Hodge Cycles on Abelian Varieties

Hodge cycles on abelian varieties

P. Deligne (notes by J.S. Milne)

July 4, 2003; June 1, 2011; October 1, 2018.

Abstract This is a TeXed copy of the article published in Deligne, P.; Milne, J.S.; Ogus, A.; Shih, Kuang-yen. Hodge cycles, motives, and Shimura varieties. LNM 900, Springer, 1982, pp. 9–100, somewhat revised and updated. See the endnotes for more details. The original M.1 article consisted of the notes (written by jsm) of the seminar “Periodes´ des Integrals´ Abeliennes”´ given by P. Deligne at I.H.E.S., 1978-79. This ﬁle is available at www. jmilne.org/math/Documents.

Contents

Introduction...... 2 1 Review of cohomology...... 5 Topological manifolds...... 5 Differentiable manifolds...... 5 Complex manifolds...... 7 Complete smooth varieties...... 7 Applications to periods...... 11 2 Absolute Hodge cycles; principle B...... 15 Definitions (k algebraically closed of finite transcendence degree).... 15 Basic properties of absolute Hodge cycles...... 16 Definitions (arbitrary k)...... 19 Statement of the main theorem...... 19 Principle B...... 19 3 Mumford-Tate groups; principle A...... 22 Characterizing subgroups by their fixed tensors...... 22 Hodge structures...... 23 Mumford-Tate groups...... 24 Principle A...... 26 4 Construction of some absolute Hodge cycles...... 28 Hermitian forms...... 28 Vd 1 Conditions for E H .A;Q/ to consist of absolute Hodge cycles.... 30 5 Completion of the proof for abelian varieties of CM-type...... 36 Abelian varieties of CM-type...... 36 Proof of the main theorem for abelian varieties of CM-type...... 37

1 CONTENTS 2

6 Completion of the proof; consequences...... 41 Completion of the proof of Theorem 2.11...... 41 Consequences of Theorem 2.11...... 42 7 Algebraicity of values of the -function...... 44 The Fermat hypersurface...... 45 Calculation of the cohomology...... 46 The action of Gal.Q=k/ on the etale´ cohomology...... 49 Calculation of the periods...... 51 The theorem...... 53 Restatement of the theorem...... 55 References...... 57 M Endnotes (by J.S. Milne)...... 60

Introduction

Let X be a smooth projective variety over C. Hodge conjectured that certain cohomology classes on X are algebraic. The main result proved in these notes shows that, when X is an abelian variety, the classes considered by Hodge have many of the properties of algebraic classes. In more detail, let X an be the complex analytic manifold associated with X, and consider n the singular cohomology groups H .X an;Q/. The variety X an being of Kahler¨ type (every n projective embedding deﬁnes a Kahler¨ structure), its cohomology groups H .X an;C/ n ' H .X an;Q/ C have canonical decompositions ˝ n an M p;q p;q def q an p H .X ;C/ H ;H H .X ;˝ an /. D D X p q n C D 2p The cohomology class cl.Z/ H .X an;C/ of an algebraic subvariety Z of codimension p 2 2p p;p in X is rational (i.e., it lies in H .X an;Q// and is of bidegree .p;p/ (i.e., it lies in H ). The Hodge conjecture states that, conversely, every element of

2p an p;p H .X ;Q/ H \ is a Q-linear combination of the classes of algebraic subvarieties. Since the conjecture is unproven, it is convenient to call these rational .p;p/-classes Hodge cycles on X. Now consider a smooth projective variety X over a ﬁeld k that is of characteristic zero, algebraically closed, and small enough to be embeddable in C. The algebraic de Rham n cohomology groups H .X=k/ have the property that, for any embedding k , C, there dR W ! are canonical isomorphisms

n n an n an H .X=k/ k; C H .X ;C/ H .X ;C/: dR ˝ ' dR ' 2p It is natural to say that t HdR .X=k/ is a Hodge cycle on X relative to if its image in 2p an p 2 H .X ;C/ is .2i/ times a Hodge cycle on X k; C. The arguments in these notes ˝ 2p show that, if X is an abelian variety, then an element of HdR .X=k/ that is a Hodge cycle on X relative to one embedding of k into C is a Hodge cycle relative to all embeddings; further, p 2p for any embedding, .2i/ times a Hodge cycle in H .X an;C/ always lies in the image CONTENTS 3

2p 1 of HdR .X=k/. Thus the notion of a Hodge cycle on an abelian variety is intrinsic to the variety: it is a purely algebraic notion. In the case that k C the theorem shows that the D image of a Hodge cycle under an automorphism of C is again a Hodge cycle; equivalently, the notion of a Hodge cycle on an abelian variety over C does not depend on the map X SpecC. Of course, all this would be obvious if only one knew the Hodge conjecture. ! In fact, a stronger result is proved in which a Hodge cycle is defined to be an element of n Q n H .X/ H .Xet;Ql /. As the title of the original seminar suggests, the stronger result dR l has consequences for the algebraicity of the periods of abelian integrals: briefly, it allows one to prove all arithmetic properties of abelian periods that would follow from knowing the Hodge conjecture for abelian varieties. M.2 —————————————————- In more detail, the main theorem proved in these notes is that every Hodge cycle on an abelian variety (in characteristic zero) is an absolute Hodge cycle — see 2 for the definitions and Theorem 2.11 for a precise statement of the result. The proof is based on the following two principles.

A. Let t1;:::;tN be absolute Hodge cycles on a smooth projective variety X and let G be the largest algebraic subgroup of GL.H .X;Q// GL.Q.1// ﬁxing the ti ; then every cohomology class t on X ﬁxed by G is an absolute Hodge cycle (see 3.8).

B. If .Xs/s S is an algebraic family of smooth projective varieties with S connected and 2 smooth and .ts/s S is a family of rational cycles (i.e., a global section of . . . ) such 2 that ts is an absolute Hodge cycle for one s, then ts is an absolute Hodge cycle for all s (see 2.12, 2.15). Every abelian variety A with a Hodge cycle t is contained in a smooth algebraic family in which t remains Hodge and which contains an abelian variety of CM-type. Therefore, Principle B shows that it suffices to prove the main theorem for A an abelian variety of CM-type (see 6). Fix a CM-field E, which we can assume to be Galois over Q, and let ˙ be a set of representatives for the E-isogeny classes over C of abelian varieties with complex multiplication by E. Principle B is used to construct some absolute Hodge classes L on A ˙ A — the principle allows us to replace A by an abelian variety of the form 2 ˚ A0 Z OE (see 4). Let G GL. A ˙ H1.A;Q// GL.Q.1// be the subgroup fixing the ˝ ˚ 2 absolute Hodge cycles just constructed plus some other (obvious) absolute Hodge cycles. It is shown that G fixes every Hodge cycle on A, and Principle A therefore completes the proof (see 5). On analyzing which properties of absolute Hodge cycles are used in the above proof, one arrives at a slightly stronger result.2 Call a rational cohomology class c on a smooth projective complex variety X accessible if it belongs to the smallest family of rational cohomology classes such that: (a) the cohomology class of every algebraic cycle is accessible; (b) the pull-back by a map of varieties of an accessible class is accessible; 2p (c) if t1;:::;tN H .X;Q/ are accessible, and if a rational class t in some H .X;Q/ 2 is fixed by an algebraic subgroup G of Aut.H .X;Q// (automorphisms of H .X;Q/ as a graded algebra) fixing the ti , then t is accessible;

1Added(jsm). This doesn’t follow directly from Theorem 2.11 (see 2.4). However, one obtains a variant of Theorem 2.11 using the above deﬁnitions simply by dropping the etale´ component everywhere in the proof (see, for example, 2.10b). 2Added(jsm). See also 9 of Milne, Shimura varieties and moduli, 2013. CONTENTS 4

(d) Principle B holds with “absolute Hodge” replaced by “accessible”. Sections 4,5,6 of these notes can be interpreted as proving that, when X is an abelian variety, every Hodge cycle is accessible. Sections 2,3 deﬁne the notion of an absolute Hodge cycle and show that the family of absolute Hodge cycles satisﬁes (a), (b), (c), and (d); therefore, an accessible class is absolutely Hodge. We have the implications: M.3

abelian varieties trivial Hodge accessible absolutely Hodge Hodge. HHHHHHHH) HHHH) HHHH) Only the ﬁrst implication is restricted to abelian varieties. The remaining three sections, 1 and 7, serve respectively to review the different cohomology theories and to give some applications of the main results to the algebraicity of products of special values of the -function.

NOTATION

We define C to be the algebraic closure of R and i C to be a square root of 1; thus i 2 is only defined up to sign. A choice of i p 1 determines an orientation of C as a real D manifold — we take that for which 1 i > 0 — and hence an orientation of every complex ^ manifold. Complex conjugation on C is denoted by or by z z. 7! Recall that the category of abelian varieties up to isogeny is obtained from the category of abelian varieties by taking the same class of objects but replacing Hom.A;B/ with Hom.A;B/ Q. We shall always regard an abelian variety as an object in the category of ˝ abelian varieties up to isogeny: thus Hom.A;B/ is a vector space over Q. If .V˛/ is a family of representations of an algebraic group G over a field k and t V˛, ˛;ˇ 2 then the subgroup of G fixing the t˛;ˇ is the algebraic subgroup H of G such that, for all k-algebras R,

H.R/ g G.R/ g.t 1/ t 1, all ˛;ˇ . D f 2 j ˛;ˇ ˝ D ˛;ˇ ˝ g

Linear duals are denoted by _. If X is a variety over a ﬁeld k and is a homomorphism k , k , then X denotes the variety X k ( X Spec.k /). W ! 0 ˝k; 0 D Spec.k/ 0 By a b we mean that a is sufﬁciently greater than b. 1 REVIEW OF COHOMOLOGY 5

1 Review of cohomology

Topological manifolds Let X be a topological manifold and F a sheaf of abelian groups on X. We set

H n.X;F / H n. .X;F // D where F F is any acyclic resolution of F . This defines H n.X;F / uniquely up to a ! unique isomorphism. When F is the constant sheaf defined by a field K, these groups can be identified with singular cohomology groups as follows. Let S .X;K/ be the complex in which Sn.X;K/ is the K-vector space with basis the singular n-simplices in X and the boundary map sends a simplex to the (usual) alternating sum of its faces. Set

S .X;K/ Hom.S .X;K/;K/ D with the boundary map for which

.˛;/ ˛./ S .X;K/ S .X;K/ K 7! W ˝ ! is a morphism of complexes, namely, that deﬁned by

.d˛/./ . 1/deg.˛/ 1˛.d/: D C PROPOSITION 1.1. There is a canonical isomorphism H n.S .X;K// H n.X;K/. ! PROOF. If U is the unit ball, then H 0.S .U;K// K and H n.S .U;K// 0 for n > 0. D D Thus, K S .U;K/ is a resolution of the group K. Let Sn be the sheaf of X associated ! with the presheaf V S n.V;K/. The last remark shows that K S is a resolution of 7! ! the sheaf K. As each Sn is ﬁne (Warner 1971, 5.32), H n.X;K/ H n. .X;S //. But the ' obvious map S .X;K/ .X;S / is surjective with an exact complex as kernel (loc. cit.), ! and so H n.S .X;K// ' H n. .X;S // H n.X;K/. ! '

Differentiable manifolds Now assume X is a differentiable manifold. On replacing “singular n-simplex” by “differentiable singular n-simplex” in the above deﬁnitions, one obtains complexes S 1.X;K/ and S .X;K/. The same argument shows that there is a canonical isomorphism 1 n def n n H .X;K/ H .S 1.X;K// ' H .X;K/ 1 D ! (Warner 1971, 5.32). n Let OX be the sheaf of C 1 real-valued functions on X, let ˝ be the OX -module 1 X 1 1 of C 1 differential n-forms on X, and let ˝ be the complex X 1

d 1 d 2 d OX ˝X ˝X : 1 ! 1 ! 1 ! 1 REVIEW OF COHOMOLOGY 6

The de Rham cohomology groups of X are deﬁned to be

n n closed n-forms HdR.X/ H . .X;˝X // f g: D 1 D exact n-forms f g 0 n If U is the unit ball, Poincare’s´ lemma shows that HdR.U / R and HdR.U / 0 for n > 0. D Dn Thus, R ˝X is a resolution of the constant sheaf R, and as the sheaves ˝X are ﬁne ! 1 n n 1 (Warner 1971, 5.28), we have H .X;R/ HdR.X/. n ' For ! .X;˝X / and Sn1.X;R/, deﬁne 2 1 2

n.n 1/ Z !; . 1/ 2C ! R. h i D 2 Stokes’s theorem states that R d! R !, and so D d d!; . 1/n !;d 0. h i C h i D The pairing ; therefore deﬁnes a map of complexes h i

f .X;˝X / S .X;R/. W 1 ! 1 n n THEOREM 1.2 (DE RHAM). The map H .X/ H .X;R/ deﬁned by f is an isomor- dR ! phism for all n. 1

PROOF. The map is inverse to the map

n n n H .X;R/ ' H .X;R/ HdR.X/ 1 ! ' deﬁned in the previous two paragraphs (Warner 1971, 5.36). (Our signs differ from the usual signs because the standard sign conventions Z Z Z Z Z d! !; pr1! pr2 ! ; etc. D d X Y ^ D X Y violate the sign conventions for complexes.) R A number !, Hn.X;Q/, is called a period of !. The map in (1.2) identiﬁes n 2 H .X;Q/ with the space of classes of closed forms whose periods are all rational. Theorem 1.2 can be restated as follows: a closed differential form is exact if all its periods are zero; there exists a closed differential form having arbitrarily assigned periods on an independent set of cycles.

REMARK 1.3 (SINGERAND THORPE 1967, 6.2). If X is compact, then it has a smooth triangulation T . Deﬁne S .X;T;K/ and S .X;T;K/ as before, but using only simplices in T . Then the map

.X;˝X / S .X;T;K/ 1 ! deﬁned by the same formulas as f above induces isomorphisms

H n .X/ H n.S .X;T;K//. dR ! 1 REVIEW OF COHOMOLOGY 7

Complex manifolds

Now let X be a complex manifold, and let ˝X an denote the complex

d 1 d 2 d OX an ˝ an ˝ an ! X ! X ! n in which ˝X an is the sheaf of holomorphic differential n-forms. Thus, locally a section of n ˝X an is of the form X ! ˛i :::i dzi ::: dzi D 1 n 1 ^ ^ n with ˛i1:::in a holomorphic function and the zi complex local coordinates. The complex form of Poincare’s´ lemma shows that C ˝ an is a resolution of the constant sheaf C, and ! X so there is a canonical isomorphism

n n H .X;C/ H .X;˝ an / (hypercohomology). ! X If X is a compact Kahler¨ manifold, then the spectral sequence

p;q q p p q E H .X;˝ an / H C .X;˝ an / 1 D X H) X degenerates, and so provides a canonical splitting M.4

n M q p H .X;C/ H .X;˝ an / (the Hodge decomposition) D X p q n C D p;q def q p q;p as H H .X;˝X an / is the complex conjugate of H relative to the real structure n D n H .X;R/ C H .X;C/ (Weil 1958). The decomposition has the following explicit ˝ ' description: the complex ˝X C of sheaves of complex-valued differential forms on 1 ˝ n the underlying differentiable manifold is an acyclic resolution of C, and so H .X;C/ n D H . .X;˝X C//; Hodge theory shows that each element of the second group is 1 ˝ represented by a unique harmonic n-form, and the decomposition corresponds to the decomposition of harmonic n-forms into sums of harmonic .p;q/-forms, p q n. 3 C D Complete smooth varieties

Finally, let X be a complete smooth variety over a field k of characteristic zero. If k C, n D then X defines a compact complex manifold X an, and there are therefore groups H .X an;Q/, n depending on the map X Spec.C/, that we shall write H .X/ (here B abbreviates Betti). ! B If X is projective, then the choice of a projective embedding determines a Kahler¨ structure on X an, and hence a Hodge decomposition (which is independent of the choice of the embedding because it is determined by the Hodge filtration, and the Hodge filtration depends only on X; see Theorem 1.4 below). In the general case, we refer to Deligne 1968, 5.3, 5.5, for the existence of the decomposition. n n For an arbitrary field k and an embedding k , C, we write H .X/ for HB .X/ p;q p;q W ! an an and H .X/ for H .X/. As defines a homeomorphism X X , it induces an n n ! isomorphism H .X/ H .X/. Sometimes, when k is given as a subfield of C, we write n n ! HB .X/ for HB .XC/.

3For a recent account of Hodge theory, see C. Voisin, Hodge Theory and Complex Algebraic Geometry, I, CUP, 2002. 1 REVIEW OF COHOMOLOGY 8

n Let ˝X=k denote the complex in which ˝X=k is the sheaf of algebraic differen- n tial n-forms, and deﬁne the (algebraic) de Rham cohomology group HdR.X=k/ to be n H .XZar;˝X=k / (hypercohomology with respect to the Zariski cohomology). For any homomorphism k , k , there is a canonical isomorphism W ! 0 H n .X=k/ k H n .X k =k /: dR ˝k; 0 ! dR ˝k 0 0 The spectral sequence

p;q q p p q E H .XZar;˝ / H C .XZar;˝ / 1 D X=k H) X=k p n n defines a filtration (the Hodge filtration) F HdR.X/ on HdR.X/ which is stable under base change.

THEOREM 1.4. When k C the obvious maps D an X XZar; ˝ an ˝ ; ! X X induce isomorphisms n n an n an H .X/ H .X / H .X ;C/ dR ! dR ' p n p n def M p ;q under which F H .X/ corresponds to F H .X an;C/ H 0 0 . dR D p p, p q n 0 0C 0D PROOF. The initial terms of the spectral sequences

p;q q p p q E H .XZar;˝ / H C .XZar;˝ / 1 D X=k H) X=k p;q q p p q E H .X;˝ an / H C .X;˝ an / 1 D X H) X are isomorphic — see Serre 1956 for the projective case and Grothendieck 1966 for the general case. The theorem follows from this because, by definition of the Hodge decomposition, n an the filtration of HdR.X / defined by the above spectral sequence is equal to the filtration of n H .X an;C/ defined in the statement of the theorem.

