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The Tate conjecture over finite fields (AIM talk)

J.S. Milne

Abstract

These are my notes for a talk at the The Tate Conjecture workshop at the AmericanInstitute of Mathematics in Palo Alto, CA, July 23–July 27, 2007, somewhat revisedand expanded. The intent of the talk was to review what is known and to suggestdirections for research.v1 (September 19, 2007): first version on the web.v2 (November 6, 2018): revised and expanded.

Contents

1 The conjecture, and some folklore 2

2 Divisors on abelian varieties 5

3 K3 surfaces 8

4 Algebraic classes have the Tannaka property 9

5 On the equality of equivalence relations 10

6 The Hodge conjecture and the Tate conjecture 11

7 Rational Tate classes 14

8 Reduction to the case of codimension 2 15

9 The Hodge standard conjecture 19

10 On the existence of a good theory of rational Tate classes 20

Conventions

All varieties are smooth and projective. Complex conjugation onC is denoted byι. Thesymbol F denotes an algebraic closure ofFp, and ℓ always denotes a prime6= p. Onthe other hand,l is allowed to equalp. For a varietyX, H∗(X) =

⊕iH

i(X) andH2∗(X)(∗) =

⊕iH

2i(X)(i); both are graded algebras. I denote a canonical (or a specif-ically given) isomorphism by≃. I assume that the reader is familiar with the basic theoryof abelian varieties as, for example, in Milne 1986.

MSC2000: 14C25, 11G25

1

1 THE CONJECTURE, AND SOME FOLKLORE 2

1 The conjecture, and some folklore

Let X be a variety overF. A modelX1 of X over a finite subfieldk1 of F gives rise to acommutative diagram:

Zr(X)cr

−−−−→ H2r(X,Qℓ(r))xx

Zr(X1)cr

−−−−→ H2r(X1,Qℓ(r)).

HereZr(∗) denotes the group of algebraic cycles of codimensionr on a variety (freeZ-module generated by the closed irreducible subvarieties ofcodimensionr) and cr is thecycle map. The image of the vertical arrow at right is contained inH2r(X,Qℓ(r))

Gal(F/k1)

andZr(X) = lim−→X1/k1

Zr(X1), and so the image of the top cycle map is contained in

H2r(X,Qℓ(r))′ def=⋃

X1/k1H2r(X,Qℓ(r))

Gal(F/k1).

In his talk at the AMS Summer Institute at Woods Hole in July, 1964, Tate conjectured thefollowing:1

CONJECTURET r(X, ℓ) (TATE). TheQℓ-vector spaceH2r(X,Qℓ(r))′ is spanned by alge-

braic classes.

The conjecture implies that, for any modelX1/k1, theQℓ-subspaceH2r(X,Qℓ(r))Gal(F/k1)

is spanned by the classes of algebraic cycles onX1; conversely, if this is true for all modelsX1/k1 over (sufficiently large) finite fieldsk1, thenT r(X, ℓ) is true.

I write T r(X) (resp.T (X, ℓ), resp.T (X)) for the conjecture thatT r(X, ℓ) is true forall ℓ (resp. allr, resp. allr andℓ).

In the same talk, Tate mentioned the following “conjecturalstatement”:

CONJECTUREEr(X, ℓ) (EQUALITY OF EQUIVALENCE RELATIONS). The kernel of the cy-cle class mapcr : Zr(X) → H2r(XF,Qℓ(r)) consists exactly of the cycles numericallyequivalent to zero.

Both conjectures are existence statements for algebraic classes. It is well known that Con-jectureE1(X) holds for allX (see Tate 1994,§5).

LetX be a variety overF. The choice of a model ofX over a finite subfield ofF definesa Frobenius mapπ : X → X. For example, for a modelX1 ⊂ Pn overFq, π acts as

(a1 : a2 : . . .) 7→ (aq1 : aq2 : . . .) : X1(F)→ X1(F), X1(F) ≃ X(F).

