l + 1).† More = S1∨ (l + 1) of S2l+1

precisely, it is proven (see [De1], § 3 and Theorem 11.3.1) that the projection of rD ◦ EM (Q[E l tors. ]0 ) to

L

∨ l lR) σ (S1 (l + 1) ⊗Q

generates HD2 (h1 (E) ⊗Ql lR, lR(l + 2)) as

a T ⊗Ql lR–vector space. Here, T := EndK (ResF/K E)⊗ZZ Q l is an algebra of dimension [F : K] over K. Its explicit description ([De1], page 10) admits the following reformulation of the above: as an lR–vector space, HD2 (h1 (E) ⊗Ql lR, lR(l + 2)) is generated by the images under pull–back by isogenies E −→ → E0 0 of rD ◦ EeM (Q[E l tors. ]0 ). Of course, these quotients E 0 of E are merely Shimura τ0 0 ). We chose not to include a precise , Xτ2,a varieties of lower level associated to (P2,a

statement “on finite level” like the ones we proved in 3.29 and 3.36. In [De2], the weak version of the Beilinson conjecture (see [De1], 1.5.2) is shown for arbitrary algebraic Hecke characters of K. Observe that if ψ denotes the b

Hecke character of h1 (E), then Symk−1 h1 (E) is the direct sum of h(ψ a ψ ), for a > b and a + b = k − 1, and Q l b



k−1 2



if k is odd. Furthermore, the Hodge–de

τ0 . Therefore, [De2], Rahm realization of h(ψ a ψ ), for a > b, is equal to W−b,−a b

Theorem 1.4.1 implies the weak Beilinson conjecture for the h(ψ a ψ ). Its proof, which runs along similar lines as [De1], shows in particular that the values at torsion points of polp,k−1−p generate τ0 Ext1HDRs (lR(0), W−p,p−(k−1) (1) ⊗Ql lR) lR

for any p. c) On non–CM curves, the values at Levi sections of the polylogarithm will in general not be sufficient to generate ∨ Ext1HDRs (lR(0), Sk−1 (k) ⊗Ql lR) , lR

see [De2], § 5. This seems to confirm that although polylogarithmic extensions could be defined in a much more general context, they behave particularly well on mixed Shimura varieties. †

Observe that Shimura’s condition, denoted (S) in [De1], is automatically satisfied - in

fact it occurs quite naturally for CM –elliptic curves that are mixed Shimura varieties.

84

Index of Notations

1

pol(0, ia , Ka,N )

8

X2,a

1

8

G2

1

fKa,N (P M

V2

P2,a

2,a , X2,a )

1

ϕN ˆ (LieV2 ) U

8

H2

1

Log(ia , Ka,N )

8

πa

2

e1

9

kτr1 ,r2

2

e2

9

r1

4

9

r2

4

γ1N γ2N

c H2

4

µKa,N ,∞

9

cCl

4

ForQl

9

ia

4

c V2 ( Z Z)

H2+

G2 (Q) l

+

g]∗ [π

8

9

a

10

4

pa,N

10

4

e0 X 2,a

10

4

M L (G2 , H2 )(C) l

4

xe

Γ(gf )

5

πe1

α1N

10

Ka,N

5

α2N

10

M Ka,N (P2,a , X2,a )(C) l

5

T

11

Λ(pf )

5

S

11

X+ 2,a

5

12

5

e Π 1

ε1

13

M L (G2 , H2 )C0l

5

ε2

13

M Ka,N (P2,a , X2,a )C0l

5

Sl

14, 33

Γ

5

19

Λ

6

Fordiff.Ka,N 0 (MCl ) (C) l

ek,l

19

prN

6

LW N

20

µKa,N ,l

6

hp,q

20

ia,v

6

fp0 ,q

21

Lv,N

7

Hp,q (Log(ia , Ka,N ))

21

P2,a (Q) l

+

85

10 10

ψp,q

22

B+

53

LFN

22

C(gf0 )(C) l

53

ΩLog,N

22

Vk−1,1 (gf0 )

53

Λm,n

23

Vk−1,1 (R)

53

Λn

23

hg(i∞)

53

q H2

23

L Mk+1 (MCl v,N , C) l

54

qCl

23

sh0

54

Al,m

24

t

54

Bl,m

24

Er,l,p

55

ΛN m,n Qk,l 1 Qk,l 2

30

Hk+1

56

30

vgf

56

30

56

Cp,q,r

30

Forhol. M Lv,N (C) l w Hk+1

QN,k,l j

32

F |[g]k+1

58

ε01

58

33

hi

59

ε02

33

HDRFs (X)

60

(PNW )−1

34

IDR,σ

60

Dp,q,r

35

cσ

61

Lim,n

35

c∗σ

61

R1k,l

35

I∞,σ

61

R2k,l RjN,k,l PNW

35

c∞,σ

61

35

cDR,σ

36

VB(M

ϕ0,−1

38

ωnk−1

63

ϕ−1,0

39

Si

67

61 Lv,N

)

62

w

67

Ωpol,N

43

Si

PNF

43

ξg,w

67

[ia,v ]C∗l pol

51

Div 0

69

L M HMQl (MCl v,N )

51

M

C

51

T τ0

71

Vn,m

51

71

[ia,v ]C∗l polk 1

52

H

52

V (gf0 )

53

B

53

τ0 P2,a 0 Xτ2,a πaτ0 τ0 Wp,q 0 hτp,q

86

ϕ(L)