It follows from the theorem and the discussion preceding it that every embedding k , C defines an isomorphism W ! n n H .X/ k; C ' H .X/ C dR ˝ ! ˝Q n and, in particular, a k-structure on H .X/ Q C. When k Q, this structure should be ˝ n D distinguished from the Q-structure defined by H .X/: the two are related by the periods. When k is algebraically closed, we write H n.X; /, or H n.X/, for H n.X ; / Af et et ZO Z n n ˝ , where H .Xet; / lim H .Xet; =m / (etale´ cohomology). If X is connected, Q ZO m Z Z 0 D H .X;Af / Af , the ring of finite adeles` for Q, which justifies the first notation. By D definition, H n.X/ depends only on X (and not on its structure morphism X Speck). et ! The map H n.X/ H n.X k / defined by an inclusion k , k of algebraically closed et ! et ˝k 0 ! 0 fields is an isomorphism (special case of the proper base change theorem Artin et al. 1973, XII). The comparison theorem (ibid. XI) shows that, when k C, there is a canonical n n n D isomorphism H .X/ Af H .X/. It follows that H .X/ Af is independent of the B ˝ ! et B ˝ morphism X SpecC, and that, over any algebraically closed field of characteristic zero, n ! Het .X/ is a free Af -module. 1 REVIEW OF COHOMOLOGY 9

n The Af -module H .X;Af / can also be described as the restricted product of the spaces n n H .X;Ql /, l a prime number, with respect to the subspaces H .X;Zl /= torsion . f g Next we define the notion of the “Tate twist” in each of the three cohomology theories. n n m For this we shall define objects Q.1/ and set H .X/.m/ H .X/ Q.1/˝ . We want 2 1 D ˝ Q.1/ to be H .P / (realization of the Tate motive in the cohomology theory), but to avoid the possibility of introducing sign ambiguities we shall define it directly,

QB.1/ 2iQ D def r Qet.1/ Af .1/ .limr / Q; r k 1 D D ˝Z D f 2 j D g r QdR.1/ k; D and so

n n m n an m H .X/.m/ H .X/ .2i/ Q H .X ;.2i/ Q/ .k C/ B D B ˝Q D D n n m n m Á H .X/.m/ H .X/ f .Af .1//˝ lim r H .Xet;r˝ / Q .k alg. closed) et D et ˝A D ˝Z H n .X/.m/ H n .X/. dR D dR These deﬁnitions extend in an obvious way to negative m. For example, we set Qet. 1/ D HomAf .Af .1/;Af / and deﬁne

n n m H .X/. m/ H .X/ Qet. 1/˝ : et D et ˝ There are canonical isomorphisms

QB.1/ Af Qet.1/ (k C, k algebraically closed/ ˝Q ! QB.1/ C QdR.1/ k C (k C) ˝ ! ˝ and hence canonical isomorphisms (the comparison isomorphisms)

n n HB .X/.m/ Q Af Het .X/.m/ (k C, k algebraically closed/ n ˝ ! n H .X/.m/ C H .X/.m/ k C (k C). B ˝Q ! dR ˝ To define the first, note that exp defines an isomorphism

z z e 2iZ=r2iZ r : 7! W ! After passing to the inverse limit over r and tensoring with Q, we obtain the required isomorphism 2iAf Af .1/. The second isomorphism is induced by the inclusions ! 2iQ , C - k: ! Although the Tate twist for de Rham cohomology is trivial, it should not be ignored. For example, when k C, D m n 1 .2i/ n HB .X/ C 7! HB .X/.m/ C ˝ ' ˝ ' ' n n HdR.X/ HdR.X/.m/ 1 REVIEW OF COHOMOLOGY 10 fails to commute by a factor .2i/m. Moreover when m is odd the top isomorphism is defined only up to sign. In each cohomology theory there is a canonical way of attaching a class cl.Z/ in H 2p.X/.p/ to an algebraic cycle Z on X of pure codimension p. Since our cohomology groups are without torsion, we can do this using Chern classes (Grothendieck 1958). Starting 2 with a functorial isomorphism c1 Pic.X/ H .X/.1/, one uses the splitting principle to W ! define the Chern polynomial P p 2p ct .E/ cp.E/t ; cp.E/ H .X/.p/; D 2 of a vector bundle E on X. The map E ct .E/ is additive, and therefore factors through 7! the Grothendieck group of the category of vector bundles on X. But, as X is smooth, this group is the same as the Grothendieck group of the category of coherent OX -modules, and we can therefore define 1 cl.Z/ cp.OZ/ D .p 1/Š (loc. cit. 4.3). In defining c1 for the Betti and etale´ theories, we begin with maps 2 an Pic.X/ H .X ;2iZ/ ! 2 Pic.X/ H .Xet;r / ! arising as connecting homomorphisms from the sequences exp 0 2i OX an O an 0 ! ! ! X ! r 0 r O O 0: ! ! X ! X ! For the de Rham theory, we note that the d log map, f df , defines a map of complexes 7! f 0 O 0 X dlog

d 1 d 2 d OX ˝ ˝ X X and hence a map 1 2 Pic.X/ H .X;O / H .X;0 O / ' X ' ! X ! 2 2 2 H .X;˝ / H .X/ H .X/.1/ ! X D dR D dR whose negative is c1. It can be checked that the three maps c1 are compatible with the comparison isomorphisms (Deligne 1971a, 2.2.5.1), and it follows formally that the maps cl are also compatible once one has checked that the Gysin maps and multiplicative structures are compatible with the comparison isomorphisms. When k C, there is a direct way of deﬁning a class cl.Z/ H2d 2p.X.C/;Q/ D 2 (singular cohomology, d dim.X/, p codim.Z//: the choice of an i p 1 deter- D D D mines an orientation of X and of the smooth part of Z, and there is therefore a topologically deﬁned class cl.Z/ H2d 2p.X.C/;Q/. This class has the property that for 2d 2p 2d2 2p Œ! H .X 1;R/ H . .X;˝X // represented by the closed form !, 2 D 1 Z cl.Z/;Œ! !. h i D Z 1 REVIEW OF COHOMOLOGY 11

2p By Poincare´ duality, cl.Z/ corresponds to a class cltop.Z/ HB .X/, whose image in 2p p 2 H .X/.p/ under the map induced by 1 .2i/ Q Q.p/ is known to be clB.Z/. The B 7! W ! above formula becomes Z Z cltop.Z/ Œw !. X [ D Z There are trace maps (d dimX) D 2d TrB H .X/.d/ ' Q W B ! 2d Tret H .X/.d/ ' Af W et ! 2d TrdR H .X/.d/ ' k W dR ! that are determined by the requirement that Tr.cl.point// 1. They are compatible with D the comparison isomorphisms. When k C, TrB and TrdR are equal respectively to the D composites

d R 2d .2i/ 1 2d 2d Œ! X ! H .X/.d/ 7! H .X/ ' H . .˝ // 7! C B B X 1 1 R Œ! X ! 2d 2d 2d 7! .2i/d H .X/.d/ H .X/ ' H . .˝ // C dR dR X 1 where we have chosen an i and used it to orientate X (the composite maps are obviously independent of the choice of i). The formulas of the last paragraph show that 1 Z Tr .cl .Z/ Œ!/ !: dR dR dimZ [ D .2i/ Z

A deﬁnition of Tret can be found in Milne 1980, VI 11.

Applications to periods We now deduce some consequences concerning periods.

PROPOSITION 1.5. Let X be a complete smooth variety over an algebraically closed ﬁeld k C and let Z be an algebraic cycle on XC of dimension r. For any C 1 differential 2r 2r r-form ! on XC whose class Œ! in HdR .XC/ lies in HdR .X/ Z ! .2i/r k: Z 2

PROOF. We first note that Z is algebraically equivalent to a cycle Z0 defined over k. In proving this, we can assume Z to be prime. There exists a smooth variety T over k, a subvariety Z X T that is flat over T , and a point SpecC T such that Z Z T SpecC ! D in X T SpecC XC. We can therefore take Z0 to be Z T Speck X T Speck X D 2r D for any point Speck T . From this it follows that cldR.Z/ cldR.Z0/ HdR .X/.r/ and ! R D r 2 TrdR.cldR.Z/ Œ!/ k. But we saw above that ! .2i/ TrdR.cldR.Z/ Œ!/. [ 2 Z D [ We next derive a classical relation between the periods of an elliptic curve. For a complete smooth curve X and an open affine subset U , the map

1 1 1 .U;˝X / meromorphic diffls, holomorphic on U HdR.X/ HdR.U / f g ! D d .U;OX / D exact differentials on U f g 1 REVIEW OF COHOMOLOGY 12 is injective with image the set of classes represented by forms whose residues are all zero (such forms are said to be of the second kind). When k C, TrdR.Œ˛ Œˇ/, where ˛ and ˇ D [ are differential 1-forms of the second kind, can be computed as follows. Let ˙ be the finite set of points where ˛ or ˇ has a pole. For z a local parameter at P ˙, ˛ can be written 2 X i ˛ ai z dz with a 1 0: D D i< 1 1 There therefore exists a meromorphic function f defined near P such that df ˛. We R D R write ˛ for any such function — it is defined up to a constant. As ResP ˇ 0, ResP . ˛/ˇ D is well-defined, and one proves that X R TrdR.Œ˛ Œˇ/ ResP ˛ ˇ. [ D P ˙ 2 Now let X be the elliptic curve

2 3 2 3 y z 4x g2xz g3z : D There is a lattice in C and corresponding Weierstrass function }.z/ such that z .}.z/ } .z/ 1/ 7! W 0 W defines an isomorphism C= X.C/. Let 1 and 2 be generators of such that the bases ! 1; 2 and 1;i of C have the same orientation. We can regard 1 and 2 as elements of f g f g H1.X;Z/, and then 1 2 1. The differentials ! dx=y and xdx=y on X pull back D D D to dz and }.z/dz respectively on C. The first is therefore holomorphic and the second has a single pole at .0 1 0/ on X with residue zero (because 0 C maps to X and 1 12 D W W 2 1 2 }.z/ a2z :::). We find that D z2 C C ÂZ Ã TrdR.Œ! Œ/ Res0 dz }.z/dz Res0.z}.z/dz/ 1. [ D D D For i 1;2, let D Z Z dx def dx !i p 3 i y D i 4x g2x g3 D Z Z x dx def xdx i p 3 i y D i 4x g2x g3 D be the periods of ! and . Under the map

1 1 H .X/ H .X;C/ dR ! ! maps to !1 !2 and maps to 1 2 , where ; is the basis dual to 10 C 20 10 C 20 f 10 20 g 1; 2 . Thus f g 1 TrdR.Œ! Œ/ D [ TrB..!1 !2 / .1 2 // D 10 C 20 [ 10 C 20 .!12 !21/TrB. / D 10 [ 20 1 .!12 !21/: D 2i 1 REVIEW OF COHOMOLOGY 13

Hence !12 !21 2i. D This is the Legendre relation. The next proposition shows how the existence of algebraic cycles can force algebraic relations between the periods of abelian integrals. Let X be an abelian variety over a subfield k of C. In each of the three cohomology theories, ^r H r .X/ H 1.X/ D and H 1.X X / H 1.X/ H 1.X/ D ˚ ˚ 1 1 Let Gm.Q/ act on QB.1/ as . There is then a natural action of GL.HB .X// Gm on r 2n 1 HB .X /.m/ for any r;n; and m. We define G to be the subgroup of GL.HB .X// Gm fixing n all the tensors of the form clB.Z/, Z an algebraic cycle on some X (see the Notations). Consider the comparison isomorphisms

1 1 an 1 H .X/ k C ' H .X ;C/ ' H .X/ C. dR ˝ ! B ˝Q

The periods pij of X are deﬁned by the equations X ˛i pj i aj D 1 1 where ˛i and ai are bases for H .X/ and H .X/ over k and Q respectively. The ﬁeld f g f g dR B k.pij / generated over k by the pij is independent of the bases chosen.

PROPOSITION 1.6. With the above deﬁnitions, the transcendence degree of k.pij / over k is dim.G/. Ä PROOF. We can replace k by its algebraic closure in C, and hence assume that each algebraic cycle on XC is equivalent to an algebraic cycle on X (see the proof of 1.5). Let P be the functor of k-algebras whose value on R is the set of isomorphisms p H 1 R H 1 R W B ˝Q ! dR ˝k mapping clB.Z/ 1 to cldR.Z/ 1 for all algebraic cycles Z on a power of X. When ˝ ˝ R C, the comparison isomorphism is such a p, and so P.C/ is not empty. It is easily seen D that P is represented by an algebraic variety that becomes a Gk-torsor under the obvious action. The bases ˛i and ai can be used to identify the points of P with matrices. The f g f g matrix pij is a point of P with coordinates in C, and so the proposition is a consequence of the following easy lemma.

N N LEMMA 1.7. Let A be the afﬁne N -space over a subﬁeld k of C, and let z A .C/. The 2 transcendence degree of k.z1;:::;zN / over k is the dimension of the Zariski closure of z N f g in A .

PROOF. Let a be the kernel of the homomorphism kŒT1;:::;TN kŒz1;:::;zN sending N ! Ti to zi . The Zariski closure of z in A is the zero set Z.a/ of a, and Z.a/ is a variety f g over k with function ﬁeld k.z1;:::;zN /. Now dim.Z.a// tr. deg. k.z1;:::;zN / (standard D k result). 1 REVIEW OF COHOMOLOGY 14

REMARK 1.8. If X is an elliptic curve, then dimG is 2 or 4 according as X has complex multiplication or not. Chudnovsky has shown that

tr. deg. k.pij / dimG k D when X is an elliptic curve with complex multiplication. Does equality hold for all abelian varieties? M.5

One of the main purposes of the seminar was to show that, in the case that X is an abelian variety, (1.5) and (1.6) make sense, and remain true, if “algebraic cycle” is replaced by “Hodge cycle”. 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 15

2 Absolute Hodge cycles; principle B

Deﬁnitions (k algebraically closed of ﬁnite transcendence degree)

Let k be an algebraically closed ﬁeld of ﬁnite transcendence degree over Q, and let X be a complete smooth variety over k. Set H n.X/.m/ H n .X/.m/ H n.X/.m/ A D dR et — it is a free k Af -module. Corresponding to an embedding k , C, there are canonical W ! isomorphisms

n n H .X/.m/ k; C ' H .X/.m/ dRW dR ˝ ! dR H n.X/.m/ ' H n.X/.m/ etW et ! et whose product we write . The diagonal embedding H n.X/.m/ , H n .X/.m/ H n.X/.m/ ! dR et induces an isomorphism

n n n H .X/.m/ .C Af / ' H .X/.m/ H .X/.m/ ˝ ! dR et (product of the comparison isomorphisms, 1). An element t H 2p.X/.p/ is a Hodge 2 A cycle relative to if 2p (a) t is rational relative to , i.e., .t/ lies in the rational subspace H .X/.p/ of H 2p.X/.p/ H 2p.X/.p/; dR et (b) the ﬁrst component of t lies in F 0H 2p.X/.p/ def F pH 2p.X/. dR D dR 2p Equivalent condition: .t/ lies in H .X/.p/ and is of bidegree .0;0/. If t is a Hodge cycle relative to every embedding k , C, then it is called an absolute Hodge cycle. W ! EXAMPLE 2.1. (a) For any algebraic cycle Z on X, t .cldR.Z/;clet.Z// is an abso- D lute Hodge cycle — the Hodge conjecture predicts there are no others. Indeed, for any k , C, .t/ clB.Z/, and is therefore rational, and it is well-known that W ! D 2p cldR.Z/ is of bidegree .p;p/ in HdR .X/. (b) Let X be a complete smooth variety of dimension d, and consider the diagonal X X. Corresponding to the decomposition 2d M H 2d .X X/.d/ H 2d i .X/ H i .X/.d/ D ˝ i 0 D we have P2d i cl./ i 0 : D D The i are absolute Hodge cycles. 2 (c) Suppose that X is given with a projective embedding, and let HdR.X/.1/ 2 2 Het.X/.1/ be the class of a hyperplane section. The hard Lefschetz theorem states that x d 2p x H 2p.X/.p/ H 2d 2p.X/.d p/; 2p d; 7! W ! Ä is an isomorphism. The class x is an absolute Hodge cycle if and only if d 2p x is an absolute Hodge cycle. 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 16

Loosely speaking, every cycle constructed from a set of absolute Hodge cycles by a canonical rational process will again be an absolute Hodge cycle.

OPEN QUESTION 2.2. Does there exist a cycle rational for every but which is not absolutely Hodge? M.6

More generally, consider a family .X˛/˛ A of complete smooth varieties over a ﬁeld k .A/ .A/2 (as above). Let .m.˛// N , .n.˛// N , and m Z, and write 2 2 2 ! ! O m.˛/ O n.˛/ TdR HdR .X˛/ HdR .X˛/_ .m/ D ˛ ˝ ˛ ! ! O m.˛/ O n.˛/ Tet Het .X˛/ Het .X˛/_ .m/ D ˛ ˝ ˛

TA TdR Tet D ! ! O m.˛/ O n.˛/ T H .X˛/ H .X˛/_ .m/ . k , C/: D ˛ ˝ ˛ W ! Then we say that t T is 2 A rational relative to if its image in TA k Af ;.;1/ C Af lies in T , ˘ ˝ a Hodge cycle relative to if it is rational relative to and its ﬁrst component lies in ˘ F 0, and an absolute Hodge cycle if it is a Hodge cycle relative to every . ˘ Note that, in order for there to exist Hodge cycles in TA, it is necessary that

Pm.˛/ Pn.˛/ 2m. D EXAMPLE 2.3. Cup product deﬁnes maps

m;n m ;n m m ;n n T .p/ T 0 0 .p / T C 0 C 0 .p p /; A A 0 ! A C 0 and hence an element of T T T , which is an absolute Hodge cycle. A_ ˝ A_ ˝ A 0 2p 2p OPEN QUESTION 2.4. Let t F H .X/.p/. If .t/ H .X/./ for all k , C, 2 dR dR 2 W ! is t necessarily the ﬁrst component of an absolute Hodge cycle?

Basic properties of absolute Hodge cycles In order to develop the theory of absolute Hodge cycles, we shall need to use the Gauss- Manin connection (Katz and Oda 1968; Katz 1970; Deligne 1971b). Let k0 be a field of characteristic zero and let S be a smooth k0-scheme (or the spectrum of a finitely generated field over k0). A k0-connection on a coherent OS -module E is a homomorphism of sheaves of abelian groups 1 E ˝ O E rW ! S=k0 ˝ S such that .fe/ df e f .e/ r D ˝ C r 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 17

for local sections f of OS and e of E. The kernel of , E , is the sheaf of horizontal r r sections of .E; /. Every k0-connection can be extended to a homomorphism of abelian r r sheaves,

n n 1 n ˝ O E ˝ C O E; r W S=k0 ˝ S ! S=k0 ˝ S ! e d! e . 1/n! .e/ ˝ 7! ˝ C ^ r and is said to be integrable if 1 0. Moreover, gives rise to an OS -linear map r r ı r D r

D D Der.S=k0/ End .E/ 7! r W ! k0 where D is the composite r 1 D 1 E r ˝ O E ˝ OS O E E: ! S=k0 ˝ S ! ˝ S '

Note that D.fe/ D.f /e f D.e/. One checks that is integrable if and only if r D C r r D D is a Lie algebra homomorphism. 7! r Now consider a proper smooth morphism X S of smooth varieties, and write n n W ! HdR.X=S/ for R .˝X=S /. This is a locally free sheaf of OS -modules with a canonical connection ; called the Gauss-Manin connection, which is integrable. It therefore deﬁnes r a Lie algebra homomorphism

n Der.S=k0/ End .H .X=S//: ! k0 dR

If k0 , k0 is an inclusion of fields and X 0=S 0 .X=S/ k0 k0 , then the Gauss-Manin ! n D ˝ connection on H .X 0=S 0/ is 1. In the case that k0 C, the relative form of Serre’s dR r ˝ D GAGA theorem (Serre 1956) shows that Hn .X=S/an Hn .X an=X an/ and gives rise to dR ' dR r a connection an on Hn .X an=S an/. The relative Poincare´ lemma shows that r dR n n an an .R C/ OS an ' HdR.X =S /; ˝ ! and it is known that an is the unique connection such that r n n an an an R C ' HdR.X =S /r . ! PROPOSITION 2.5. Let k0 C have finite transcendence degree over Q, and let X be a complete smooth variety over a field k that is finitely generated over k0. Let be the n r Gauss-Manin connection on HdR.X/ relative to the composite X Speck Speck0. If n ! ! t H .X/ is rational relative to all embeddings of k into C, then t is horizontal for , i.e., 2 dR r t 0. r D

PROOF. Choose a regular k0-algebra A of ﬁnite-type and a smooth proper map XA W ! SpecA whose generic ﬁbre is X Speck and which is such that t extends to an element of n ! .SpecA;H .X=SpecA/. After a base change relative to k0 , C, we obtain maps dR !

XS S SpecC;S SpecA ; ! ! D C n an an and a global section t 0 t 1 of HdR.XS =S /. We have to show that . 1/t 0 0, or D ˝ n an def n an r ˝ D equivalently, that t 0 is a global section of H .XS ;C/ R C. D 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 18

An embedding k , C gives rise to an injection A, C (i.e., a generic point of SpecA W ! ! in the sense of Weil) and hence a point s of S. The hypotheses show that, at each of these n an n an n an an points, t.s/ H .Xs ;Q/ HdR.Xs /. Locally on S, HdR.Xs =S / will be the sheaf 2 n n of holomorphic sections of the trivial bundle S C and H .X an;C/ will be the sheaf of locally constant sections. Thus, locally, t 0 is a function m s .t1.s/;:::;tm.s// S S C : 7! W ! Each ti .s/ is a holomorphic function which, by hypothesis, takes real (even rational) values on a dense subset of S. It is therefore constant.

REMARK 2.6. In the situation of (2.5), assume that t H n .X/ is rational relative to one 2 dR and horizontal for . An argument similar to the above then shows that t is rational relative r to all embeddings that agree with on k0.

COROLLARY 2.7. Let k0 k be algebraically closed fields of finite transcendence degree n over Q, and let X be a complete smooth variety over k0. If t HdR.Xk/ is rational relative 2 n n to all k , C, then it is defined over k0, i.e., it is in the image of H .X/ H .Xk/. W ! dR ! dR

PROOF. Let k be a subﬁeld of k which is ﬁnitely generated over k0 and such that t 0 2 H n .X k /. The hypothesis implies that t 0 where is the Gauss-Manin connection dR ˝k0 0 r D r for X Speck Speck . Thus, for any D Der.k =k /, .t/ 0. But X arises k0 0 0 0 0 D k0 ! ! 2 n r n D from a variety over k0, and so Der.k =k0/ acts on H .Xk / H .X/ k k through k , 0 dR 0 D dR ˝ 0 0 0 i.e., D 1 D. Thus the corollary follows from the next well-known lemma. r D ˝

LEMMA 2.8. Let k0 k be as above, and let V V0 k , where V0 is a vector space 0 D ˝k0 0 over k0. If t V is ﬁxed (i.e., killed) by all derivations of k =k0, then t V0. 2 0 2 Let C p .X/ be the subset of H 2p.X/.p/ of absolute Hodge cycles. It is a ﬁnite- AH A dimensional vector space over Q.