Any such map will be called aFrobenius mapofX (or aq-Frobenius mapif it is defined bya model overFq). If π1 andπ2 arepn1- andpn2-Frobenius maps ofX, thenπn2N

1 = πn1N2

for someN > 1.2 For a Frobenius mapπ of X, we use a subscripta to denote thegeneralized eigenspace with eigenvaluea, i.e.,

⋃N Ker

((π − a)N

).

1Tate’s talk is included in the mimeographed proceedings of the conference, which were distributed by theAMS to only a select few. Despite their great historical importance — for example, they contain the only writ-ten account by Artin and Verdier of their duality theorem, and the only written account by Serre and Tate oftheir lifting theorem — the AMS has ignored requests to make the proceedings more widely available. Fortu-nately, Tate’s talk was reprinted in the proceedings of an earlier conference (Arithmetical algebraic geometry.Proceedings of a Conference held at Purdue University, December 5–7, 1963. Edited by O. F. G. Schilling,Harper & Row, Publishers, New York 1965).

2Because any two models ofX become isomorphic over a finite subfield ofF; when the modelX1/k1 isreplaced byX1K/K, then its Frobenius mapπ is replaced byπ[K : k1].

1 THE CONJECTURE, AND SOME FOLKLORE 3

CONJECTURESr(X, ℓ) (PARTIAL SEMISIMPLICITY ). Every Frobenius mapπ of X actssemisimply onH2r(X,Qℓ(r))1 (i.e., it acts as1).

Weil proved that, for an abelian varietyA overF, the Frobenius maps act semisimply onH1(A,Qℓ). It follows that they act semisimply on all the cohomology groupsH i(A,Qℓ) ≃∧iH1(A,Qℓ). In particular, ConjectureS(X) holds whenX is an abelian variety overF.

From now on, I’ll write T rℓ(X) for H2r(X,Qℓ(r))

′ and call its elements theTate

classesof degreer on X. Note thatT ∗ℓ(X)

def=⊕

r Trℓ(X) is a gradedQℓ-subalgebra

of H2∗(X,Qℓ(∗)).

ASIDE 1.4. One can ask whether the Tate conjecture holds integrally, i.e., whether the map

cr : Zr(X)⊗ Zℓ → H2r(XF,Zℓ(r))′ (1)

is surjective for all varietiesX overF. ClearlyH2r(XF,Zℓ(r))′ contains all torsion classes, and

essentially the same argument that shows that not all torsion classes in Betti cohomology are alge-braic, shows that not all torsion classes in etale cohomology are algebraic.3 However, I don’t knowof any varieties overF for which the map (1) is not surjective modulo torsion. It is known that ifT r(X, ℓ) andEr(X, ℓ) hold for a singleℓ, then the map (1) is surjective for all but possibly finitelymanyℓ. See Milne and Ramachandran 2004,§3.

Folklore

The next three theorems are folklore.

THEOREM 1.5. Let X be a variety overF of dimensiond, and letr ∈ N. The followingstatements are equivalent:

(a) T r(X, ℓ) andEr(X, ℓ) are true for a singleℓ.

(b) T r(X, ℓ), Sr(X, ℓ), andT d−r(X, ℓ) are true for a singleℓ.

(c) T r(X, ℓ), Er(X, ℓ), Sr(X, ℓ), T d−r(X, ℓ), andEd−r(X, ℓ) are true for allℓ, and theQ-subspaceAr

ℓ(X) of T rℓ(X) generated by the algebraic classes is aQ-structure on

T rℓ (X), i.e.,Ar

ℓ(X)⊗Q Qℓ ≃ Trℓ(X).

(d) the order of the pole of the zeta functionZ(X, t) at t = q−r is equal to the rank ofthe group of numerical equivalence classes of algebraic cycles of codimensionr.

The proof is explained in Tate 1994,§2.

THEOREM 1.6. LetX be a variety overF of dimensiond. If S2d(X ×X, ℓ) is true, thenevery Frobenius mapπ acts semisimply onH∗(X,Qℓ).