70

71 71 72 72

Lτ0

73

Fp,q,r

77

τ0 Ka,N L τ0

73

0 ∗ [iτa,v ] polp,k−1−p

78

73

cσ

78

0 iτa,v

74

c−r

78

iτa0

74

78

0 Lτv,N τ0 Ka,N

74

P

M

τ0

(T , {τ0 })(C) l

k,s

A(τ )

80

74

( , )τ

80

τ0 pol(0, iτa0 , Ka,N )

74

Ga,b

80

M HF

74

cσ,lR

81

Q lX

75

c−r,lR

81

76

ω

82

GK (F (P ) ,→ C) l

76

dσ,lR

82

ησ

77

dp,lR

82

τσ

77

M

τ0 0 (P2,a , Xτ2,a )

GQ(τ l 0 ) (O

M

τ L 0 v,N

,→ C) l

87

References [BL]

A.A. Beilinson, A. Levin, “The Elliptic Polylogarithm”, in U. Jannsen, S.L. Kleiman, J.–P. Serre, “Motives”, Proc. of Symp. in Pure Math. 55, Part II, AMS 1994, pp. 123–190.

[BLp]

A.A. Beilinson, A. Levin, “Elliptic Polylogarithm”, typewritten preliminary version of [BL], preprint, MIT 1992.

[BLpp]

A.A. Beilinson, A. Levin, “Elliptic Polylogarithm”, handwritten preliminary version of [BLp], June 1991.

[Br]

K.S. Brown, “Cohomology of Groups”, LNM 87, Springer–Verlag 1982.

[BZ]

J.–L. Brylinski, S. Zucker, “An Overview of Recent Advances in Hodge Theory”, in W. Barth, R. Narasimhan, “Several Complex Variables VI”, Encycl. of Math. Sciences 69, Springer–Verlag 1990.

[D1]

P. Deligne, “Equations Diff´erentielles a` Points Singuliers R´eguliers”, LNM 163, Springer–Verlag 1970.

[D2]

P. Deligne, “Formes modulaires et repr´esentations l–adiques”, Expos´e 355, in “S´eminaire Bourbaki vol. 1968/69”, LNM 179, Springer–Verlag, pp. 139–172.

[De1]

C. Deninger, “Higher regulators and Hecke L–series of imaginary quadratic fields, I”, Inv. math. 96 (1989), pp. 1–69.

[De2]

C. Deninger, “Higher regulators and Hecke L–series of imaginary quadratic fields, II”, Ann. of Math. 132 (1990), pp. 131–158.

[GS]

P. Griffiths, W. Schmid, “Recent developments in Hodge theory: A discussion of techniques and results”, in W.L. Baily, Jr. et al., “Discrete subgroups of Lie groups and applications to moduli”, Proceedings of the 1973 Bombay Colloquium, Oxford Univ. Press 1975, pp. 31–127.

88

[J]

U. Jannsen, “Mixed Motives and Algebraic K–Theory”, LNM 1400, Springer–Verlag 1990.

[Ka]

M. Kashiwara, “A Study of Variation of Mixed Hodge Structure”, Publ. RIMS, Kyoto Univ. 22 (1986), pp. 991–1024.

[K]

D.S.

Kubert,

“Product

formulae

on

elliptic

curves”,

Inv. math. 117 (1994), pp. 227–273. [KL]

D.S. Kubert, S. Lang, “Modular units”, Grundlehren der math. Wiss. 244, Springer–Verlag 1981.

[L]

S. Lang, “Elliptic functions”, Addison–Wesley 1973.

[P]

R. Pink, “Arithmetical compactification of Mixed Shimura Varieties”, thesis, Bonner Mathematische Schriften 1989.

[S]

Morihiko Saito, “Modules de Hodge Polarisables”, Publ. RIMS, Kyoto Univ. 24 (1988), pp. 849–995.

[SGA1]

A. Grothendieck et al., “Revˆetements Etales et Groupe Fondamental”, LNM 224, Springer–Verlag 1971.

[SchSch]

N. Schappacher, A.J. Scholl, “The boundary of the Eisenstein symbol”, Math. Ann. 290 (1991), pp. 303–321, 815.

[SZ]

J. Steenbrink, S. Zucker, “Variation of mixed Hodge structure, I”, Inv. math. 80 (1985), pp. 489–542.

[V]

J.–L. Verdier, “Sur les int´egrales attach´ees aux formes automorphes (d’apr`es G. Shimura)”, S´eminaire Bourbaki vol. 1960/61, Expos´e 216.

[W1]

J. Wildeshaus, “Polylogarithmic Extensions on Mixed Shimura varieties”, Schriftenreihe des Mathematischen Instituts der Universit¨at M¨ unster, 3. Serie, Heft 12, 1994.

[W2]

J. Wildeshaus, “Mixed structures on fundamental groups”, preprint, 1994.

[W3]

J. Wildeshaus, “The canonical construction of mixed sheaves on mixed Shimura varieties”, preprint, 1994. 89

[W4]

J. Wildeshaus, “Polylogarithmic Extensions on Mixed Shimura varieties, Part I: Construction and basic properties”, preprint, 1994.

[W5]

J. Wildeshaus, “Polylogarithmic Extensions on Mixed Shimura varieties, Part II: The classical polylogarithm”, preprint, 1994.

[Za]

D. Zagier, “The Bloch–Wigner–Ramakrishnan polylogarithm function”, Math. Ann. 286 (1990), pp. 613–624.

[Z]

S. Zucker, “Hodge theory with degenerating coefficients: L2 – cohomology in the Poincar´e metric”, Ann. of Math. 109 (1979), pp. 415–476.

90