PROPOSITION 2.9. Let k be an algebraically closed ﬁeld of ﬁnite transcendence degree over Q.

(a) For any smooth complete variety X deﬁned over an algebraically closed subﬁeld k0 of k, the canonical map

H 2p.X/.p/ H 2p.X /.p/ A ! A k induces an isomorphism C p .X/ C p .X /. AH ! AH k (b) Let X0 be a smooth complete variety defined over a subfield k0 of k whose algebraic p closure is k, and let X X0 k. Then Gal.k=k0/ acts on C .X/ through a finite D ˝k0 AH quotient.

PROOF. (a) The map is injective, and a cycle on X is absolutely Hodge if and only if it is absolutely Hodge on Xk, and so it remains to show that an absolute Hodge cycle t on Xk 2p arises from a cycle on X. But (2.7) shows that tdR arises from an element of HdR .X/.p/, 2p 2p and Het .X/.p/ Het .Xk/.p/ is an isomorphism. ! 2p 2p (b) It is obvious that the action of Gal.k=k0/ on HdR .X/.p/ Het .X/.p/ stabilizes p CAH.X/. We give three proofs that it factors through a ﬁnite quotient. 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 19

p 2p 2p S 2p (i) Note that C .X/ H .X/ is injective. Clearly, H .X/ H .X0 ki / AH ! dR dR D dR ˝ where the ki run over over the finite extensions of k0 contained in k. Thus, all elements of a p 2p finite generating set for CAH.X/ lie in HdR .X0 ki / for some i. p 2p ˝ (ii) Note that CAH.X/ H .Xet;Q`/.p/ is injective for all `. The subgroup H of p ! Gal.k=k0/ fixing CAH.X/ is closed. Thus, the quotient of Gal.k=k0/ by H is a profinite group, which is countable because it is a finite subgroup of GLm.Q/ for some m. It follows that it is finite (the Cantor diagonalization argument shows that an infinite profinite group is uncountable). p (iii) A polarization of X gives a positive definite form on CAH.X/, which is stable under Gal.k=k0/. This shows that the action factors through a finite quotient.

REMARK 2.10. (a) The above results remain valid for a family of varieties .X˛/˛ rather than a single X. (b) Proposition 2.9 would remain true if we had defined an absolute Hodge cycle to be 0 2p 2p an element t of F H .X/.p/ such that, for all k , C; .t/ H .X/. dR W ! dR 2 Definitions (arbitrary k) Proposition 2.9 allows us to define the notion of an absolute Hodge cycle on any smooth complete variety X over a field of characteristic zero. When k is algebraically closed, we choose a model X0=k0 of X0 over an algebraically closed subfield k0 of k of finite 2p transcendence degree over Q, and we define t H .X/.p/ to be an absolute Hodge cycle 2 A if it lies in the subspace H 2p.X /.p/ of H 2p.X/.p/ and is an absolute Hodge cycle there. A 0 A The proposition shows that this definition is independent of the choice of k0 and X0. (This definition is forced on us if we want (2.9a) to hold without restriction on the transcendence degrees of k and k0.) When k is not algebraically closed, we choose an algebraic closure k of it, and define an absolute Hodge cycle on X to be an absolute Hodge cycle on X k ˝k that is fixed by Gal.k=k/. One can show (assuming the axiom of choice) that if k is algebraically closed and of 2p 2p cardinality not greater than that of C, then an element t of H .X/.p/ Het .X/.p/ is dR an absolute Hodge cycle if it is rational relative to all embeddings k , C and tdR 0 2p W ! 2 F H .X/.p/. If k C, then the first condition has to be checked only for isomorphisms dR D of C. When k C, we define a Hodge cycle to be a cohomology class that is Hodge relative to the inclusion k , C. ! Statement of the main theorem M.7 MAIN THEOREM 2.11. Let X be an abelian variety over an algebraically closed field k, 2p and let t H .X/.p/. If t is a Hodge cycle relative to one embedding k , C, then it 2 A W ! is a Hodge cycle relative to every embedding, i.e., it is an absolute Hodge cycle.

The proof will occupy 2–6 of the notes.

Principle B We begin with a result concerning families of varieties parametrized by smooth algebraic varieties over C. Let X S be a proper smooth map of smooth varieties over C with S W ! 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 20 connected. We set n n m H .X=S/.m/ lim.R r˝ / Q: et D et ˝Z r and Hn .X=S/.m/ Hn .X=S/.m/ Hn .X=S/.m/. A D dR et 2p THEOREM 2.12 (PRINCIPLE B). Let t be a global section of H .X=S/.p/ such that A 0 2p tdR 0. If .tdR/s F H .Xs/.p/ for all s S and ts is an absolute Hodge cycle in r D 2 dR 2 H 2p.X /.p/ for one s, then t is an absolute Hodge cycle for all s. A s s

PROOF. Suppose that ts is an absolute Hodge cycle for s s1, and let s2 be a second point D of S. We have to show that ts2 is rational relative to every isomorphism C C. On W ! applying , we obtain a morphism X S and a global section t of H2p.X=S/.p/. W ! A We know that .t/s1 is rational, and we have to show that .t/s2 is rational. Clearly, only translates the problem, and so we can omit it. First consider the component tdR of t. By assumption, tdR 0, and so tdR is a global 2p r D section of H .X an;C/. Since it is rational at one point, it must be rational at every point. 2p def 2p an 2p Next consider tet. As H .X=S/.p/ R Q.p/ and Het .X=S/ are local systems, B D for any point s S there are isomorphisms 2

2p 2p 1.S;s/ .S;H .X=S/.p// ' H .Xs/.p/ B ! B 2p 2p 1.S;s/ .S;H .X=S/.p// ' H .Xs/.p/ : et ! et Consider

2p 2p 2p .S;H .X=S/.p// .S;H .X=S/.p// Af ' .S;Het .X=S/.p// B B ˝

' ' '

2p 1.S;s/ 2p 1.S;s/ 2p 1.S;s/ H .Xs/.p/ H .Xs/.p/ Af ' Het .Xs/.p/ B B ˝

2p 2p 2p H .Xs/.p/ H .Xs/.p/ Af ' Het .Xs/.p/ B B ˝

2p 2p We have tet .S;Het .X=S/.p// and are told that its image in Het .Xs1 /.p/ lies in 2p 2 HB .Xs1 /.p/. On applying the next lemma (with Z A and z 1), we ﬁnd that tet lies in 2p 2p D D .S;HB .X=S/.p//, and is therefore in HB .Xs/.p/ for all s.

LEMMA 2.13. Let W, V be an inclusion of vector spaces. Let Z be a third vector space ! and let z be a nonzero element of Z. Embed V in V Z by v v z. Then ˝ 7! ˝ .W Z/ V W (inside V Z). ˝ \ D ˝

PROOF. Choose a basis .ei /i I for W and extend it to a basis .ei /I J for V . Every x V Z has a unique expression2 t 2 ˝ P x i I J ei zi ; .zi Z, ﬁnite sum/: D 2 t ˝ 2 If x W Z, then zi 0 for i I , and if x V , then zi z for all i. 2 ˝ D … 2 D 2 ABSOLUTE HODGE CYCLES; PRINCIPLE B 21

0 2p REMARK 2.14. The assumption in the theorem that .tdR/s F H .Xs/.p/ for all s is 2 dR unnecessary: it is implied by the condition that tdR 0 (Deligne 1971a, 4.1.2, Theor´ eme` r D de la partie ﬁxe). M.8

We shall need a slight generalization of Theorem 2.12.

THEOREM 2.15. Let X S again be a smooth proper map of smooth varieties over W ! 2p C with S connected, and let V be a local subsystem of R Q.p/ such that Vs consists of .0;0/-cycles for all s and consists of absolute Hodge cycles for at least one s. Then Vs consists of absolute Hodge cycles for all s.

PROOF. If V is constant, so that every element of Vs extends to a global section, then this is a consequence of Theorem 2.12, but the following argument reduces the proof of the general case to that case. 2p 2p At each point s S, R Q.p/s has a Hodge structure. Moreover, R Q.p/ has a 2 polarization, i.e., there is a form

2p 2p R Q.p/ R Q.p/ Q. p/ W ! 2p which at each point defines a polarization on the Hodge structure R Q.p/s. On 2p 2p 0;0 R Q.p/ .R C.p// \ the form is symmetric, bilinear, rational, and positive definite. Since the action of 1.S;s0/ preserves the form, the image of 1.S;s0/ in Aut.Vs0 / is finite. Thus, after passing to a finite covering of S, we can assume that V is constant.

REMARK 2.16. Both Theorem 2.12 and Theorem 2.15 generalize, in an obvious way, to families ˛ X˛ S. W ! 3 MUMFORD-TATE GROUPS; PRINCIPLE A 22

3 Mumford-Tate groups; principle A

Characterizing subgroups by their ﬁxed tensors

Let G be a reductive algebraic group over a field k of characteristic zero, and let .V˛/˛ A be Q 2 a faithful family of finite-dimensional representations over k of G, so that G GL.V˛/ .A/ ! is injective. For all m; n N , we can form 2 m;n N m.˛/ N n.˛/ T V˛˝ .V / , D ˛ ˝ ˛ ˛_ ˝ which is again a finite-dimensional representation of G. For an algebraic subgroup H of G, m;n we write H 0 for the subgroup of G fixing all tensors that occur in some T and are fixed by H. Clearly, H H , and we shall need criteria guaranteeing their equality. 0 PROPOSITION 3.1. The notations are as above. (a) Every finite-dimensional representation of G is contained in a direct sum of representations T m;n. (b) (Chevalley’s Theorem). Every subgroup H of G is the stabilizer of a line D in some finite-dimensional representation of G. (c) If H is reductive, or if X .G/ X .H / is surjective, then H H . (Here X .G/ k ! k D 0 k denotes Homk.G;Gm/, so the hypotheses is that every k-character of H extends to a k-character of G.)

4 PROOF. (a) Let W be a representation of G, and let W0 denote the underlying vector space of W with G acting trivially (i.e., gw w, all g G, w W ): Then G W W D 2 2 ! defines a map W W0 kŒG which is G-equivariant (Waterhouse 1979, 3.5). Since !dimW ˝ W0 kŒG kŒG , it suffices to prove (a) for the regular representation. There is a ˝ finite sum V V˛ such that G GL.V / is injective (because G is noetherian). The map D ˚ ! GL.V / End.V / End.V / ! _ identifies GL.V / (and hence G) with a closed subvariety of End.V / End.V / (ibid.). _ There is therefore a surjection

Sym.End.V // Sym.End.V // kŒG; _ ! where Sym denotes the symmetric algebra, and (a) now follows from the fact that representations of reductive groups in characteristic zero are semisimple (see Deligne and Milne 1982, 2). (b) Let I be the ideal of regular functions on G that are zero on H . Then, in the regular representation of G on kŒG, H is the stabilizer of I . There exists a ﬁnite-dimensional subspace V of kŒG that is G-stable and contains a generating set for I (Waterhouse 1979, 3.3). Then H is the stabilizer of the subspace I V in V , and hence of Vd .I V/ in Vd V , \ \ where d dim .I V/. D k \ (c) According to (b), H is the stabilizer of a line D in some representation V of G, which (according to (a)) can be taken to be a direct sum of T m;n’s.

4Added(jsm): For a detailed proof of (a), see Milne, Algebraic Groups, 2017, 4.14, 22.42; for a detailed proof of (b), see ibid. 4.27 3 MUMFORD-TATE GROUPS; PRINCIPLE A 23

If H is reductive, then V W D for some H-stable W and V W D . Now D ˚ _ D _ ˚ _ H is the group fixing a generator of D D in V V . ˝ _ ˝ _ If every k-character of H extends to a k-character of G, then the one-dimensional representation of H on D can be regarded as the restriction to H of a representation of G. Now H is the group fixing a generator of D D in V D . ˝ _ ˝ _ REMARK 3.2. (a) It is clearly necessary to have some condition on H in order to have H H . For example, let B be a Borel subgroup of a reductive group G, and let v V be 0 D 2 fixed by B. Then g gv defines a map of algebraic varieties G=B V , which must be 7! ! constant because G=B is complete and V is affine. Thus, v is fixed by G, and so B G. 0 D However, the above argument proves the following: let H 0 be the group fixing all tensors fixed by G occurring in any representation of G (equivalently, any representation occurring as a subquotient of some T m;n); then H H . D 0 (b) In fact, in all our applications of (3.1c), H will be the Mumford-Tate group of a polarizable Hodge structure, and hence will be reductive. However, the Mumford-Tate groups of mixed Hodge structures (even polarizable) will not in general be reductive, but will satisfy the second condition in (3.1c) (with G GL). D (c) The theorem of Haboush (Demazure 1976) can be used to show that the second form of (3.1c) holds when k has nonzero characteristic. (d) In (3.1c) it suffices to require that X .G/ X .H / has finite cokernel, i.e., a k ! k nonzero multiple of each k-character of H extends to a k-character of G.

Hodge structures

Let V be a ﬁnite-dimensional vector space over Q.A Q-rational Hodge structure of weight n on V is a decomposition V L V p;q such that V q;p is the complex conjugate of C D p q n V p;q. Such a structure determines a cocharacterC D

Gm GL.V / W ! C such that .z/vp;q z pvp;q; vp;q V p;q: D 2 The complex conjugate .z/ of .z/ has the property .z/ vp;q z qvp;q. Since .z/ D and .z/ commute, their product determines a homomorphism of real algebraic groups

p;q p q p;q h C GL.V /; h.z/v z z v . W ! R D n Conversely, a homomorphism h C GL.V / whose restriction to R is r r idV W ! R 7! defines a Hodge structure of weight n on V . p L p0;q0 Let F V p p V , so that D 0 F pV F p 1V C is a decreasing (Hodge) filtration on VC. Let Q.1/ denote the vector space Q with the Hodge structure for which Q.1/C 1; 1 D Q.1/ . It has weight 2 and h.z/ 1 zz 1. For any integer m, D def m m; m Q.m/ Q.1/˝ Q.m/ D D has weight 2m. (Strictly speaking, we should define Q.1/ 2iQ:::.) D 3 MUMFORD-TATE GROUPS; PRINCIPLE A 24

q REMARK 3.3. The notation h.z/ vp;q z pz vp;q is the negative of that used in Deligne D 1971b, Saavedra Rivano 1972, and elsewhere. It is perhaps justiﬁed by the following. Let A be an abelian variety over C. The exact sequences

0 Lie.A_/_ H1.A;C/ Lie.A/ 0 ! ! ! ! and

1 1 1 0 1 0 F H .A;C/ H .A;C/ F =F 0

1;0 0 1 0;1 1 H H .A;˝ / H H .A;OX / D D 1 are canonically dual. Since H .A;C/ has a natural Hodge structure of weight 1 with 0 1 .1;0/-component H .˝ /, H1.A;C/ has a natural Hodge structure of weight 1 with . 1;0/-component Lie.A/. Thus h.z/ acts on Lie.A/, the tangent space to A at zero, as multiplication by z.

Mumford-Tate groups

Let V be a Q-vector space with Hodge structure h of weight n. For m1;m2 N and m3 Z, m1 m2 m3 2 2 T V ˝ V _˝ Q.1/˝ has a Hodge structure of weight .m1 m2/n 2m3. An D ˝ ˝ p;q element of TC is said to be rational of bidegree .p;q/ if it lies in T T . We let Gm 1 \ 2 act on Q.1/ as . The action of GL.V / on V and the action of Gm on Q.1/ define an action of GL.V / Gm on T . The Mumford-Tate group G of .V;h/ is the subgroup of GL.V / Gm fixing all rational tensors of type .0;0/ belonging to any T . Thus the projection on the first factor identifies G.Q/ with the set of g GL.V / for which there exists p 2 m1 m2 a .g/ Q with the property that gt .g/ t for all t V ˝ V _˝ of type .p;p/. 2 D 2 ˝ PROPOSITION 3.4. The group G is the smallest algebraic subgroup of GL.V / Gm defined over Q for which .Gm/ G . C

PROOF. Let H be the intersection of all Q-rational subgroups of GL.V / Gm that, over C, contain .Gm/. For any t T , t is of type .0;0/ if and only if it is ﬁxed by .Gm/ or, 2 equivalently, it is ﬁxed by H. Thus G H in the notation of (3.1), and the next lemma D 0 completes the proof.

LEMMA 3.5. With H as above, every Q-character of H extends to a Q-character of GL.V / Gm. PROOF. Let H GL.W / be a representation of dimension one deﬁned over Q, i.e., a W ! Q-character. The restriction of the representation to Gm is isomorphic to Q.n/ for some n. After tensoring W with Q. n/, we can assume that 1, i.e., .Gm/ acts trivially. ı D But then H must act trivially, and the trivial character extends to the trivial character.

PROPOSITION 3.6. If V is polarizable, then G is reductive.

PROOF. Choose an i and write C h.i/ (C is often called the Weil operator). For vp;q D 2 V p;q, C vp;q i p qvp;q, and so C 2 acts as . 1/n on V , where n p q is the weight D C D C of V . 3 MUMFORD-TATE GROUPS; PRINCIPLE A 25

Recall that a polarization of V is a morphism V V Q. n/ such that the W ! real-valued form .x;Cy/ on VR is symmetric and positive deﬁnite. Under the canonical isomorphism Hom.V V;Q. n// V _ V _. n/; ˝ ! ˝ corresponds to a tensor of bidegree .0;0/ (because it is a morphism of Hodge structures) and therefore is ﬁxed by G:

n .g1v;g1v0/ g2 .v;v0/, all .g1;g2/ G.Q/ GL.V / Q; .v;v0/ V: D 2 2

Recall that if H is a real algebraic group and is an involution of HC, then the real-form of H defined by is a real algebraic group H endowed with an isomorphism H .H / C ! C under which complex conjugation on H .C/ corresponds to .complex conjugation/ on ı H.C/. We are going to use the following criterion: a connected algebraic group H over R is reductive if it has a compact real-form H . To prove the criterion, it suffices to show that H is reductive. On any finite-dimensional representation V of H , there is an H -invariant positive definite symmetric form, namely, Z u;v 0 hu;hv dh; h i D H h i where ; is any positive definite symmetric form on V . If W is an H -stable subspace h i of V , then its orthogonal complement is also H -stable. Thus every finite-dimensional representation of H is semisimple, and this implies that H is reductive (Deligne and Milne 1982, 2). We shall apply the criterion to the special Mumford-Tate group of .V;h/,

0 def G Ker.G Gm/. D ! 1 1 1 Let G be the smallest Q-rational subgroup of GL.V / Gm such that G contains h.U /, R where U 1. / z zz 1 . Then G1 G, and in fact G1 G0. Since G1 R C R D f 21 j D g 0 1 0 h.C/ G and h.U / Ker.h.C/ Gm/, it follows that G G , and therefore G D R D ! D is connected. 0 2 n Since C h.i/ acts as 1 on Q.1/, C G .R/. Its square C acts as . 1/ on V and D 0 2 therefore lies in the centre of G .R/. The inner automorphism adC of GR defined by C is 0 therefore an involution. For u;v V , and g G .C/, we have 2 C 2 .u;C v/ .gu;gC v/ .gu;CC 1gC v/ .gu;C g v/ D D D 1 def where g C gC .adC /.g/. Thus, the positive definite form .u;v/ .u;C v/ on D D 0 D VR is invariant under the real-form of G defined by adC , and so this real-form is compact.