PROOF. If a occurs as an eigenvalue ofπ onHr(X,Qℓ), then1/a occurs as an eigenvalueof π onH2d−r(X,Qℓ(d)) (by Poincare duality), and

Hr(X,Qℓ)a ⊗H2d−r(X,Qℓ(d))1/a ⊂ H

2d(X ×X,Qℓ(d))1

(Kunneth formula), from which the claim follows. ✷

3The proof shows that the odd dimensional Steenrod operations are zero on the torsion algebraic classes butnot on all torsion cohomology classes.

1 THE CONJECTURE, AND SOME FOLKLORE 4

An ℓ-adic Tateq-structureis a finite dimensionalQℓ-vector space together with a linear(Frobenius) mapπ whose characteristic polynomial has rational coefficientsand whoseeigenvalues areWeil q-numbers, i.e., algebraic numbersα such that, for some integermcalled theweight of α, |ρ(α)| = qm/2 for every homomorphismρ : Q[α] → C and, forsome integern, qnα is an algebraic integer. When the eigenvalues ofπ are all algebraicintegers, the Tate structure is said to beeffective. For example, for a varietyX overFq,Hr(XF,Qℓ(s)) is a Tateq-structure of weightr − 2s, which is effective ifs = 0.

A Tatepn1-structureπ1 and a Tatepn2-structureπ2 on aQℓ-vector spaceV are said tobe equivalent ifπn1N

1 = πn2N2 for someN . This is an equivalence relation, and anℓ-adic

Tate structureis a finite dimensionalQℓ-vector space together with an equivalence class ofTateq-structures. For example, for a varietyX overF,Hr(X,Qℓ(s)) is a Tate structure ofweightr − 2s, which is effective ifs = 0.

LetX be a smooth projective variety overF. For eachr, let F raH

i(X,Qℓ) denote thesubspace ofH i(X,Qℓ) of classes with support in codimensionr, i.e.,

F raH

i(X,Qℓ) =⋃

UKer(H i(X,Qℓ)→ H i(U,Qℓ))

whereU runs over the open subvarieties ofX such thatX r U has codimension≥ r. IfZ = X r U has codimensionr andZ → Z is a desingularization ofZ, then

H i−2r(Z,Qℓ)(−r)→ H i(X,Qℓ)→ H i(U,Qℓ)

is exact (see Deligne 1974, 8.2.8; a similar proof applies toetale cohomology). This showsthatF r

aHi(X,Qℓ) is an effective Tate substructure ofH i(X,Qℓ) such thatF r

aHi(X,Qℓ)(r)

is still effective.

CONJECTURE(GENERALIZED TATE CONJECTURE). For a smooth projective varietyXoverF, every Tate substructureV ⊂ H i(X,Qℓ) such thatV (r) is effective is contained inF raH

i(X,Qℓ) (cf. Grothendieck 1968, 10.3).

THEOREM 1.8. LetX be a variety overF. If the Tate conjecture holds for all varieties ofthe formA×X withA an abelian variety (and someℓ), then the generalized Tate conjectureholds forX (and the sameℓ).

The proof is explained in Milne and Ramachandran 2006, 1.10.

REMARK 1.9. There arep-analogues of all of the above conjectures and statements. LetW (k) be the ring of Witt vectors with coefficients in a perfect fieldk, and letB(k) be itsfield of fractions. Letσ be the automorphism ofW (k) (or B(k)) that acts asx 7→ xp

on k. For a varietyX over k, let Hrp(X) denote the crystalline cohomology group with

coefficients inB(k). It is a finite dimensionalB(k)-vector space with aσ-linear FrobeniusmapF . Define

T rp(X) =

⋃X1/k1

H2rp (X1)(r)

Fn1=1 (pn1 = |k1|).

This is a finite dimensionalQp-vector space, which the Tate conjectureT r(X, p) says isspanned by algebraic classes.