EXAMPLE 3.7. (Abelian varieties of CM-type). A CM-field is a quadratic totally imaginary extension of a totally real field, and a CM-algebra is a finite product of CM-fields. Let E be a CM-algebra, and let E be the involution of E such that E for all E C. ı D ı W ! Let I S Hom.E;Q/ Hom.E;C/ spec.E /: D D DD C A CM-type for E is a subset ˙ S such that S ˙ ˙ (disjoint union). D t 3 MUMFORD-TATE GROUPS; PRINCIPLE A 26

˙ To the pair .E;˙/, there is attached an abelian variety A with A.C/ C =˙.OE / where ˙ D OE , the ring of integers in E, is embedded in C by the map u .u/ ˙ . Obviously, 7! 2 E acts on A. Moreover, from the choice of a nonzero element of H1.A;Q/, we obtain an isomorphism H1.A;Q/ E, and ' S ˙ ˙ H1.A/ C E C ' C C C ˝ ' ˝Q ! D ˚ u 1 .u/ S ˝ 7! 2 ˙ ˙ with C the . 1;0/-component of H1.A/ C and C the .0; 1/-component. Thus, .z/ ˙ ˙ ˝ acts as z on C and as 1 on C . Let G be the Mumford-Tate group of H1.A/. The actions of .C/ and E on H1.A/ ˝ C commute. As E is its own commutant in GL.H1.A//, this implies that .C/ .E C/ and G is the smallest algebraic subgroup of E Q such that G.C/ contains ˝ def .C/. In particular, G is a torus, and can be described by its cocharacter group Y.G/ D Hom . ;G/. Q Gm Clearly, S Y.G/ Y.E/ Y.Gm/ Z Z. D P S Note that Y.G/ is equal to s ˙ es e0, where .es/s S Z is the basis dual to the 2 2 C 2 basis S of X.E/ and e0 is the element 1 of the last copy of Z. The following are obvious: S (a) Z Z =Y.G/ is torsion-free; (b) Y.G/; 2 (c) Y.G/ is stable under Gal.Q=Q/; thus Y.G/ is the Gal.Q=Q/-module generated by ; (d) since 1 on S, C D ˚P S « Y.G/ nses n0e0 Z Z ns ns constant . C 2 j C D

Let F be the subalgebra of E whose elements are ﬁxed by E (thus, F is a product of totally real ﬁelds). Then (d) says that ˚ « G.Q/ .x;y/ E Q NmE=F .x/ Q : 2 j 2 Principle A M.9 THEOREM 3.8 (PRINCIPLE A). Let .X˛/˛ be a family of varieties over C, and consider n˛ n˛ spaces T obtained by tensoring spaces of the form HB .X˛/, HB .X˛/_, and Q.1/. Let ti Ti , i 1;:::;N (Ti of the above type) be absolute Hodge cycles, and let G be the 2 D subgroup of Q n˛ GL.HB .X˛// Gm ˛;n˛

ﬁxing the ti . If t belongs to some T and is ﬁxed by G, then it is an absolute Hodge cycle.

We ﬁrst need a lemma.

LEMMA 3.9. Let G be an algebraic group over Q, and let P be a G-torsor of isomorphisms ˛ ˛ ˛ ˛ H H where .H /˛ and .H /˛ are families of Q-rational representations of G. Let ! T and T be like spaces of tensors constructed out of H and H respectively. Then P G deﬁnes a map T T . ! 3 MUMFORD-TATE GROUPS; PRINCIPLE A 27

PROOF. Locally for the etale´ topology on Spec.Q/, points of P define isomorphisms G T T . The restriction to T of such a map is independent of the point. Thus, by etale´ ! G descent theory, they define a map of vector spaces T T . ! PROOFOF THEOREM 3.8. We remove the identification of the ground field with C. Thus, the ground field is now a field k equipped with an isomorphism k C. Let k C be W ! W ! a second isomorphism. We can assume that t and the ti all belong to the same space T . The canonical inclusions of cohomology groups

H .X˛/, H .X˛/ .C Af / -H .X˛/ ! ˝ induce maps T , T .C Af / -T . ! ˝ We shall regard these maps as inclusions. Thus,

.t1;:::;tN ;t/ T T .C Af /; ˝ .t1;:::;tN / T T .C Af /. ˝ Let P be the functor of Q-algebras such that

P.R/ p H R H R p maps ti (in T ) to ti (in T ), i 1;:::;N : D f W ˝ ! ˝ j D g

The existence of the canonical inclusions mentioned above shows that P.C Af / is nonempty, and it is easily seen that P is a G-torsor. G On applying the lemma (and its proof) in the above situation, we obtain a map T T ! such that G T T

T T .C Af / ˝ G commutes. This means that T T , and therefore t T . 2 0 It remains to show that the component tdR of t in T C TdR lies in F TdR. But for a ˝ D rational s TdR, 2 0 s F TdR s is ﬁxed by .C/. 2 ” 0 0 Thus, .ti /dR F , i 1;2;:::;N , implies G .C/, which implies that tdR F . 2 D 2 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 28

4 Construction of some absolute Hodge cycles

Hermitian forms

Recall that a number field E is a CM-field if, for each embedding E, C, complex ! conjugation induces a nontrivial automorphism e e on E that is independent of the 7! embedding. The fixed field of the automorphism is then a totally real field F over which E has degree two. A bi-additive form V V E W ! on a vector space V over a CM-field E is Hermitian if

.ev;w/ e.v;w/; .v;w/ .w;v/; all e E, v;w V . D D 2 2 For any embedding F, R we obtain a Hermitian form in the usual sense on the vector W ! space V V F; R, and we let a and b denote the dimensions of the maximal subspaces D ˝ of V on which is positive definite and negative definite respectively. If d dimV , D then defines a Hermitian form on Vd V that, relative to some basis vector, is of the form .x;y/ f xy. The element f is in F , and is independent of the choice of the basis vector 7! up to multiplication by an element of NmE=F E. It is called the discriminant of . Let .v1;:::;vd / be an orthogonal basis for , and let .vi ;vi / ci ; then a is the number of i D Q for which ci > 0, b the number of i for which ci < 0, and f ci (mod Nm E /. D E=F If is nondegenerate, then f F =NmE , and 2

b a b d; sign.f / . 1/ , all : (1) C D D

PROPOSITION 4.1. Suppose given nonnegative integers .a ;b / F, C and an element f F =NmE satisfying (1). Then there exists a non-degenerate HermitianW ! form on an 2 E-vector space V of dimension d with invariants .a ;b / and f ; moreover, .V;/ is unique up to isomorphism.

PROOF. The result is due to Landherr 1936. Today one prefers to regard it as a consequence of the Hasse principle for simply connected semisimple algebraic groups and the classiﬁcation of Hermitian forms over local ﬁelds.

COROLLARY 4.2. Let .V;/ be a non-degenerate Hermitian space, and let d dimV . The D following conditions are equivalent: d=2 (a) a b for all and disc.f / . 1/ D D I (b) there exists a totally isotropic subspace of V of dimension d=2.

PROOF. Let W be a totally isotropic subspace of V of dimension d=2. The map v 7! . ;v/ V W induces an antilinear isomorphism V=W W . Thus, a basis e1;:::;e W ! _ ! _ d=2 of W can be extended to a basis ei of V such that f g

.ei ;e d i / 1; 1 i d=2; 2 C D Ä Ä .ei ;ej / 0; j i d=2: D ¤ ˙ 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 29

It is now easy to check that .V;/ satisﬁes (a). Conversely, .Ed ;/ where X ..ai /;.bi // ai b d i a d i bi ; D 2 C 2 1 i d=2 C C Ä Ä is, up to isomorphism, the only Hermitian space satisfying (a), and it also satisﬁes (b).

A Hermitian form satisfying the equivalent conditions of the corollary will be said to be split (because then the algebraic group AutE .V;/ over F is split). We shall need the following lemma from linear algebra.

LEMMA 4.3. Let k be a field, and let V be a free finitely generated module over an etale´ k-algebra k0 (i.e., k0 is a finite product of finite separable field extensions of k). (a) The map f Trk =k f Homk .V;k0/ Homk.V;k/ 7! 0 ı W 0 ! is an isomorphism of k-vector spaces. (b) Vn V is, in a natural way, a direct summand of Vn V . k0 k

PROOF. (a) As the pairing Trk =k k k k is nondegenerate, the map f Trk =k f is 0 W 0 0 ! 7! 0 ı injective, and it is onto because the two spaces have the same dimension over k. (b) There are obvious maps Vn V Vn V k k0 Vn ! Vn k V _ k V _ ! 0 where V Homk .V;k / Homk.V;k/. The pairing _ D 0 0 ' Vn V Vn V k _ ! determined by .f1 fn;v1 vn/ det. fi vj / ^ ^ ˝ ˝ D h j i Vn Vn induces an isomorphism . V _/ . V/_ (Bourbaki, N., Algebre´ Multilineaire,´ Her- ' Vn Vn mann, 1958, 8), and so the second map gives rise to a map k V k V , which is left 0 ! inverse to the first. Alternatively, and more elegantly, descent theory shows that it suffices to prove the S L proposition with k0 k , S Homk.k0;k/. Then V s S Vs with the Vs subspaces of D D D 2 P the same dimension, and the map in (a) becomes f .fs/ fs, which is obviously an D 7! isomorphism. For (b), note that5 ! Vn M OVns MVn Vn k V Vs k Vs k V . D k D 0 P s S s S ns n D 2 2 P The first sum is over the set of families .ns/s S such that ns N and s S ns n, while 2 2 2 D the second is over the set of such families with ns n for some s. D 5Because V is a free kS -module, S M M V W k W k Vs D ˝k D s S ˝k D s S 2 2 for some k-vector space W . Correspondingly,

^n ^n Á S M ^n Á M ^n V W k W k Vs: kS D k ˝k D s S k ˝k D s S k 2 2 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 30

Vd 1 Conditions for E H .A;Q/ to consist of absolute Hodge cycles Let A be an abelian variety over C and let E End.A/ be a homomorphism with E a CM- W ! ﬁeld (in particular, this means that v.1/ idA). Let d be the dimension of H1.A;Q/ over E, D so that dŒE Q 2dimA. When H1.A;R/ is identiﬁed with the tangent space to A at zero, W D it acquires a complex structure; we denote by J the R-linear endomorphism “multiplication 1 by i” of H1.A;R/. If h C GL.H .A;R// is the homomorphism determined by the 1 W ! 1 Hodge structure on H .A;R/, then h.i/ J under the isomorphism GL.H .A;R// 1 $ ' GL.H1.A;R// determined by H .A;R/ H1.A;R/_. ' Corresponding to the decomposition Y e z .:::;e z;:::/ E C ' C;S Hom.E;C/; ˝ 7! W ˝Q ! S D 2 there is a decomposition

1 M 1 H .A/ C ' H (E-linear isomorphism) B ˝ ! S B; 2 1 1 such that e E acts on the complex vector space HB; as e. Each HB; has dimension d, 2 1 and (as E respects the Hodge structure on HB .A/) acquires a Hodge structure

H 1 H 1;0 H 0;1 : B; D B; ˚ B; 1;0 0;1 Let a dimH and b dimH ; thus a b d. D B; D B; C D Vd 1 d d d PROPOSITION 4.4. The subspace E HB .A/ of H .A;Q/ is purely of bidegree . 2 ; 2 / if M.10 d and only if a b for all S. D 2 D 2 d d PROOF. Note that H d .A; / V H 1.A; /, and so (4.3) canonically identiﬁes V H 1.A/ Q Q Q E B d ' with a subspace of HB .A/. As in the last line of the proof of (4.3), we have Á Vd H 1 Vd H 1 E B C E C B C ˝ ' ˝ ˝ M Vd H 1 ' S B; M 2 Vd .H 1;0 H 0;1/ ' S B; ˚ B; M 2 Va H 1;0 Vb H 0;1; ' S B; ˝ B; 2 Va 1;0 Vb 0;1 and HB; and HB; are purely of bidegree .a ;0/ and .0;b / respectively.

Vd 1 Á d Thus, in this case, E HB .A/ . 2 / consists of Hodge cycles, and we would like to show that it consists of absolute Hodge cycles. In one special case, this is easy.

d LEMMA 4.5. Let A0 be an abelian variety of dimension 2 and let A A0 Q E. Then the Vd 1 d d d D ˝ subspace E H .A;Q/. 2 / of H .A;Q/. 2 / consists of absolute Hodge cycles. 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 31

PROOF. There is a commutative diagram

d d d d H .A0/. / E H .A0/. / E B 2 ˝Q A 2 ˝Q

' ' Á Á Vd H 1.A E/ . d / Vd H 1.A E/ . d / H d .A E/. d / E B 0 Q 2 E A A 0 Q 2 A 0 2 ˝ ˝ ˝ ˝

1 1 in which the vertical maps are induced by H .A0/ E ' H .A0 E/. From this, and ˝ ! ˝ similar diagrams corresponding to isomorphisms C C, one sees that W ! d d d d H .A0/. / E, H .A0 E/. / A 2 ˝ ! A ˝ 2 induces an inclusion d d C .A0/ E, C .A0 E/: AH ˝ ! AH ˝ d d d d d But C .A0/ H .A0/. / since H .A0/. / is a one-dimensional space generated by AH D B 2 B 2 the class of any point on A0. This completes the proof as the above diagram shows that the d d d d Vd 1 d image of HB .A0/. 2 / in HB .A/. 2 / is E HB .A/. 2 /. M.11 In order to prove the general result, we need to consider families of abelian varieties (ultimately, we wish to apply (2.15)), and for this we need to consider polarized abelian varieties. A polarization on A is determined by a Riemann form, i.e., a Q-bilinear alternating form on H1.A;Q/ such that the form .z;w/ .z;J w/ on H1.A;R/ is 7! symmetric and definite; two Riemann forms and 0 on H1.A;Q/ correspond to the same polarization if and only if there is an a Q such that 0 a . We shall consider only 2 D triples .A;;v/ in which the Rosati involution defined by induces complex conjugation on E. (The Rosati involution e t e End.A/ End.A/ is determined by the condition 7! W ! t .ev;w/ .v; ew/; v;w H1.A;Q/:/ D 2 LEMMA 4.6. Let f E be such that f f , and let be a Riemann form for A. There 2 D exists a unique E-Hermitian form on H1.A;Q/ such that .x;y/ TrE= .f .x;y//. D Q We first need:

SUBLEMMA 4.7. Let V and W be ﬁnite-dimensional vector spaces over E, and let V W W Q be a Q-bilinear form such that .ev;w/ .v;ew/ for e E. Then there exists a ! D 2 unique E-bilinear form such that .v;w/ Tr .v;w/. D E=Q PROOF. The condition says that deﬁnes a Q-linear map V E W Q. Let be the ˝ ! element of Hom .V E W;E/ corresponding to under the isomorphism (see 4.3(a)) Q ˝ HomE .V E W;E/ Hom .V E W;Q/. ˝ ' Q ˝

PROOFOF LEMMA 4.6. We apply (4.7) with V H1.A;Q/ W , but with with E acting D D through complex conjugation on W . This gives a sesquilinear 1 such that .x;y/ 1 D Tr 1.x;y/. Let f 1, so that .x;y/ Tr .f .x;y//. Since is sesquilin- E=Q D D E=Q ear it remains to show that .x;y/ .x;y/. As .x;y/ .y;x/ for all x;y D D 2 H1.A;Q/, Tr.f .x;y// Tr.f .y;x// Tr.f .y;x//: D D 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 32

On replacing x by ex with e E, we ﬁnd that 2 Tr.fe.x;y// Tr.fe.y;x//: D On the other hand, Tr.fe.x;y// Tr.fe.x;y//; D and so Tr.fe.y;x// Tr.fe.x;y//: D As fe is an arbitrary element of E, the non-degeneracy of the trace implies that .x;y/ D .y;x/. Finally, the uniqueness of is obvious from (4.7).

THEOREM 4.8. Let A be an abelian variety over C, and let v E End.A/ be a homomor- 1 W ! phism with E a CM-ﬁeld. Let d dimE H .A;Q/. Assume there exists a polarization D for A such that (a) the Rosati involution of induces complex conjugation on E;

(b) there exists a split E-Hermitian form on H1.A;Q/ and an f E with f f def 2 D such that .x;y/ Tr .f .x;y// is a Riemann form for . D E=Q Vd 1 Á d d d Then the subspace E H .A;Q/ . 2 / of H .A;Q/. 2 / consists of absolute Hodge cycles.

PROOF. In the course of the proof, we shall see that (b) implies that A satisﬁes the equivalent statements of (4.4). Thus, the theorem will follow from (2.15) and (4.5) once we have shown that there exists a connected smooth variety S over C and an abelian scheme Y over S together with an action v of E on Y=S such that6

(a) for all s S, .Ys;vs/ satisfies the equivalent conditions in (4.4); 2 (b) for some s0 S, Ys is isomorphic to A0 E, some A0, with e E acting as id e; 2 0 ˝Q 2 ˝ (c) for some s1 S, .Ys ;vs / .A;v/. 2 1 1 D We shall first construct an analytic family of abelian varieties satisfying these conditions, and then pass to the quotient by a discrete group to obtain an algebraic family. Let H H1.A;Q/ regarded as an E-space and chose , , f , and as in the statement D of the theorem. We choose i p 1 so that .x;h.i/y/ is positive definite. D Consider the set of all quadruples .A1;1;v1;k1/ in which A1 is an abelian variety over C, v1 is an action of E on A1, 1 is a polarization of A, and k1 is an E-linear isomorphism H1.A1;Q/ H carrying a Riemann form for 1 into c for some c Q. From such a ! 2 quadruple, we obtain a complex structure on H.R/ (corresponding via k1 to the complex structure on H1.A1;R/ Lie.A1/) such that: D (a) the action of E commutes with the complex structure; (b) is a Riemann form relative to the complex structure. Conversely, a complex structure on H R satisfying (a) and (b) determines a quadruple ˝ .A1;1;v1;k1/ with H1.A1;Q/ H (as an E-module), Lie.A1/ H R (endowed with D D ˝ the given complex structure), 1 the polarization with Riemann form , and k1 the identity map. Moreover, two quadruples .A1;1;v1;k1/ and .A2;2;v2;k2/ are isomorphic if and only if they define the same complex structure on H . Let X be the set of complex structures on H satisfying (a) and (b). Our first task is to turn X into an analytic manifold in such a way that the family of abelian varieties that it parametrizes becomes an analytic family.

6Added: we also need that there exists a locally constant subsytem of Hodge cycles. 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 33

2 A point of X is determined by an R-linear map J H R H R, J 1, such W ˝ ! ˝ D that

(a0) J is E-linear, and (b0) .x;Jy/ is symmetric and definite. Note that .x;Jy/ is symmetric if and only if .J x;Jy/ .x;y/. Let F be the real D subfield of E, and fix an isomorphism L E Q R T C;T Hom.F;R/ ˝ ! 2 D such that .f 1/ .if / with f R, f > 0. Corresponding to this isomorphism, there is ˝ 7! 2 a decomposition L H Q R T H ˝ ' 2 in which each H is a complex vector space. Condition (a0) implies that J J , where 2 D ˚ J is a C-linear isomorphism H H such that J 1. Let ! D

H H H , D C ˚ where H and H are the eigenspaces of J with eigenvalues i and i respectively. The C C compatibility of and implies L .H; / R T .H ; / ˝ ' 2 with an R-bilinear alternating form on H such that

.zx;y/ .x;zy/; z C: D 2 The condition .J x;Jy/ .x;y/ D implies that HC is the orthogonal complement of H relative to :

H H H : D C ? We also have L .H;/ R T .H ; / ˝ ' 2 and .x;y/ Tr .if .x;y//. As D C=R P .x;y/ Tr .if .x;y//; D C=R we ﬁnd that

.x;J x/ > 0 all x Tr .if .x ;J x // > 0 all x ; ” C=R TrC=R.i .x ;J x // all x ;; ” is positive deﬁnite on HC, and ” is negative deﬁnite on H.

1;0 0; 1 This shows, in particular, that H H and H H each have dimension d=2 C D D (cf. 4.4). Let X and X be the sets of J X for which .x;Jy/ is positive definite and C 2 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 34 negative definite respectively. Then X is a disjoint union X X X . As J is determined D C t by its i eigenspace we see that X can be identified with C C

.V / T V a maximal subspace of H such that > 0 on V : f 2 j g This is an open connected complex submanifold of a product of Grassman manifolds Q X C T Grassd=2.V /. 2

Moreover, there is an analytic structure on X C V.R/ such that X C V.R/ X C is ! analytic and the inverse image of J X C is V.R/ with the complex structure provided by J . 2 On dividing V.R/ by an OE -stable lattice V.Z/ in V , we obtain the sought analytic family B of abelian varieties. Note that A is a member of the family. We shall next show that there is also an abelian variety of the form A0 E in the family. To do this, we only have to show that there ˝ exists a quadruple .A1;1;1;k1/ of the type discussed above with A1 A0 E. Let A0 D ˝ be any abelian variety of dimension d=2 and deﬁne 1 E End.A0 E/ so that e E W ! ˝ 2 acts on H1.A0 E/ H1.A0/ E through its action on E. A Riemann form 0 on ˝ D ˝ A0 extends in an obvious way to a Riemann form 1 on A1 that is compatible with the action of E. We deﬁne 1 to be the corresponding polarization, and let 1 be the Hermitian form on H1.A0 E;Q/ such that 1 TrE= .f 1/ (see 4.6). If I0 H1.A0;Q/ is ˝ D Q a totally isotropic subspace of H1.A0;Q/ of (maximum) dimension d=2, then I0 E ˝ is a totally isotropic subspace of dimension d=2 over E, which (by 4.2) shows that the Hermitian space .H1.A0 E;Q/;1/ is split. There is therefore an E-linear isomorphism ˝ k1 .H1.A0 E;Q/;1/ .H;/ which carries 1 TrE= .f 1/ to TrE= .f 1/. W ˝ ! D Q D Q This completes this part of the proof. Let n be an integer 3, and let be the set of OE -isomorphisms g V.Z/ V .Z/ W ! 1 preserving and such that .g 1/V .Z/ nV .Z/. Then acts on X C by J g J g 7! ı ı and (compatibly) on B. On forming the quotients, we obtain a map

B X n ! n C which is an algebraic family of abelian varieties. In fact, X is the moduli variety for n C quadruples .A1;1;1;k1/ in which A1, 1, and 1 are essentially as before, but now k1 is a level n structure k1 An.C/ H1.A;Z=nZ/ V.Z/=nV .Z/ W D ! I the map X X can be interpreted as “regard k1 modulo n”. To prove these facts, C ! n C one can use the theorem of Baily and Borel (1966) to show that X is algebraic, and a n C theorem of Borel (1972) to show that B is algebraic — see 6 where we discuss a similar n question in greater detail.

REMARK 4.9. With the notations of the theorem, let G be the algebraic group over Q such that

G.Q/ g GLE .H / .g/ Q such that .gx;gx/ .g/ .x;y/; x;y H . D f 2 j 9 2 D 8 2 g

The homomorphism h C GL.H R/ deﬁned by the Hodge structure on H1.A;Q/ W ! ˝ factors through GR, and X can be identiﬁed with the G.R/-conjugacy class of the homomorphisms C G containing h. Let K be the compact open subgroup of G.Af / of g such ! R 4 CONSTRUCTION OF SOME ABSOLUTE HODGE CYCLES 35

that .g 1/V .Z/ nV .Z/. Then X Cis a connected component of the Shimura variety O O n ShK .G;X/. The general theory shows that ShK .G;X/ is a ﬁne moduli scheme (see Deligne 1971c, 4, or Milne and Shih 1982, 2) and so, from this point of view, the only part of the above proof that is not immediate is that the connected component of ShK .G;X/ containing A also contains a variety A0 E. ˝

REMARK 4.10. It is easy to construct algebraic cycles on A0 E: every Q-linear map ˝ E Q deﬁnes a map A0 E A0 Q A0 , and we can take cl./ to be the image W ! d ˝ ! d˝ D of the class of a point in H .A0/ H .A0 E/. More generally, we have ! ˝

Sym.Hom -linear.E;Q// algebraic cycles on A0 E . Q ! f ˝ g r If E Q , this gives the obvious cycles. D REMARK 4.11. The argument in the proof of Theorem 4.8 is similar to, and was suggested by, an argument of B. Gross (1978). 5 COMPLETION OF THE PROOF FOR ABELIAN VARIETIES OF CM-TYPE 36

5 Completion of the proof for abelian varieties of CM- type

Abelian varieties of CM-type

The Mumford-Tate, or Hodge, group of an abelian variety A over C is defined to be the Mumford-Tate group of the rational Hodge structure H1.A;Q/: it is therefore the subgroup of GL.H1.A;Q// Gm fixing all Hodge cycles on A and its powers (see 3). In the language of Tannakian categories, the category of rational Hodge structures is Tannakian with an obvious fibre functor, and the Mumford-Tate group of A is the group associated with the Tannakian subcategory generated by H1.A;Q/ and Q.1/. An abelian variety A is said to be of CM-type if its Mumford-Tate group is commutative. Q Since every abelian variety A is a product A A˛ of simple abelian varieties (up to D isogeny) and A is of CM-type if and only if each A˛ is of CM-type (the Mumford-Tate group of A is contained in the product of the Mumford-Tate groups of the A˛ and projects onto each), in understanding this concept we can assume A is simple.

PROPOSITION 5.1. A simple abelian variety A over C is of CM-type if and only if E D EndA is a commutative field over which H1.A;Q/ has dimension 1. Then E is a CM-field, and the Rosati involution on E End.A/ defined by any polarization of A is complex D conjugation.

PROOF. Let A be an abelian variety such that End.A/ contains a field E for which H1.A;Q/ has dimension 1 as an E-vector space. As .Gm/ commutes with E R in End.H1.A;R//, ˝ we have that .Gm/ .E R/ and so the Mumford-Tate group of A is contained in E. ˝ Conversely, let A be simple and of CM-type, and let Gm GL.H1.A;C// be defined W ! by the Hodge structure on H1.A;C/ (see 3). As A is simple, E End.A/ is a field (possibly D noncommutative) of degree dimH1.A;Q/ over Q. As for any abelian variety, End.A/ is Ä the subalgebra of End.H1.A;Q// of elements preserving the Hodge structure or, equivalently, that commute with .Gm/ in GL.H1.A;C//. If G is the Mumford-Tate group of A, then GC is generated by the groups .Gm/ Aut.C/ (see 3.4). Therefore E is the commutant f j 2 g L of G in End.H1.A;Q//. By assumption, G is a torus, and so H1.A;C/ X.G/ H . D 2 The commutant of G therefore contains etale´ commutative algebras of rank dimH1.A;Q/ over Q. It follows that E is a commutative field of degree dimH1.A;Q/ over Q (and that it is generated as a Q-algebra by G.Q/; in particular, h.i/ E R). 2 ˝ Let be a Riemann form corresponding to some polarization on A. The Rosati involution e e on End.A/ E is determined by the condition 7! D .x;ey/ .ex;y/; x;y H1.A;Q/. D 2 It follows from .x;y/ .h.i/x;h.i/y/ D that h.i/ h.i/ 1 . h.i//: D D The Rosati involution therefore is nontrivial on E, and E has degree 2 over its fixed field F . There exists an ˛ F such that 2 E FŒp˛; p˛ p˛; D D 5 COMPLETION OF THE PROOF FOR ABELIAN VARIETIES OF CM-TYPE 37 and ˛ is uniquely determined up to multiplication by a square in F . If E is identified with H1.A;R/ through the choice of an appropriate basis vector, then

.x;y/ Tr ˛xy ; x;y E; D E=Q 2 (cf. 4.6). The positivity condition on implies that

2 TrE R=R.f x / > 0; x 0; x F R; f ˛=h.i/, ˝ ¤ 2 ˝ D which implies that F is totally real. Moreover, for every embedding F, R, we must W ! have .˛/ < 0, for otherwise E F; R R R with .r1;r2/ .r2;r1/, and the positivity ˝ D D condition is impossible. Thus, .˛/ < 0, and is complex conjugation relative to any embedding of E into C. This completes the proof.

Proof of the main theorem for abelian varieties of CM-type

Let .A˛/ be a ﬁnite family of abelian varieties over C of CM-type. We shall show that every element of a space

N 1 m˛ N 1 n˛ T H .X˛/ H .X˛/ .m/ A D ˛ A ˝ ˝ ˛ A _˝ that is a Hodge cycle (relative to id C C) is an absolute Hodge cycle. According to W ! (3.8) (Principle A), to do this it suffices to show that the following two subgroups of Q GL. H1.A˛;Q// Gm are equal: GH group fixing all Hodge cycles; D GAH group fixing all absolute Hodge cycles. D Obviously GH GAH . After breaking up each A˛ into its simple factors, we can assume A˛ is itself simple. Let E˛ be the CM-field End.A˛/ and let E be the smallest Galois extension of Q containing all E˛; it is again a CM-field. Let B˛ A˛ E E. It suffices to prove the theorem for the D ˝ ˛ family .B˛/ (because the Tannakian category generated by the H1.B˛/ and Q.1/ contains every H1.A˛/; cf. Deligne and Milne 1982). In fact, we consider an even larger family. Fix E, a CM-field Galois over Q, and consider the family .A˛/ of all abelian varieties with complex multiplication by E (so H1.A˛/ has dimension 1 over E) up to E-isogeny. This family is indexed by S, the set of CM-types for E. Thus, if S Hom.E;C/, then each element of S is a set ˚ S such that S ˚ ˚ D D t (disjoint union). We often identify ˚ with the characteristic function of ˚, i.e., we write

1 if s ˚ ˚.s/ 2 D 0 if s ˚: … With each ˚ we associate the isogeny class of abelian varieties containing the abelian variety ˚ C =˚.OE / where OE is the ring of integers in E and

˚ ˚.OE / .e/ ˚ C e OE : D f 2 2 j 2 g With this new family, we have to show that GH GAH . We begin by determining GH D (cf. 3.7). The Hodge structure on each H1.A˚ ;Q/ is compatible with the action of E. This 5 COMPLETION OF THE PROOF FOR ABELIAN VARIETIES OF CM-TYPE 38

Q H Q implies that, as a subgroup of ˚ S GL.H1.A˚ // Gm, G commutes with ˚ S E Q 2 H 2 and is therefore contained in E Gm. In particular, G is a torus and can be described H def H H by its group of cocharacters Y.G / Hom al .Gm;G / or its group of characters X.G /. D Q Note that H Q S S Y.G / Y. ˚ S E Gm/ Z Z. 2 D There is a canonical basis for X.E/, namely the set S, and therefore a canonical basis for Q Q X. ˚ S E Gm/ which we denote ..xs;˚ /;x0/. We denote the dual basis for Y. E 2 H P Gm/ by .ys;˚ ;y0/. The element Y.G / equals ˚.s/ys;˚ y0 (see 3.7). As 2 s;˚ C GH is generated by . / Aut. / , Y.GH / is the Gal. al= /-submodule of C Gm C Q Q Q f j 2 g al 1 Y. E Gm/ generated by . (Here, Gal.Q =Q/ acts on S by s s ; it acts on Q S S D ı Y. E Gm/ Z Z through its action on S, ys;˚ ys;˚ ; these actions ˚ S D D factor through2 Gal.E=Q/). To begin the computation of GAH , we make a list of the tensors we know to be absolute Hodge cycles on the A˛. (a) The endomorphisms E End.A˚ / for each ˚. (More precisely, we mean the classes cl .e/ H .A˚ / H .A˚ /, e graph of e, e E.) A 2 A ˝ A D 2 (b) Let .A˚ ; E, End.A˚ // correspond to ˚ S, and let Gal.E=Q/. Deﬁne W ! 2 2 ˚ s s ˚ . There is an isomorphism A˚ A˚ induced by D f j 2 g !

˚ .:::;z./;:::/ .:::;z./;:::/ ˚ C 7! C

˚ ˚ C =˚.OE / C =˚.OE / whose graph is an absolute Hodge cycle. (Alternatively, we could have used the fact that 1 .A˚ ; E End.A˚ //, where , is of type ˚ to show that A˚ and A˚ W ! D ı are isomorphic.) Ld (c) Let .˚i /1 i d be a family of elements of S and let A i 1 Ai where Ai A˚i . Ä Ä Ld D D D Then E acts on A and H1.A;Q/ i 1 H1.Ai ;Q/ has dimension d over E. Under the P D D P assumption that ˚i constant (so that ˚i .s/ d=2, all s S), we shall apply (4.8) i D i D 2 to construct absolute Hodge cycles on A. For each i, there is an E-linear isomorphism M H1.Ai ;Q/ C H1.Ai /s ˝Q ! s S 2 such that s E acts on H1.Ai /s as s.e/. From the deﬁnitions one sees that 2 ( 1;0 H1.Ai /s ; s ˚i ; H1.Ai /s 0; 1 2 D H1.Ai /s ; s ˚i : … Thus, with the notations of (4.4), P as i ˚i .s/ D P P bs .1 ˚i .s// ˚i .s/ as: D i D i D P The assumption that ˚i constant therefore implies that D

as bs d=2; all s: D D 5 COMPLETION OF THE PROOF FOR ABELIAN VARIETIES OF CM-TYPE 39

For each i, choose a polarization i for Ai whose Rosati involution stabilizes E, and let i be the corresponding Riemann form. For any totally positive elements fi in F (the L maximal totally real subﬁeld of E) fi i is a polarization for A. Choose vi 0, D i ¤ vi H1.Ai ;Q/; then vi is a basis for H1.Ai ;Q/ over E. There exist i E such that 2 f g 2 i i and i .xvi ;yvi / TrE=Q.i xy/ for all x;y E. Thus i , where i .xvi ;yvi / i D D 2 D xy, is an E-Hermitian form on H1.Ai ;Q/ such that i .v;w/ TrE= .1i .v;w//. The 1 D Q E-Hermitian form on H1.A;Q/ P P P . xi vi ; yi vi / fi i .xi vi ;yi vi / D i def has the property that .v;w/ Tr .1.v;w// and is the Riemann form of . The D E=Q Q i discriminant of is fi . /. On the other hand, if s S restricts to on F , then i 1 2 sign.disc.// . 1/bs . 1/d=2. D D Thus, disc./ . 1/d=2f D for some totally positive element f of F . After replacing one fi with fi =f , we have that disc./ . 1/d=2, and that is split. Hence (4.8) applies. D d P In summary: let A i 1A˚i be such that i ˚i constant; then D ˚ D D Vd 1 d Á d d H .A;Q/. / H .A;Q/. / E 2 2 consists of absolute Hodge cycles. AH AH Q Since G ﬁxes the absolute Hodge cycles of type (a), G E Gm. It is ˚ therefore a torus, and we have an inclusion

AH Q S S Y.G / Y. E Gm/ Z Z D and a surjection, Q S S AH X. E Gm/ Z Z X.G /: D Q Let W be a space of absolute Hodge cycles. The action of the torus E Gm on Q W C decomposes it into a sum W indexed by the X. E Gm/ of subspaces ˝ ˚ AH 2 W on which the torus acts through . Since G fixes the elements of W , the for which AH W 0 map to zero in X.G /. ¤ On applying this remark with W equal to the space of absolute Hodge cycles described AH in (b), we find that xs;˚ xs;˚ maps to zero in X.G /, all Gal.E=Q/, s S, and 2 2 ˚ S. As Gal.E=Q/ acts simply transitively on S, this implies that, for a fixed s0 S, 2 AH 2 X.G / is generated by the image of xs ;˚ ;x0 ˚ S . f 0 j 2 g Let d.˚/ 0 be integers such that Pd.˚/˚ d=2 (constant function on S) where D d Pd.˚/. Then (c) shows that the subspace D def O E d.˚/ Vd L d.˚/ W H1.A˚ ;Q/˝ . d=2/ H1. A ;Q/. d=2/ D E D E ˚ L d.˚/ of Hd . A˚ ;Q/. d=2/ consists of absolute Hodge cycles. The remark then shows that P AH d.˚/xs;˚ d=2 maps to zero in X.G / for all s. Let Q P X X. E Gm/= Z.xs;˚ xs;˚ / D 5 COMPLETION OF THE PROOF FOR ABELIAN VARIETIES OF CM-TYPE 40 and regard xs ;˚ ;x0 ˚ S f 0 j 2 g as a basis for X. We know that

Q AH X. E Gm/ X.G / factors through X, and that therefore Y Y.GAH / ( Y.GH /) where Y is the submodule Q of Y. E Gm/ dual to X. H H LEMMA 5.2. The submodule Y.G /? of X orthogonal to Y.G / is equal to

nP d ˇ P d P o d.˚/xs ;˚ x0 ˇ d.˚/˚ , d.˚/ d 0 2 ˇ D 2 D I it is generated by the elements

P d P d d.˚/xs ;˚ x0; d.˚/˚ ; d.˚/ 0 all ˚: 0 2 D 2 H PROOF. As Y.G / is the Gal.E=Q/-submodule of Y generated by , we see that

P d H x d.˚/xs ;˚ x0 Y.G / D 0 2 2 ? P if and only if ;x 0 all Gal.E=Q/. But ˚.s/ys;˚ y0 and P h i D 2 P 1 D d C D ˚.s/ys;˚ x0, and so ;x d.˚/˚. s0/ . The first assertion is now C h i D 2 obvious. H As ˚ ˚ 1, xs ;˚ xs ;˚ x0 Y.G / and has positive coefficients d.˚/. By C D 0 C 0 2 ? adding enough elements of this form to an arbitrary element x Y.GH / we obtain an 2 ? element with coefficients d.˚/ 0, which completes the proof of the lemma. The lemma shows that Y.GH / Ker.X X.GAH // Y.GAH / . Hence Y.GH / ? D ? Y.GAH / and it follows that GH GAH ; the proof is complete. M.12 D 6 COMPLETION OF THE PROOF; CONSEQUENCES 41

6 Completion of the proof; consequences

Completion of the proof of Theorem 2.11

Let A be an abelian variety over C and let t˛, ˛ I , be Hodge cycles on A (relative to 2 id C C). To prove the Main Theorem 2.11, we have to show that the t˛ are absolute W ! Hodge cycles. Since we know the result for abelian varieties of CM-type, Theorem 2.15 shows that it remains only to prove the following proposition. PROPOSITION 6.1. There exists a connected smooth algebraic variety S over C and an abelian scheme Y S such that W ! (a) for some s0 S, Ys A; 2 0 D (b) for some s1 S, Ys is of CM-type; 2 1 (c) the t˛ extend to elements that are rational and of bidegree .0;0/ everywhere in the family.

The last condition means the following. Suppose that t˛ belongs to a tensor space 1 m.˛/ 1 m.˛/ T˛ HB .A/˝ :::; then there is a section t of R Q ˝ ::: over the universal D ˝ ˝ covering S of S (equivalently, over a finite covering of S) such that for s0 mapping to s0, Q 1 m.˛/ Q ts0 t˛, and for all s S, ts HB .Ys/˝ ::: is a Hodge cycle. Q D Q 2 Q Q 2 Q ˝ We sketch a proof of Proposition 6.1. The parameter variety S will be a Shimura variety and (b) will hold for a dense set of points s1. We may suppose that one of the t˛ is a polarization for A. Let H H1.A;Q/ and D let G be the subgroup of GL.H / Gm fixing the t˛. The Hodge structure on H defines 0 a homomorphism h0 C G.R/. Let G Ker.G Gm/; then ad.h0.i// is a Cartan W ! D ! involution on G0 because the real form of G0 corresponding to it fixes the positive definite C C form .x;h.i/y/ on H R where is a Riemann form for . In particular, G is reductive ˝ (see 3.6). Let X h C G.R/ h is conjugate to h0 under G.R/ . D f W ! j g Each h X defines a Hodge structure on H of type . 1;0/;.0; 1/ relative to which each 2 0 0; 1 f g t˛ is of bidegree .0;0/. Let F .h/ H H C. Since G.R/=K X, where K D ˝ 1 ! 1 is the centralizer of h0, there is an obvious real differentiable structure on X, and the tangent space to X at h0, Tgth0 .X/ Lie.GR/=Lie.K /. In fact, X is a Hermitian symmetric D 1 domain. The Grassmannian,

def Grassd .H C/ W H C W of dimension d . dimA/ ˝ D f ˝ j D g is a complex analytic manifold (even an algebraic variety). The map 0 X Grassd .H C/; h F .h/; W ! ˝ 7! is a real differentiable map, and is injective (because the Hodge ﬁltration determines the Hodge decomposition). The map on tangent spaces factors into

0 Tgth0 .X/ Lie.GR/=Lie.K / End.H C/=F End.H C/ Tgt.h0/.Grass/ D 1 ˝ ˝ D

0 Lie.GC/=F .Lie.GC// 6 COMPLETION OF THE PROOF; CONSEQUENCES 42

— the maps are induced by G.R/, G.C/, GL.H C/ and the filtrations on Lie.G / ! ! ˝ C and End.H C/ are those corresponding to the Hodge structure defined by h0. Thus, d ˝ identifies Tgth0 .X/ with a complex subspace of Tgt.h0/.Grass/, and so X is an almost- complex (in fact, complex) manifold (see Deligne 1979b, 1.1, for more details). (There is an alternative, more group-theoretic, description of the complex structure; see Knapp 1972, 2.4, 2.5.) 0 To each point h of X, we can attach a complex torus F .h/ H C=H.Z/, where H.Z/ n ˝ is some fixed lattice in H. For example, to h0 is attached

0 F .h0/ H C=H.Z/ Tgt0.A/=H.Z/; n ˝ D which is an abelian variety representing A. From the definition of the complex structure on X, it is clear that these tori form an analytic family B over X. Let g G.Q/ .g 1/H.Z/ nH.Z/ D f 2 j g some fixed integer n. For a suitably large n 3, will act freely on X, and so X will n again be a complex manifold. The theorem of Baily and Borel (1966) shows that S X D n is an algebraic variety. The group acts compatibly on B, and on forming the quotients, we obtain a complex analytic map Y S with Y B. For s S, Ys is a polarized complex torus (hence an W ! D n 2 abelian variety) with level n structure (induced by H1.Bh;Z/ H.Z/ where h maps to s). ! The solution Mn of the moduli problem for polarized abelian varieties with level n-structure in the category of algebraic varieties is also a solution in the category of complex analytic manifolds. There is therefore an analytic map S Mn such that Y is the pull-back of W ! the universal family on Mn. A theorem of Borel (1972, 3.10) shows that is automatically algebraic, from which it follows that Y=S is an algebraic family. For some connected component S of S, 1.S / S will satisfy (a) and (c) of the ı ı ! ı proposition. To prove (b) we shall show that, for some h X close to h0, B is of CM-type 2 h (cf. Deligne 1971c, 5.2). Recall (5) that an abelian variety is of CM-type if and only if its Mumford-Tate group is a torus. From this it follows that B , h X, is of CM-type if and only if h factors through h 2 the real points of a subtorus of G defined over Q. Let T be a maximal torus, defined over R, of the algebraic group K . (See Borel and 1 Springer 1966 for a proof that T exists, or apply the argument that follows.) Since h0.C/ is contained in the centre of K , h0.C/ T.R/. If T 0 is a torus in GR containing T , then 1 T 0 will centralize h0 and so T 0 K ; T is therefore maximal in GR. For a general (regular) 1 element of Lie.T /, T is the centralizer of . Choose a 0 Lie.G/ that is close to in 2 Lie.GR/, and let T0 be the centralizer of 0 in G. Then T0 is a maximal torus of G that is 1 defined over Q, and, because T0 is close to T , T0 gTg for some g G.R/. Now R R R D 2 h ad.g/ h0 factors through T0 , and so B is of CM-type. D ı R h This completes the proof of the main theorem. M.13

Consequences of Theorem 2.11 We end this section by giving two immediate consequences. 2p Let X be a complete smooth variety over a ﬁeld k and let H .Xet;Q`/.p/, ` 2 ¤ char.k/. Tate’s conjecture states that is in the Q`-span of the algebraic classes if there 6 COMPLETION OF THE PROOF; CONSEQUENCES 43

exists a subfield k0 of k finitely generated over the prime field, a model X0 of X over k0, 2p and a H .X0 k0;Q`/.p/ mapping to that is fixed by Gal.k0=k0/. 2 ˝ COROLLARY 6.2. Let A be an abelian variety over C. If Tate’s conjecture is true for A, then so also is the Hodge conjecture.