2 DIVISORS ON ABELIAN VARIETIES 5

Motivic interpretation

Let Mot(F) be the category of motives overF defined using algebraic cycles modulo nu-merical equivalence. It is known thatMot(F) is a semisimple Tannakian category (Jannsen1992). Etale cohomology defines a functorωℓ onMot(F) if and only if ConjectureE(X, ℓ)holds for all varietiesX. Assuming this, conjectureT (X, ℓ) holds for allX if and only if,for all X andY , the image of the map

Hom(X,Y )⊗Qℓ → HomQℓ(ωℓ(X), ωℓ(Y )) (2)

consists of the homomorphismsα : ωℓ(X) → ωℓ(Y ) such thatα ◦ πX = πY ◦ α for someFrobenius mapsπX andπY of X andY (necessarilyq-Frobenius maps for the sameq).In other words, conjecturesE(X, ℓ) andT (X, ℓ) hold for allX if and only if ℓ-adic etalecohomology defines an equivalence fromMot(F) ⊗Q Qℓ to the category ofℓ-adic Tatestructures.

2 Divisors on abelian varieties

Tate (1966) proved the Tate conjecture for divisors on abelian varieties overF, in the form:

THEOREM 2.1. For all abelian varietiesA andB overFq, the map

Hom(A,B)⊗ Zℓ → HomQℓ(TℓA,TℓB)Gal(F/Fq) (3)

is an isomorphism.

SKETCH OF PROOF. It suffices to prove this withA = B (becauseHom(A,B) is a directsummand ofEnd(A × B)). Choose a polarization onA, of degreed2 say. It defines anondegenerate skew-symmetric form onVℓA, and a maximal isotropic subspaceW of VℓAstable underGal(F/Fq) will have dimensiong = 1

2 dimVℓA. Forn ∈ N, let

X(n) = TℓA ∩W + ℓnTℓA ⊂ TℓA.

For eachn, there exists an abelian varietyA(n) and an isogenyA(n)→ AmappingTℓA(n)isomorphically ontoX(n). There are only finitely many isomorphism classes of abelianvarieties in the set{A(n)} because eachA(n) has a polarization of degreed2, and hencecan be realized as a closed subvariety ofP3gd−1 of degree3gd(g!). Thus two of theA(n)’sare isomorphic, and we have constructed a (nonobvious) isogeny. From this beginning, Tatewas able to deduce the theorem by exploiting the semisimplicity of the Frobenius map.✷

COROLLARY 2.2. For varietiesX andY overF,

T 1(X × Y, ℓ) ⇐⇒ T 1(X, ℓ) + T 1(Y, ℓ).

PROOF. Compare the decomposition

NS(X × Y ) ≃ NS(X)⊕NS(Y )⊕Hom(Alb(X),Pic0(Y ))

with the similar decomposition ofH2(X × Y,Qℓ(1)) given by the Kunneth formula. ✷

COROLLARY 2.3. The Tate conjectureT 1(X) is true whenX is a product of curves andabelian varieties overF.

2 DIVISORS ON ABELIAN VARIETIES 6

PROOF. LetA be an abelian variety overF. Choose a polarizationλ : A → A∨ of A, andlet † be the Rosati involution onEnd0(A) defined byλ. The mapD 7→ λ−1 ◦ λD definesan isomorphism

NS(A)⊗Q ≃ {α ∈ End0(A) | α† = α}.

Similarly,

T 1ℓ (A) ≃ {α ∈ EndQℓ

(VℓA) | α† = α andαπ = πα for some Frobenius mapπ},

and soT 1 for abelian varieties follows from Theorem 2.1. SinceT 1 is obvious for curves,the general statement follows from Corollary 2.2. ✷

As E1(X, ℓ) is true for all varietiesX, the equivalent statements in Theorem 1.5 holdfor products of curves and abelian varieties in the caser = 1. According to some morefolklore (Tate 1994, 5.2),T 1(X) andE1(X) hold for any varietyX for which there existsa dominant rational mapY → X with Y a product of curves and abelian varieties.