PROOF. We first show that, for any complete smooth variety X over C, Tate’s conjecture implies that all absolute Hodge cycles on X are algebraic. Let X0 be a model of X over p a subfield k0 of C finitely generated over Q. According to Proposition 2.9, CAH.X/ p D CAH.X0 k0/ and, after we have replaced k0 by a finite extension, Gal.k0=k0/ will act ˝ p p p trivially on CAH.X0 k0/. Let Calg.X/ denote the Q-subspace of CAH.X/ spanned by the ˝ p algebraic cycles on X. Tate’s conjecture implies that the Q`-span of CAH.X/ is contained in p p p p p the Q`-span of Calg.X/. Hence Calg.X/ Q` CAH.X/ Q`, and so Calg.X/ CAH.X/. ˝ D ˝2p p;p D Now let A be an abelian variety over C, and let t H .A;Q/ H . The image t 0 2p 2 \ of t in H .A/.p/ is a Hodge cycle relative to id C C, and so Theorem 2.11 shows that p A W ! t 0 C .A/. It is therefore in the Q-span of the algebraic cycles. 2 AH REMARK 6.3. The last result was first proved independently by Pjatecki˘ı Sapiroˇ 1971 and Deligne (unpublished) by an argument similar to that which concluded the proof of the main theorem. (Corollary 6.2 is easy to prove for abelian varieties of CM-type; in fact, Pohlmann 1968 shows that the two conjectures are equivalent in that case.) We mention also that Borovoi 1977 shows that, for an abelian variety X over a field k, the Q`-subspace of 2p H .Xet;Q`/.p/ generated by cycles that are Hodge relative to an embedding k , C is W ! independent of the embedding. M.14 A COROLLARY 6.4. Let A be an abelian variety over C and let G be the Mumford-Tate A group of A. Then dim.G / tr:deg k.pij / where pij are the periods of A. k PROOF. Same as that of Proposition 1.6. 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 44

7 Algebraicity of values of the -function

The next result generalizes Proposition 1.5.

PROPOSITION 7.1. Let k be an algebraically closed subﬁeld of C, and let V be a complete smooth variety of dimension n over k. If H B .V / maps to an absolute Hodge cycle 2 2r under

r 1 .2i/ H B .V / 7! H B .V /. r/ ' H 2n 2r .V /.n r/ , H 2n 2r .V /.n r/ 2r ! 2r ! B ! A C 2r then, for any C 1 differential r-form ! on VC whose class Œ! in HdR .V=C/ lies in H 2r .V=k/, dR Z ! .2i/r k. 2

PROOF. Proposition 2.9 shows that arises from an absolute Hodge cycle 0 on V=k. Let 2n 2r . 0/dR be the component of 0 in HdR .V=k/. Then, as in the proof of (1.5), Z r r 2n r ! .2i/ TrdR.. 0/dR Œ!/ .2i/ HdR .V=k/ .2i/ k. D [ 2 D

In the most important case of the proposition, k will be the algebraic closure Q of Q in C, and it will then be important to know not only that the period Z def r P.;!/ .2i/ ! D is algebraic, but also in which field it lies in. We begin by describing a general procedure for finding this field and then illustrate it by an example in which V is a Fermat hypersurface and the period is a product of values of the -function. Let V now be a complete smooth variety over a number field k C, and let S be a finite abelian group acting on V over k. Let V V k Q. When ˛ S C is a character of S D ˝ W ! taking values in k and H is a k-vector space on which S acts, we let

H˛ v H sv ˛.s/v, all s S . D f 2 j D 2 g 2r Assume that all Hodge cycles on VC are absolutely Hodge and that H .V .C/;C/˛ has r dimension 1 and is of bidegree .r;r/. Then .CAH.V/ k/˛ has dimension one over k. 2r 2r ˝ The actions of S and Gal.Q=k/ on HdR .V =Q/ HdR .V=k/ k Q commute because the ' ˝ r latter acts through its action on Q; they therefore also commute on CAH.V/ k, which 2r r ˝ embeds into H .V =Q/. It follows that Gal.Q=k/ stabilizes .C .V/ k/˛ and, as this dR AH ˝ has dimension 1, there is a character Gal.Q=k/ k such that W ! 1 r ./ ; Gal.Q=k/; C .V/ k ˛: D 2 2 AH ˝ B PROPOSITION 7.2. With the above assumptions, let H2r .V / and let ! be a C 1- 2r 2 2r differential 2r-form on V.C/ whose class Œ! in HdR .V=C/ lies in HdR .V=k/˛; then P.;!/ lies in an abelian algebraic extension of k, and

.P.;!// ./P.;!/ for all Gal.Q=k/: D 2 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 45

2r r PROOF. Consider Œ! HdR .V=C/˛ .CAH.V/ C/˛; then Œ! z for some z C, r 2 D ˝ D 2 .C .V/ k/˛. Moreover, 2 AH ˝ Â Ãr Z def 1 r P.;!/ ! z . .2i/ / zk; D 2i D ˝ 2 where we are regarding as an element of H 2r .V /.r/ k H B .V /. r/ k. Thus B ˝ D 2r _ ˝ P.;!/ 1Œ! C r .V/ k. 2 AH ˝ As 2r r Œ! H .V =Q/ C .V/ Q; 2 dR D AH ˝ this shows that P.;!/ Q. Moreover, 2 .P.;!/ 1Œ!/ ./ 1.P.;!/ 1Œ!/. D On using that Œ! Œ!, we deduce that D .P.;!// ./ P.;!/. D

r 2r REMARK 7.3. (a) Because CAH.V/ injects into H .V et;Q`/.r/, can be calculated from 2r the action of Gal.Q=k/ on H .V et;Q`/˛.r/. r B (b) The argument in the proof of the proposition shows that .2i/ H2r .V /. r/ 1 2r ˝ 2 and P.;!/ Œ! H .V =Q/ are different manifestations of the same absolute Hodge 2 dR cycle.

The Fermat hypersurface We shall apply (7.2) to the Fermat hypersurface

d d d V X0 X1 Xn 1 0 W C C C C D def 2i=d of degree d and dimension n, which we shall regard as a variety over k Q.e /. As D above, we let V V k Q, and we shall often drop the the subscript on V . D ˝ C It is known that the motive of V is contained in the category of motives generated by abelian varieties (see Deligne and Milne 1982, 6.26), and therefore Theorem 2.11 shows that every Hodge cycle on V is absolutely Hodge (ibid. 6.27). Let d be the group of dth roots of 1 in C, and let

n 1 MC S =(diagonal). D d i 0 D Then S acts on V=k according to the formula:

.::: i :::/.::: xi :::/ .::: i xi :::/; all .::: xi :::/ V.C/. W W W W D W W W W 2 The character group of S will be identiﬁed with

˚ n 2 ˇ P « X.S/ a .Z=dZ/ C ˇ a .a0;:::;an 1/; i ai 0 D 2 D C D I 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 46 here a X.S/ corresponds to the character 2 a def Qn 1 ai .0 :::/ i C0 i . D W 7! D D For a Z=dZ, we let a denote the representative of a in Z with 1 a d, and for 2 h i1 P Ä h i Ä a X.S/ we let a d ai N. 2 h i D h i 2 If H.V / is a cohomology group on which there is a natural action of k, we have a decomposition

M a H.V / H.V /a; H.V /a v v v; S . D D f j D 2 g

Let .Z=dZ/ act on X.S/ in the obvious way,

u .a0;:::/ .ua0;:::/; D and let Œa denote the orbit of a. The irreducible representations of S over Q (and hence the Q idempotents of QŒS) are classified by the these orbits, and so QŒS QŒa where QŒa is D a a field whose degree over Q is equal to the order of Œa. The map S C induces 7! W !L an embedding QŒa , k. Every cohomology group decomposes as H.V / H.V /Œa ! D where M H.V /Œa C .H.V / C/a . ˝ D ˝ 0 a Œa 02 Calculation of the cohomology n M.15 PROPOSITION 7.4. The dimension of H .V;C/a is 1 if no ai 0 or if all ai 0 and n is n D D even; otherwise H .V;C/a 0. D PROOF. The map d d n 1 n 1 .x0 x1 :::/ .x0 x1 :::/ P C P C W W 7! W W W ! n n n P defines a finite surjective map V P where P . P / is the hyperplane Xi 0. W ! L D There is an action of S on C, which induces a decomposition C . C/a. The r r n ' isomorphism H .V;C/ ' H .P ; C/ is compatible with the actions of S, and so gives ! rise to isomorphisms r r n H .V;C/a ' H .P ;. C/a/: ! Clearly . C/0 C, and so D r n r n H .P ;. C/0/ H .P ;C/ for all r: '

For a 0, the sheaf . C/a is locally constant of dimension 1, except over the hyperplanes ¤ Hi Xi 0 corresponding to the i for which ai 0, where it is ramiﬁed. It follows that W D ¤ r n H .P ;. C/a/ 0; r n; a 0; D ¤ ¤ n n n and so . 1/ dimH .P ;. C/a/ is equal to the Euler-Poincare´ characteristic of . C/a (a 0). We have ¤ EP.P n;. / / EP.P n S H ; /. C a ai 0 i C D X ¤ 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 47

Suppose ﬁrst that no ai is zero. Then

P n n .x0 ::: xn xi / .x0 ::: xn/ P ' P W W W $ W W W $ induces n Sn 1 n Sn n 1 P i C0 Hi P i 0 Hi P ; X D $ X D [ n 1 n where Hi denotes the coordinate hyperplane in P C or P . As

n S n 1 n 1 S n S .P Hi P / .P Hi / P Hi ; X [ t X D X n S n and P Hi , being topologically isomorphic to .C/ , has Euler-Poincare´ characteristic X zero, we see that

n Sn 1 n 1 Sn n 0 n EP.P C Hi / EP.P Hi / ::: . 1/ EP.P / . 1/ : X D X D D D n S r n r If some, but not all, ai are zero, then P Hi .C/ C with r 1, and so n S r n r X EP.P Hi / 0 1 0: X D D REMARK 7.5. Note that the above argument shows that the primitive cohomology of V ,

n M n H .V;C/prim H .V;C/a. D a 0 ¤ n n The action of S on H .V;C/ respects the Hodge decomposition, and so H .V;C/a is purely of bidegree .p;q/ for some p;q with p q n. C D n PROPOSITION 7.6. If no ai 0, then H .V;C/a is of bidegree .p;q/ with p a 1. D D h i PROOF. We apply the method of Grifﬁths 1969, 8. When V is a smooth hypersurface in n 1 P C , Grifﬁths shows that the maps in

n 1 n 1 0 n 1 n 1 n 2 n 1 n 2 n 1 H C .P C ;C/ H C .P C V;C/ H C .P C ;C/ H C .P C ;C/ X V

' H n.V /. 1/ induce an isomorphism

n 1 n 1 n H C .P C V;C/ ' H .V /. 1/prim X ! and that the Hodge ﬁltration on H n.V /. 1/ has the following explicit interpretation: identify n 1 n 1 n 1 n 1 n 1 n H C .P C V;C/ with .P C V;˝ C /=d .P C V;˝ / and let X X X n 1 ˚ n 1 n 1 ˇ « ˝pC .V / ! .P C V;˝ C / ˇ ! has a pole of order p on V D 2 X Ä I then the map n 1 n R ˝pC .V / H .V;C/ W ! determined by 1 Z ;R.!/ !; all Hn.V;C/; h i D 2i 2 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 48 induces an isomorphism

n 1 n n p n n p 1 n ˝pC .V /=d˝p 1 F H .V /. 1/prim F C H .V /prim. ! D (For example, when p 1, R is the residue map D ˝n 1.V / F nH n.V / H 0.V;˝n//: 1 C ! D n 1 Let f be the irreducible polynomial deﬁnining V . As ˝ nC 1 .n 2/ OPn 1 has basis P C C C P i !0 . 1/ Xi dX0 ::: dXi ::: dXn; D ^ ^ ^ ^ p any differential form ! equals P!0=f with P a homogeneous polynomial of degree p deg.f / .n 2/ lies in ˝n 1.V /. In particular, when V is our Fermat surface, C pC

a0 1 an 1 1 X0h i Xnh 1C i ! C !0 d d a b D .X0 Xn 1/h i C C C a0 an 1 X h i X h C i 0 n 1 P i dX0 dXi C . 1/ ::: ::: d d a X0 Xi D .X0 Xn 1/h i ^ ^ ^ C C C n 1 1 a lies in ˝ aC .V /. For S, Xi i Xi , and so ! !. This shows that h i 2 D D n n a 1 n H .V;C/ a F h iC H .V;C/: b Since a 1 n 1 a , we can rewrite this inclusion as h i D C h i n a 1 n H .V;C/a F h i H .V;C/. n n Thus H .V;C/a is of bidegree .p;q/ with p a 1. The complex conjugate of H .V;C/a n h i is H .V;C/ a, and is of bidegree .q;p/. Hence n p q a 1 n 1 a D h i D C h i and so p a 1. Ä h i Recall that H n.V / L H n.V / ; thus (7.4) shows that H n.V / has dimen- B Œa a Œa B a0 B Œa D 02 sion 1 over QŒa when no ai is zero and otherwise n n H .V / H .V /prim 0. B Œa \ B D n n n COROLLARY 7.7. Let a be such that no ai 0. Then H .V / is purely of type . ; / if D B Œa 2 2 and only if ua is independent of u. h i n PROOF. As a a n 2, ua is constant if and only if ua 2 1 for all u h i C h i D C h ni h i D C 2 .Z=dZ/, i.e., if and only if a0 1 for all a0 Œa. Thus the corollary follows from h i D 2 C 2 the proposition.

n COROLLARY 7.8. If no ai 0 and ua is constant, then C .V/ has dimension one D h i AH Œa over QŒa:

PROOF. This follows immediately from (7.7) because, as we have remarked, all Hodge cycles on V are absolutely Hodge. 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 49

The action of Gal.Q=k/ on the etale´ cohomology

Let p be a prime ideal of k not dividing d, and let Fq be the residue field of p. Then d q 1 j and reduction modulo p defines an isomorphism d F whose inverse we denote t. Fix ! d an a .a0;:::;an 1/ X.S/ with all ai nonzero, and define a character "i Fq d by D C 2 W ! .1 q/=d ai "i .x/ t.x / ; x 0: D ¤ Q Q n 1 As "i 1, the product "i .xi / is well-defined for x .x0 ::: xn 1/ P C .Fq/, and D D W W C 2 we define a Jacobi sum n 1 n P QC J."0;:::;"n 1/ . 1/ "i .xi / C D n x P .Fq / i 0 2 D n P n 1 where P is the hyperplane Xi 0 in P C . (As usual, we set "i .0/ 0.) Let be a D D nontrivial additive character Fq C and define Gauss sums W ! P g.p;ai ; / "i .x/ .x/ D x Fq 2 n 1 a QC g.p;a/ qh i g.p;ai ; /: D i 0 D a 1 LEMMA 7.9. The Jacobi sum J."0;:::;"n 1/ qh i g.p;a/. C D PROOF. We have n 1 a QC P qh ig.p;a/ . "i .x/ .x// D i 0 x Fq D 2 Ân 1 Ã n P QC P . 1/ "i .xi / . xi /; x .x0;:::/ D n 2 i 0 D x FqC 2 D Ân 1 Ã n P P QC P . 1/ "i .xi / . xi / : D n 1 x P .Fq / Fq i 0 2 C 2 D Q We can omit the from "i .xi /, and so obtain

n 1 ! a n P QC P P qh ig.p;a/ . 1/ "i .xi / . . xi // : D n 1 x P .Fq / i 0 Fq 2 C D 2 Since n 1 n 1 ! P QC QC P "i .xi / "i .x// 0; x i 0 D i 0 x Fq D D D 2 we can replace the sum over Fq by a sum over Fq. From 2 2 ( P P P q if xi 0 . xi / D D P Fq 0 if xi 0 2 ¤ we deduce ﬁnally that

n 1 a n P QC qh ig.p;a/ . 1/ q "i .xi // D n x P .Fq / i 0 2 D qJ."0;:::;"n/. D 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 50

Note that this shows that g.p;a/ is independent of and lies in k. Let ` be a prime number such that ` − d, p − `, and d ` 1. Then Q` contains a primitive j dth root of 1 and so, after choosing an embedding k , Q`, we can assume g.p;a/ Q`. ! 2 ab PROPOSITION 7.10. Let Fp Gal.Q=k/ be a geometric Frobenius element of p − d; for n 2 all v H .V et;Q`/a, 2 a 1 Fpv q g.p;a/v. D h i PROOF. As p − d, V reduces to a smooth variety Vp over Fq and the proper-smooth base n n change theorem shows that there is an isomorphism H .V;Q`/ H .V p;Q`/ compatible n ! with the action of S and carrying the action of Fp on H .V;Q`/ into the action of the Frobe- n n nius endomorphism Frob on H .V p;Q`/. The comparison theorem shows that H .V;Q`/a has dimension 1, and so it remains to compute n n Tr.Fp H .V;Q`/a/ Tr.Frob H .V p;Q`/a/: j D j n Let Vp P be as before. Then W ! n n n H .V p;Q`/a H .P ;. Q`/a/, D and the Lefschetz trace formula shows that n n n X . 1/ Tr.Frob H .P ;. Q`/a/ Tr.Frob .. Q`/a/x/ (2) j D j x P n.F/ 2 where .. Q`/a/x is the stalk of . Q`/a at x. n d Fix an x P .Fq/ with no xi zero, and let y Vp.Fq/ map to x; thus yi xi all i. 2 2 1 D 1 .x/ Then .x/ y S , and . Q`/x is the vector space Q` . D f j 2 g If denotes the arithmetic Frobenius automorphism (i.e., the generator z zq of 7! Gal.Fq=Fq/), then q 1 q 1 q d d .yi / y x yi t.x /yi ; 0 i n 1; D i D i D i Ä Ä and so 1 q Frob.y/ y where .::: t.x d / :::/ S: D D W i W 2 Thus Frob acts on . Q`/x as , and for v .. Q`/a/x, we have 2 n 1 av a QC Frob.v/ v ; "i .xi / k Q`. D D D i 0 2 D Consequently, n 1 QC Tr.Frob .. Q`/a/x/ "i .xi /. j D i 0 D If some xi 0, then both sides are zero (. Q`/a is ramiﬁed over the coordinate hyper- D planes), and so, on summing over x and applying (2) and (7.9), we obtain the proposition.

COROLLARY 7.11. Let a be such that no ai is zero and ua is constant. Then, for all n n h i v H .V et;Q`/a. /, 2 2 Fpv g.p;a/v: D n PROOF. The hypotheses on a imply that a 2 1. Therefore, when we write v v0 1 n h i D C D ˝ with v0 H .V et;Q`/a, 2 n n Fpv Fpv0 Fp1 q 2 g.p;a/v0 q 2 g.p;a/v. D ˝ D ˝ D 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 51

Calculation of the periods Recall that that the -function is deﬁned by Z 1 t s dt .s/ e t ; s > 0; D 0 t and satisﬁes the following equations

.s/ .1 s/ .sins/ 1 D .1 s/ s .s/: C D

The last equation shows that, for s Q, the class of .s/ in C=Q depends only on the 2 class of s in Q=Z. Thus, for a X.S/, we can deﬁne 2 n 1 a QC ai .a/ .2i/h i . d / C=Q: Q D i 0 2 D Let V o denote the open afﬁne subvariety of V with equation

d d Y1 Yn 1 1 (so Yi Xi =X0). C C D D Denote by the n-simplex

˚ n 1 ˇ P « .t1;:::;tn 1/ R C ˇ ti 0; ti 1 C 2 D o and deﬁne 0 V .C/ to be W ! 1 1 1 d d 2i=2d pd d .t1;:::;tn 1/ ."t1 ;:::;"tn 1/; " e 1; ti > 0: C 7! C D D P LEMMA 7.12. Let a0;:::;an 1 be positive integers such that ai 0 mod d. Then C Á Z dY dY 1 n 1 a1 an 1 1 n a0 QC ai Y1 Yn C1 ::: .1 / d 0./ C Y1 ^ ^ Yn D 2i i 0 D where e2i=d . D

PROOF. Write !0 for the integrand. Then Z Z !0 0.!0/ 0./ D Z 1 1 d a1 d an 1 n dt1 dtn ."t1 / ."tn 1/ C d ::: D C t1 ^ ^ tn Z b1 bn 1 dt1 dtn c t1 tn C1 ::: D C t1 ^ ^ n

a a 1 n where bi ai =d and c " 1 n 1 . / . On multiplying by D D CC C d Z 1 t b1 bn 1 .1 b0/ .1 b1 bn 1/ e t CC C dt D C C C C D 0 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 52 we obtain Z Z Z 1 t b1 bn 1 b1 bn 1 dt1 dtn .1 b0/ !0 c e t CC C t1 tn C1 ::: dt. 0./ D 0 C t1 ^ ^ n ^

If, on the inner integral, we make the change of variables si tti , the integral becomes D Z Z 1 t b1 bn 1 ds1 dsn c e s1 :::sn C1 ::: dt 0 .t/ C s1 ^ ^ sn ^ where P .t/ .s1;:::;sn 1/ si 0; si t . D f C j D g P We now let t si , and we obtain D Z Z Z 1 1 s1 sn 1 b1 1 bn 1 ds1 dsn 1 .1 b0/ ! c e C s1 :::snC1 C ::: C 0./ D 0 0 C s1 ^ ^ sn 1 C c .b1/ .b2/::: .bn/ .1 bn 1/ D C C cbn 1 .b1/::: .bn 1/. D C C The formula recalled above shows that

.1 b0/ =.sinb0/ .b0/, D and so

1 a0 sinb0 c .1 b0/ " .b0/ mod Q D ib0 ib0 ! 1 2ib0=2 e e e .b0/ D 2i

1 2a0 .1 " / .b0/: D 2i The lemma is now obvious.

The group algebra QŒS acts on the Q-space of differentiable n-simplices in V.C/. For a X.S/ and .1;:::;;:::/ ( e2i=d in the ith position), deﬁne 2 i D D n 1 QC .1 / 1 ./ V o. / i 0 C D i 0 D where 0 and are as above.

o PROPOSITION 7.13. Let a X.S/ be such that no ai is zero, and let ! be the differential 2

a10 an0 1 dY1 dYn Y1 :::Yn C1 ::: C Y1 ^ ^ Yn o on V , where a represents ai , and a 0. Then i0 i0 (a) !o a!o; D n 1 R o 1 QC ai ai (b) ! 2i .1 / d . D i 0 D 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 53

PROOF. (a) This is obvious since

Â Ã 1 i Yi Yi : D 0 (b)

Z Z n 1 o QC o ! .1 i /! D 0./ i 1 D n 1 Z QC .1 ai / !o D i 1 0./ D 1 n 1 QC ai ai .1 / d . D 2i i 0 D

REMARK 7.14. From the Gysin sequence

n 2 o n n o .C /H .V V ;C/ H .V;C/ H .V ;C/ 0 X ! ! ! we obtain an isomorphism

n n o H .V;C/prim H .V ;C/, ! which shows that there is an isomorphism

n n o o n o n 1 H .V=k/prim H .V =k/ .V ;˝ /=d .V ;˝ /: dR ! dR D C o o n The class Œ! of the differential ! lies in HdR.V=k/a. Correspondingly, we get a C 1 differential n-form on V.C/ such that n n (a) the class Œ! of ! in HdR.V=C/ lies in HdR.V=k/a, and Z 1 n 1 n 1 (b) ! QC .1 ai / ai , where QC .1 / 1 ./. d i 0 D 2i i 0 D i 1 D D n Note that, if we regard V as a variety over Q, then Œ! even lies in HdR.V=Q/.

The theorem Recall that for a X.S/, we set 2 n 1 a QC ai .a/ .2i/h i . d /. C=Q/ Q D i 0 2 D and for p a prime ideal of k not dividing d, we set

n 1 a QC g.p;a/ qh i g.p;ai ; / D i 0 D P 1 q Áai g.p;ai ; / t x d .x/ D x Fq 2 where q is the order of the residue ﬁeld of p. 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 54

THEOREM 7.15. Let a X.S/ have no ai 0 and be such that ua a . n=2 1/ 2 D h i D h i D C for all u .Z=dZ/. 2 def 2i (a) Then .a/ and generates an abelian extension of k .e d /: Q Q Q 2 ab D (b) If Fp Gal.Q=k/ is the geometric Frobenius element at p, then 2

Fp. .a// g.p;a/ .a/: Q D Q

def (c) For all Gal.Q=Q/, a./ .a/= .a/ lies in k; moreover, for all u .Z=dZ/; 2 D Q Q 2

u.a.// ua./ D where u is the element of Gal.k=Q/ deﬁned by u.

PROOF. Choose H B .V / and ! as in (7.14). Then all the conditions of (7.2) are fulﬁlled 2 n with ˛ the character a. Moreover, (7.14) and (7.11) show respectively that

Qn 1 ai P.;!/ .a/ . a/; where .a/ i C0 .1 /; D Q D D and 1 .Fp/ g.p;a/ . D As .a/ k, (7.2) shows that . a/ generates an abelian algebraic extension of k and that 2 Q 1 Fp . a/ g.p;a/ . a/: Q D Q It is clear from this equation that g.p;a/ has absolute value 1 (in fact, it is a root of 1); thus

g.p;a/ 1 g.p;a/ g.p; a/. D D This proves (a) and (b) for a and hence for a. To prove (c) we have to regard V as a variety over Q. If S is interpreted as an algebraic group, then its action on V is rational over Q. This means that

.x/ ./.x/; Gal.Q=Q/; S.Q/; x V.Q/ D 2 2 2 and implies that

n . / ./. /; Gal.Q=Q/; S.Q/; C .V/. D 2 2 2 AH n Therefore Gal.Q=Q/ stabilizes CAH.V/Œa and, as this is a one-dimensional vector space n over QŒa, there exists for any CAH.V/Œa a crossed homomorphism Gal.Q=Q/ 2 W n ! QŒa such that . / ./ for all . On applying the canonical map C .V/Œa D AH ! C n .V/ k to this equality, we obtain AH ˝ Œa . 1/ ./a. 1/: ˝ D ˝ n=2 B n n We take to be the image of .2i/ Hn .V /. 2 / in CAH.V/Œa. Then (cf. 1 ˝ 2 7.3), . 1/dR P.;!/ Œ! if Œ! is as in (7.14). Hence ˝ D

a P.;!/ .a/ ./ a./ . D P.;!/ D .a/ 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 55

On comparing

a . a/ a./ ./ and D . a/ ua . ua/ ua./ ./ , D . ua/ and using that .. ua// .u.. a/// u.. a//; D D one obtains (c) of the theorem.

REMARK 7.16. (a) The ﬁrst statement of the theorem, that .a/ is algebraic, has an ele- Q mentary proof; see the appendix by Koblitz and Ogus to Deligne 1979a. M.16 (b) Part (b) of the theorem has been proved up to sign by Gross and Koblitz (1979, 4.5) using p-adic methods.

REMARK 7.17. Let Id be the group of ideals of k prime to d, and consider the character

def Q ri Q ri a p g.a;a/ g.pi ;a/ I k : D i 7! D W d ! When a satisﬁes the conditions of the theorem, then this is an algebraic Hecke character (Weil 1952, 1974; see also Deligne 1972, 6) . This means that there exists an ideal m of k (dividing a power of d) and a homomorphism alg k k that is algebraic (i.e., W ! deﬁned by a map of tori) and such that, for all x k totally positive and 1 mod m, 2 Á g..x/;a/ alg.x/. There is then a unique character D ab a Gal.Q=k/ k W ! such that a.Fp/ g.p;a/ for all p prime to d. Part (b) of the theorem can be stated as D

. .a// a./ .a/, all Gal.k=k/: Q D Q 2 (There is an elegant treatment of algebraic Hecke characters in Serre 1968, Chapter II. Such a character with conductor dividing a modulus m corresponds to a character of the torus Sm (ibid. p. II-17) . The map alg is

k Tm , Sm k . ! ! ! One deﬁnes from a character of the idele` class group as in (ibid. II 2.7). Weil’s deter- 1 mination of alg shows that is of ﬁnite order; in particular, it is trivial on the connected 1 ab component of the idele` class group, and so gives rise to a character a Gal.Q=k/ k:/ W ! Restatement of the theorem

1 1 1 For b d Z=Z, we write b for the representative of b in d Z with d b 1. Let P2 h i Ä h1 i Ä b n.b/ıb be an element of the free abelian group generated by the set d Z=Z 0 , D P X f g and assume that n.b/ ub c is an integer independent of u Z=dZ. Deﬁne h i D 2 1 Y .b/ . b /n.b/. Q D .2i/c h i b 7 ALGEBRAICITY OF VALUES OF THE -FUNCTION 56

Let p be a prime ideal of k, not dividing d, and let Fq be the residue field at p. For a non-trivial additive character of Fq, define 1 Y p p n.b/ p P b.1 q/ g. ;b/ c g. ;b; / , where g. ;b; / t.x / .x/. D q D x Fq b 2 As in (7.17), p g.p;b/ defines an algebraic Hecke character of k and a character 7! b Gal.Q=Q/ C such that b.Fp/ g.p;b/ for all p − b. W ! D def THEOREM 7.18. Assume that b Pn.b/ı satisfies the condition above. D b (a) Then .b/ kab, and for all Gal.Q=k/ab; Q 2 2

.b/ b./ .b/: Q D Q

(b) For Gal.Q=Q/, let b./ .b/= .b/; then b./ k, and, for any u 2 D Q Q 2 2 .Z=dZ/, u.b.// ub./. D PROOF. Suppose ﬁrst that n.b/ 0 for all b. Let n 2 Pn.b/, and let a be an .n 2/- a C D C tuple in which each a Z=dZ occurs exactly n d times. Write a .a0;:::;an 1/. Then 2 D C P P ai d. n.b/b/ dc .modd/ 0; D D D and so a X.S/. Moreover, 2 def 1 P P ua uai n.b/ ub c h i D d h i D h i D for all u Z=dZ. Thus ua is constant, and c a . We deduce that .a/ .b/, 2 h i D h i Q D Q g.p;a/ g.p;b/, and a b. Thus, in this case, (7.18) follows immediately from (7.15) D D and (7.17). P Let b be arbitrary. For some N , b N b0 has positive coefﬁcients, where b0 ı . C D b Thus (7.18) is true for b N b0. Since C

.b1 b2/ .b1/ .b2/ mod Q Q C D Q Q and g.p;b1 b2/ g.p;b1/g.p;b2/ C D this completes the proof.

REMARK 7.19. (a) Part (b) of the theorem determines .ub/ (up to multiplication by a rational number) starting from .b/. (b) Conjecture 8.11 of Deligne 1979a is a special case of part (a) of the above theorem. The more precise form of the conjecture, ibid. 8.13, can be proved by a modiﬁcation of the above methods. REFERENCES 57

FINALNOTE The original seminar of Deligne comprised fifteen lectures, given between 29/10/78 and 15/5/79. The first six sections of these notes are based on the first eight lectures of the seminar, and the seventh section on the last two lectures. The remaining five lectures, which the writer of these notes was unable to attend, were on the following topics: 6/3/79 review of the proof that Hodge cycles on abelian varieties are absolutely Hodge; discussion of the expected action of the Frobenius endomorphism on the image of an absolute Hodge cycle in crystalline cohomology; 13/3/79 definition of the category of motives using absolute Hodge cycles; semisimplicity of the category; existence of the motivic Galois group G; 20/3/79 fibre functors in terms of torsors; the motives of Fermat hypersurfaces and K3- surfaces are contained in the category generated by abelian varieties; 27/3/79 Artin motives; the exact sequence

1 Gı G Gal.Q=Q/ 1; ! ! ! ! 1 indentification of Gı with the Serre group, and description of the Gı-torsor ./; 1 3/4/79 action of Gal.Q=Q/ on Gı; study of G Q`; Hasse principle for H .Q;Gı/. ˝Q Most of the material in these five lectures is contained in Deligne and Milne 1982, 6, and Deligne 1982. The writer of these notes is indebted to P. Deligne and A. Ogus for their criticisms of the first draft of the notes and to Ogus for his notes on which section seven is largely based.

References

ARTIN,M.,GROTHENDIECK,A., AND VERDIER,J. 1972, 1973. Theorie´ des topos et cohomologie etale´ des schemas.´ Lecture Notes in Math. 269, 270, 305. Springer-Verlag, Berlin. SGA4.

BAILY,JR., W. L. AND BOREL,A. 1966. Compactiﬁcation of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84:442–528.

BOREL,A. 1972. Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geometry 6:543–560.

BOREL,A. AND SPRINGER, T. A. 1966. Rationality properties of linear algebraic groups, pp. 26–32. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965). Amer. Math. Soc., Providence, R.I.

BOROVOI, M. V. 1977. The Shimura-Deligne schemes MC.G; h/ and the rational cohomology classes of type .p; p/ of abelian varieties, pp. 3–53. In Problems of group theory and homological algebra, No. 1 (Russian). Yaroslav. Gos. Univ., Yaroslavl0.

DELIGNE, P. 1968. Theor´ eme` de Lefschetz et criteres` de deg´ en´ erescence´ de suites spectrales. Inst. Hautes Etudes´ Sci. Publ. Math. 35:259–278.

DELIGNE, P. 1971a. Theorie´ de Hodge. II. Inst. Hautes Etudes´ Sci. Publ. Math. 40:5–57.

DELIGNE, P. 1971b. Travaux de Grifﬁths, pp. Lecture Notes in Math., Vol. 180. In Seminaire´ Bourbaki, 22eme` annee´ (1969/70), Exp. No. 376. Springer, Berlin. REFERENCES 58

DELIGNE, P. 1971c. Travaux de Shimura, pp. 123–165. Lecture Notes in Math., Vol. 244. In Seminaire´ Bourbaki, 23eme` annee´ (1970/71), Exp. No. 389. Springer, Berlin.

DELIGNE, P. 1972. La conjecture de Weil pour les surfaces K3. Invent. Math. 15:206–226.

DELIGNE, P. 1979a. Valeurs de fonctions L et periodes´ d’integrales,´ pp. 313–346. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII. Amer. Math. Soc., Providence, R.I.

DELIGNE, P. 1979b. Variet´ es´ de Shimura: interpretation´ modulaire, et techniques de construction de modeles` canoniques, pp. 247–289. In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII. Amer. Math. Soc., Providence, R.I.

DELIGNE, P. 1982. Motifs et groupes de Taniyama, pp. 261–279. In Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York.

DELIGNE, P. AND MILNE,J.S. 1982. Tannakian categories, pp. 101–228. In Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics 900. Springer-Verlag, Berlin.

DELIGNE, P., MILNE,J.S.,OGUS,A., AND SHIH,K.-Y. 1982. Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics 900. Springer-Verlag, Berlin.

DEMAZURE,M. 1976. Demonstration´ de la conjecture de Mumford (d’apres` W. Haboush), pp. 138–144. Lecture Notes in Math., Vol. 514. In Seminaire´ Bourbaki (1974/1975: Exposes´ Nos. 453–470), Exp. No. 462. Springer, Berlin.

GRIFFITHS, P. A. 1969. On the periods of certain rational integrals. I, II. Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90:496–541.

GROSS,B.H. 1978. On the periods of abelian integrals and a formula of Chowla and Selberg. Invent. Math. 45:193–211.

GROSS,B.H. AND KOBLITZ,N. 1979. Gauss sums and the p-adic -function. Ann. of Math. (2) 109:569–581.

GROTHENDIECK, A. 1958. La theorie´ des classes de Chern. Bull. Soc. Math. France 86:137–154.

GROTHENDIECK,A. 1966. On the de Rham cohomology of algebraic varieties. Inst. Hautes Etudes´ Sci. Publ. Math. 29:95–103.

KATZ,N.M. 1970. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Etudes´ Sci. Publ. Math. 39:175–232.

KATZ,N.M. AND ODA, T. 1968. On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8:199–213.

KNAPP, A. W. 1972. Bounded symmetric domains and holomorphic discrete series, pp. 211–246. Pure and Appl. Math., Vol. 8. In Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970). Dekker, New York.

LANDHERR, W. 1936. Aquivalenz Hermitescher Formen uber¨ einem beliebigen algebraischen Zahlkorper.¨ Abh. Math. Semin. Hamb. Univ. 11:245–248.

MILNE,J.S. 1980. Etale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J. REFERENCES 59

MILNE,J.S. AND SHIH,K.-Y. 1982. Conjugates of Shimura varieties, pp. 280–356. In Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics. Springer-Verlag, Berlin.

PJATECKI˘I Sˇ APIRO,I.I. 1971. Interrelations between the Tate and Hodge hypotheses for abelian varieties. Mat. Sb. (N.S.) 85(127):610–620.

POHLMANN,H. 1968. Algebraic cycles on abelian varieties of complex multiplication type. Ann. of Math. (2) 88:161–180.

SAAVEDRA RIVANO,N. 1972. Categories´ Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Springer-Verlag, Berlin.

SERRE, J.-P. 1955–1956. Geom´ etrie´ algebrique´ et geom´ etrie´ analytique. Ann. Inst. Fourier, Grenoble 6:1–42.

SERRE, J.-P. 1968. Abelian l-adic representations and elliptic curves. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. W. A. Benjamin, Inc., New York-Amsterdam.

SINGER,I.M. AND THORPE,J.A. 1967. Lecture notes on elementary topology and geometry. Scott, Foresman and Co., Glenview, Ill.

WARNER, F. W. 1971. Foundations of differentiable manifolds and Lie groups. Scott, Foresman and Co., Glenview, Ill.-London.

WATERHOUSE, W. C. 1979. Introduction to afﬁne group schemes, volume 66 of Graduate Texts in Mathematics. Springer-Verlag, New York.

WEIL, A. 1952. Jacobi sums as “Grossencharaktere”.¨ Trans. Amer. Math. Soc. 73:487–495.

WEIL,A. 1958. Introduction a` l’etude´ des variet´ es´ kahl¨ eriennes.´ Publications de l’Institut de Mathematique´ de l’Universite´ de Nancago, VI. Actualites´ Sci. Ind. no. 1267. Hermann, Paris.

WEIL,A. 1974. Sommes de Jacobi et caracteres` de Hecke. Nachr. Akad. Wiss. Gottingen¨ Math.-Phys. Kl. II pp. 1–14. M ENDNOTES (BY J.S. MILNE) 60

M Endnotes (by J.S. Milne)

M.1. (p.1) The following changes from the original have been made: Numerous minor improvements to the exposition. ˘ Numerous misprints ﬁxed; major corrections have been noted in these endnotes. ˘ Part of the general introduction to the volume Deligne et al. 1982 (LNM 900) has been ˘ placed at the start of the introduction.

Some changes of notation have been made — the footnote DR has been replaced by dR, ˘ f A has been replaced by Af , and (isomorphism) has been distinguished from ' (canonical isomorphism). These endnotes and all footnotes marked “Added” have been added. ˘ The original numbering has been retained. ˘ M.2. (p.3) For a description of these consequences, see

Deligne, Pierre, Cycles de Hodge absolus et periodes´ des integrales´ des variet´ es´ abeliennes.´ Abelian functions and transcendental numbers (Colloq., Ecole´ Polytech., Palaiseau, 1979). Mem.´ Soc. Math. France (N.S.) 1980/81, no. 2, 23–33.

For applications of the results of these notes to the periods of motives attached to Hecke characters, see

Schappacher, Norbert, Periods of Hecke characters. Lecture Notes in Mathematics, 1301. Springer-Verlag, Berlin, 1988.

2p M.3. (p.4) Say that a cohomology class in H .A;Q/.p/ is a split Weil class if there exists a CM-ﬁeld E, ˘ a homomorphism E End.A/, and ˘ W ! a polarization of A satisfying the conditions (a,b) of (4.8) ˘ V2p 1 2p such that the class lies in the subspace E H .A;Q/.p/ of H .A;Q/.p/. By assumption, the algebraic classes are accessible. The proof of Theorem 4.8 will show that all split Weil classes are accessible once we check that the family in the proof contains an abelian variety for which the Hodge conjecture is true. But, in the proof, we can take A0 to be any abelian variety of dimension d=2, and it is well-known that the Hodge conjecture holds for powers of an elliptic curve (see p. 107 of Tate, Algebraic cycles and poles of zeta functions, 1965). Now the argument in 5 shows that all Hodge classes on abelian varieties of CM-type are accessible, and Proposition 6.1 shows the same result for all abelian varieties. Condition (a) is checked in (2.1), (b) is obvious from the deﬁnition of absolute Hodge cycle, (c) is proved in (3.8), and (d) is proved in (2.12).

M.4. (p.7) The “and so” is misleading since the degeneration of the spectral sequence by itself does not provide the Hodge decomposition. See mathoverﬂow questions 28265 and 311700 for a discussion of this. p n From the spectral sequence, we get a descending ﬁltration F on the groups H .X;C/ such that H n .H n F p/ .H n F q/ D \ ˚ \ M ENDNOTES (BY J.S. MILNE) 61 for all n;p;q with p q n 1. This implies that C D C H n H p;q D ˚ with H p;q H p q F p F q H q.X;˝p/: D C \ \ D M.5. (p. 14) Grothendieck conjectured that the only relations between the periods come from algebaic cycles.

. . . it is believed that if [the elliptic curve] is algebraic (i.e., its coefﬁcients g2 and g3 are algebraic), then !2 and !3 are transcendental, and it is believed that if X has no complex multiplication, then !1 and !2 are algebraically independent. This conjecture extends in an obvious way to the set of periods .!1;!2;1;2/ and can be rephrased also for curves of any genus, or rather for abelian varieties of dimension g, involving 4g periods. (Grothendieck 1966, p102).

Also:

For the period matrix itself, Grothendieck has made a very interesting conjecture concerning its relations, and his conjecture applies to a general situation as follows. Let V be a projective, nonsingular variety defined over the rational numbers. One can define the cohomology of V with rational coefficients in two ways. First, by means of differential forms (de Rham), purely algebraically, thereby obtaining a vector space Hdiff.V;Q/ over Q. Secondly, one can take the singular cohomology Hsing.V;Q/ with rational coefficients, i.e., the singular cohomology of the complex manifold VC. Let us select a basis for each of these vector spaces over Q, and let us tensor these spaces over C. Then there is a unique (period) matrix ˝ with complex coefficients which transforms one basis into the other. Any algebraic cycle on V or the products of V with itself will give rise to a polynomial relation with rational coefficients among the coefficients of this matrix. Grothendieck’s conjecture is that the ideal generated by these relations is an ideal of definition for the period matrix. (S. Lang, Introduction to Transcendental Numbers, Addison-Wesley, 1966, pp42–43; Collected Works, Vol. I, pp. 443–444.)

M.6. (p. 16) So far as I know, both (2.2) and (2.4) remain open.

M.7. (p. 19) (2.11) The theorem extends to one-motives (Theor´ eme` 2.2.5 of Brylinski, Jean-Luc, “1-motifs” et formes automorphes. Journees´ Automorphes (Dijon, 1981), 43–106, Publ. Math. Univ. Paris VII, 15, Univ. Paris VII, Paris, 1983.)

M.8. (p. 21) (2.14) By using the full strength of Deligne’s results on cohomology, it is possible to avoid the use of the Gauss-Manin connection in the proof of Theorem 2.12 (Blasius, Don, A p-adic property of Hodge classes on abelian varieties. Motives (Seattle, WA, 1991), 293–308, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994, Theorem 3.1).

THEOREM (Deligne 1971a). Let X S be a smooth proper morphism of smooth vari- W ! eties over C. (a) The Leray spectral sequence

r s r s H .S;R Q/ H C .X;Q/ ) M ENDNOTES (BY J.S. MILNE) 62

degenerates at E2; in particular, the edge morphism

n n H .X;Q/ .S;R Q/ ! is surjective. (b) If X is a smooth compactiﬁcation of X with X X a union of smooth divisors with X normal crossings, then the canonical morphism

n n H .X;Q/ .S;R Q/ ! is surjective. n 0 n n 0 (c) Let .R Q/ be the largest constant local subsystem of R Q (so .R Q/s n n 0 D .S;R Q/ for all s S.C/). For each s S, .R Q/s is a Hodge substructure n n 2 2 n of .R Q/s H .Xs;Q/, and the induced Hodge structure on .S;R Q/ is D independent of s. In particular, the map n n H .X;Q/ H .Xs;Q/ ! n 0 has image .R Q/s , and its kernel is independent of s. THEOREM. Let X S be a smooth proper morphism of complex varieties with S W ! 2n smooth and connected. Let .S;R Q.n//. 2 (a) If s is a Hodge cycle for one s S.C/, then it is a Hodge cycle for every s S.C/; 2 2 (b) If s is an absolute Hodge cycle for one s S.C/, then it is an absolute Hodge class 2 for every s S.C/. 2 PROOF. (Blasius 1994, 3.1.) According to Deligne’s theorem, for s;t S.C/, there is a 2 commutative diagram:

2n H .Xs/.n/

injective

2n onto 2n H .X;Q/.n/ .S;R Q.n// injective

2n H .Xt /.n/

2n 2n Let .S;R Q.n//. According to (c) of the theorem, .S;R Q.n// has a 2 Hodge structure for which the diagonal maps are morphisms of Hodge structures. It follows that if s is a Hodge cycle, then so also are and t . Identify H.X/ A with HA.X/. Let be an automorphism of C. If s is a Hodge ˝ 2n cycle on Xs relative to , then there is a s H .Xs/.n/ such that s 1 . s 1/ 2n 2 ˝ D ˝ in H .Xs/. Since . s 1/ is in the image of A ˝ 2n 2n H .X/.n/ A H .Xs/.n/ A; ˝ ! ˝ s is in the image of 2n 2n H .X/.n/ H .Xs/.n/ ! M ENDNOTES (BY J.S. MILNE) 63

2n (apply 2.13) — let H .X/.n/ map to s . Because s and t have a common Q 2n2 pre-image in .S;R Q.n//, . s 1/ and . t 1/ have a common pre-image 2n ˝ ˝ in .S;R Q.n// A. Therefore (see the diagram), 1 maps to . t 1/ in 2n ˝ Q ˝ ˝ H .Xt / A, and so t 1 is a Hodge cycle relative to . ˝ ˝ M.9. (p. 26) (3.8) The motivic signiﬁcance of Principle A is the following: by the usual method (e.g., Saavedra 1972, VI 4.1) we can deﬁne a category of motives using the absolute Hodge classes as correspondences; this will be a pseudo-abelian rigid tensor category, and it will be Tannakian if and only if Principle A holds for all the varieties on which the category is based.

M.10. (p. 30) (4.4) Let E be a CM-ﬁeld, and let E End.A/ be a homomorphism. The W ! pair .A;/ is said to be of Weil type if Tgt0.A/ is a free E C-module. The proposition ˝Q shows the following:

Vd 1 d If .A;/ is of Weil type, then the subspace E H .A;Q/ of H .A;Q/ consists of Hodge classes.

When E is quadratic over Q, these Hodge classes were studied by Weil (Abelian varieties and the Hodge ring, 1977c in Collected Papers, Vol. III, Springer-Verlag, pp421–429), and for this reason are called Weil classes. A polarization of an abelian variety .A;/ of Weil type is a polarization of A whose Rosati involution stabilizes E and induces complex conjugation on it. The special Mumford-Tate group of a general polarized abelian variety .A;;/ of Weil 1 type is SU./ where is the E-Hermitian form on H .A;Q/ deﬁned by the polarization. If the special Mumford-Tate group of .A;/ equals SU./, then the Q-algebra of Hodge cycles is generated by the divisor classes and the Weil classes (but not by the divisor classes alone). When E is quadratic over Q, these statements are proved in Weil (ibid.), but the same argument works in general. For more on Weil classes, see Moonen, B. J. J.; Zarhin, Yu. G. Weil classes on abelian varieties. J. Reine Angew. Math. 496 (1998), 83–92. Zarhin, Yu. G. and Moonen, B. J. J., Weil classes and Rosati involutions on complex abelian varieties. Recent progress in algebra (Taejon/Seoul, 1997), 229–236, Contemp. Math., 224, Amer. Math. Soc., Providence, RI, 1999. In a small number of cases, the Weil classes are known to be algebraic even when they are not contained in the Q-algebra generated by the divisor classes: Schoen, Chad, Hodge classes on self-products of a variety with an automorphism. Compositio Math. 65 (1988), no. 1, 3–32. Addendum, ibid., 114 (1998), no. 3, 329–336. van Geemen, Bert, An introduction to the Hodge conjecture for abelian varieties. Algebraic cycles and Hodge theory (Torino, 1993), 233–252, Lecture Notes in Math., 1594, Springer, Berlin, 1994.

7 Vd 1 d M.11. (p. 31) (4.5) It is tempting to think that E H .A;Q/. 2 / must be the E-subspace d d ŒE Q 1 ŒE Q of H .A;Q/. / spanned by the class of the algebraic cycle A W 0 A W . 2 0 f g 0 7Indeed, this is asserted in Blasius 1994, p. 305. If it had been true, it would have been possible to deduce the Hodge conjecture for Weil classes on abelian varieties from the main theorem of Bloch, Semi-regularity and de Rham cohomology, 1972 (Milne, unpublished manuscript, 1994). M ENDNOTES (BY J.S. MILNE) 64

However, this is not true. For example, let E be the subfield of Q generated by p n and 2 0 n let be the given embedding of E into Q. Let p n E act on A A0 as 1 0 , and let 1 2 D V H .A0;Q/. Then, V E V V , and D ˝ ' ˚ Vd Á Vd Vd Vd .V E/ .V V / V V : E ˝ ' E ˚ ' E ˚ E 1 Let e1;:::;ed be a basis for V H .A0;Q/ (first copy of A0), and let f1;:::;fd be the D same basis for the second copy. The elements ei p nfi form a basis for V , and so C e1 p nf1 e2 p nf2 ::: C ^ C ^ Vd 1 is an E-basis for H .A;Q/ (not e1 e2 :::). When d 2, the elements E ^ ^ D e1 e2 nf1 f2; p n.e1 f2 e2 f1/ ^ ^ ˝ C ˝ V2 1 form a Q-basis for E H .A;Q/, and the Weil classes are represented by the algebraic cycles .0 A0/ n.A0 0/ and the .1;1/-components of the diagonal. See Murty, Hodge and Weil classes on abelian varieties, 2000. It seems not to be known whether, in the Vd 1 d situation of the lemma, the space E H .A;Q/. 2 / always consists of algebraic classes. M.12. (p. 40) The proof shows that, in certain a Tannakian sense, the Q-algebra of Hodge classes on an abelian variety of CM type is generated by the divisor classes and the split Weil classes. Yves Andre´ (Une remarque a` propos des cycles de Hodge de type CM, 1992) proves a more precise result. As in the text, let A be an abelian variety over C with complex multiplication by a CM-field E. We suppose that E is Galois over Q. Denote the inclusion E End0.A/ by , and let be the CM-type of .A;/. Then .A;/ is of CM-type . ! P Let J 1;:::;2r be a subset of Gal.E=Q/ such that i r, and let E act on def D2r f g D AJ A by x .:::;.i /.x/;:::/. Then the space D 7! ^2r 1 H .AJ ; /.r/ E Q 2r of Weil classes in H .AJ ;Q.r// consists of absolute Hodge classes (Theorem 4.8). Let fJ A AJ denote the diagonal map. Andre´ shows that every Hodge class on A of W ! codimension r is a sum of classes of the form fJ. / with a Weil class on AJ . M.13. (p. 42) We discuss some simplifications and applications of the proof of Theorem 2.11.

A CRITERION FOR A FAMILY OF HODGECLASSESTOCONTAINALL HODGECLASSES

THEOREM. Suppose that for each abelian variety A over C we have a Q-subspace C.A/ of the Hodge classes on A. Assume: (a) C.A/ contains all algebraic classes on A; (b) pull-back by a homomorphism ˛ A B maps C.B/ into C.A/; W ! (c) let A S be an abelian scheme over a connected smooth variety S over C, and W ! 2p let t .S;R Q.p//; if ts is a Hodge cycle for all s and lies in C.As/ for one s, 2 then it lies in C.As/ for all s. Then C.A/ contains all the Hodge classes on A.

PROOF. The proof of Theorem 4.8 shows that C.A/ contains all split Weil classes on A (see endnote M.3), and then Andre’s´ improvement of 5 (see endnote M.12) proves the theorem for all abelian varieties of CM-type. Now Proposition 6.1 completes the proof. M ENDNOTES (BY J.S. MILNE) 65

ALGEBRAICCLASSES In Steenbrink (Some remarks about the Hodge conjecture. Hodge theory (Sant Cugat, 1985), 165–175, Lecture Notes in Math., 1246, Springer, Berlin, 1987) we find the following: [Grothendieck (1966), footnote 13] stated a conjecture which is weaker than the Hodge .p;p/ conjecture: (VHC) Suppose that f X S is a smooth projective morphism with S con- W ! 0 2p nected, smooth. Suppose that H .S;R f QX / is of type .p;p/ every- 2 where, and for some s0 S, .s0/ is the cohomology class of an algebraic cycle 2 of codimension p on Xs . Then .s/ is an algebraic cycle class for all s S. 0 2 This “variational Hodge conjecture” . . . . In fact, Grothendieck (1966, footnote 13) asks whether the following statement is true: (VHCo) Let S be a connected reduced scheme of characteristic zero, and let 2p X S be a proper smooth morphism; then a section z of R .˝X=S / is W ! algebraic on every fibre if and only if it is horizontal for the canonical integral connection and is algebraic on one fibre.

THEOREM. If the variational Hodge conjecture (either statement (VHC) or (VHCo)) is true for abelian varieties, then so also is the Hodge conjecture.

PROOF. Assume (VHC), and let C.A/ be the Q-span of the classes of algebraic cycles on A. Then the preceding theorem immediately shows that C.A/ contains all Hodge classes on A. The proof that (VHCo) implies the Hodge conjecture is similar, but requires the remark that all of 2–6 still applies when the etale´ component is omitted. Although, we didn’t need Principle A for the last theorem, it should be noted that it does 2p hold for the algebraic classes on abelian varieties (those in the Q-subspace of H .A;Q.p// spanned by the classes of algebraic cycles). This is a consequence of the following three results (cf. endnote M.9): numerical equivalence coincides with homological equivalence on complex abelian ˘ varieties (Lieberman, David I., Numerical and homological equivalence of algebraic cycles on Hodge manifolds. Amer. J. Math. 90 1968 366–374, MR37 #5898); the category of motives defined using algebraic cycles modulo numerical equivalence is ˘ an abelian category (even semisimple) (Jannsen, Uwe, Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107 (1992), no. 3, 447–452.); every abelian tensor category over a field of characteristic zero whose objects have ˘ finite dimension is Tannakian (Theor´ eme` 7.1 of Deligne, P., Categories´ tannakiennes. The Grothendieck Festschrift, Vol. II, 111–195, Progr. Math., 87, Birkhauser¨ Boston, Boston, MA, 1990).

DE RHAM-HODGECLASSES (BLASIUS)

For a complete smooth variety X over Q and an embedding Q Qp, there is a natural W ! isomorphism 2r 2r I H .X;Qp/.r/ p BdR H .X/.r/ BdR W et ˝Q ! dR ˝Qp (Faltings, Tsuji) compatible with cycle maps. Call an absolute Hodge class on X de Rham if, for all , I. p 1/ dR 1. The following is proved in Blasius 1994. ˝ D ˝ M ENDNOTES (BY J.S. MILNE) 66

THEOREM. Every absolute Hodge class on an abelian variety over Q is de Rham.

PROOF. The functor from abelian varieties over Q to abelian varieties over C is fully faithful and the essential image contains the abelian varieties of CM-type. Using this, one sees by the same arguments as above, that the theorem follows from the next result.

THEOREM (BLASIUS 1994, 3.1). Let X S be a smooth proper morphism of smooth W ! 2n 2n varieties over Q C with S connected, and let .SC;R C Q.n//. If s HB .Xs/.n/ 2 2 is absolutely Hodge and de Rham for one s S.Q/, then it is absolutely Hodge and de Rham 2 for every s.

PROOF. Let s;t S.Q/ and assume s is absolutely Hodge and de Rham. We know (see 2 endnote M.8) that t is absolutely Hodge, and we have to prove it is de Rham. Let Q , Qp be an embedding. For a smooth compactiﬁcation X of X (as in endnote W ! M.8) over Q, we have a commutative diagram

2n I 2n H .X;Qp/.n/ BdR H .X/.n/ BdR et ˝ dR ˝

2n I 2n H .Xs;Qp/.n/ BdR H .Xs/.n/ BdR: et ˝ dR ˝ 2n There exists HB .X/.n/ mapping to (see the diagram in endnote M.8). Let p Q 2 2n 2n Q and dR be the images of in H .X;Qp/.n/ and H .X/.n/. Because s is de Rham, Q Q et dR I. p 1/ differs from dR 1 by an element of Q ˝ Q ˝ 2n 2n Ker.H .X/.n/ H .Xs/.n/ BdR. dR ! dR ˝ But this kernel is independent of s, and so t is also de Rham.

MOTIVATED CLASSES (ABDULALI,ANDRE´)

Recall that Grothendieck’s Lefschetz standard conjecture says that the Q-space of algebraic classes on a smooth algebraic variety is invariant under the Hodge -operator. Abdulali (Algebraic cycles in families of abelian varieties. Canad. J. Math. 46 (1994), no. 6, 1121– 1134) shows that if the Q-spaces of algebraic cycles in the L2-cohomology of Kuga ﬁbre varieties (not necessarily compact) are invariant under the Hodge -operator, then the Hodge conjecture is true for all abelian varieties. Andre´ (Pour une theorie´ inconditionnelle des motifs, Inst. Hautes Etudes´ Sci. Publ. Math. No. 83 (1996), 5–49) proves a more precise result: every Hodge class on an abelian variety A is a sum of classes of the form p .˛ Lˇ/ in which ˛ and ˇ are algebraic classes [ on a product of A with an abelian variety and certain total spaces of compact pencils of abelian varieties. In outline, the proofs are similar to that of Theorem 2.11. M.14. (p. 43) Since Theorem 2.11 is true for one-motives (see endnote M.7), so also is the A corollary. This raises the question of whether dim.G / tr:deg k.pij / for all one-motives. D k For a discussion of the question, and its implications, see Bertolin, C., Periodes´ de 1-motifs et transcendance. J. Number Theory 97 (2002), no. 2, 204–221.

M.15. (p. 46) (7.4) There are similar calculations in M ENDNOTES (BY J.S. MILNE) 67

Ogus, A., Grifﬁths transversality in crystalline cohomology. Ann. of Math. (2) 108 (1978), no. 2, 395–419, MR 80d:14012 (3), Ran, Ziv Cycles on Fermat hypersurfaces. Compositio Math. 42 (1980/81), no. 1, 121–142, and, in a more general setting,

Aoki, Noboru, A note on complete intersections of Fermat type. Comment. Math. Univ. St. Paul. 35 (1986), no. 2, 231–245.

M.16. (p. 55) (7.16) For an elementary proof that . ; .a// is Galois over , see Q d Q Q Das, Pinaki, Algebraic gamma monomials and double coverings of cyclotomic ﬁelds. Trans. Amer. Math. Soc. 352 (2000), no. 8, 3557–3594.