ASIDE 2.4. The reader will have noted the similarity of (2) and (3).Tate (1994) describes how hewas led to his conjecture partly by his belief that (3) was true. Today, one would say that if (3)is true, then so must (2) because “everything that’s true forabelian varieties is true for motives”.4

However, when Tate was thinking about these things, motivesdidn’t exist. Apparently, the firsttext in which the notion of amotif appears is Grothendieck’s letter to Serre of August 16, 1964(Grothendieck and Serre 2001, p173, p276).

ASIDE 2.5. Theorem 2.1 has been extended to abelian varieties overfields finitely generated overthe prime field by Zarhin and Faltings (Zarhin 1974a,b; Faltings 1983; Faltings and Wustholz 1984).

Abelian varieties with no exotic Tate classes

THEOREM 2.6. The Tate conjectureT (A) holds for any abelian varietyA such that, forsomeℓ, theQℓ-algebraT ∗

ℓ(A) is generated byT 1ℓ (A); in fact, the equivalent statements of

(1.5) hold for such anA and allr ≥ 0.

PROOF. If theQℓ-algebraT ∗ℓ(A) is generated byT 1

ℓ (A), then, because the latter is spannedby algebraic classes (2.3), so is the former. ThusT r(A, ℓ) holds for allr, and asS(A) isknown, this implies that the equivalent statements of (1.5)hold forA. ✷

For an abelian varietyA, let L∗ℓ(A) be theQ-subalgebra ofH2∗(A,Qℓ(∗)) generatedby the divisor classes. ThenL∗ℓ(A) · Qℓ ⊂ T

∗ℓ (A). The elements ofL∗ℓ(A) are called the

Lefschetz classesonA, and the Tate classes not inL∗ℓ(A) are said to beexotic.An abelian varietyA is said to havesufficiently many endomorphismsif End0(A)

contains an etaleQ-subalgebra of degree2 dimA overQ. Tate’s theorem (2.1) implies thatevery abelian variety overF has sufficiently many endomorphisms.

LetA be an abelian variety with sufficiently many endomorphisms over an algebraicallyclosed fieldk, and letC(A) be the centre ofEnd0(A). The Rosati involution† defined byany polarization ofA stabilizesC(A), and its restriction toC(A) is independent of thechoice of the polarization. DefineL(A) to be the algebraic group overQ such that, for anycommutativeQ-algebraR,

L(A)(R) = {a ∈ C(A)⊗R | aa† ∈ R×}.

4This reasoning is circular: what we hope to be true for motives is partly based on our hope that the Tateand Hodge conjectures are true.

2 DIVISORS ON ABELIAN VARIETIES 7

It is a group of multiplicative type (not necessarily connected), which acts in a natural wayon the cohomology groupsH2∗(An,Qℓ(∗)) for all n. It is known that the subspace fixedbyL(A) consists of the Lefschetz classes,

H2∗(An,Qℓ(∗))L(A) = L∗ℓ(A

n) ·Qℓ, all n andℓ, (4)

(see Milne 1999a). Letπ be a Frobenius endomorphism ofA. Some powerπN of π lies inC(A), hence inL(A)(Q), and (4) shows that no power ofA has an exotic Tate class if andonly if πN is Zariski dense inL(A). This gives the following explicit criterion:

2.7 LetA be a simple abelian variety overF, let πA be aq-Frobenius en-domorphism ofA lying in C(A), and let(πi)1≤i≤2s be the roots inC of theminimum polynomial ofπA, numbered so thatπiπi+s = q. Then no powerof A has an exotic Tate class (and so the Tate conjecture holds forall powersof A) if and only if {π1, . . . , πs, q} is a Z-linearly independent inC× (i.e.,πm11 · · · π

mss = qm,mi,m ∈ Z, impliesm1 = · · · = ms = 0 = m).

Kowalski (2005, 2.2(2), 2.7) verifies the criterion for any simple ordinary abelian varietyA such that the Galois group ofQ[π1, . . . , π2g] overQ is the full group of permutations of{π1, . . . , π2g} preserving the relationsπiπi+g = q.

More generally, letP (A) be the smallest algebraic subgroup ofL(A) containing aFrobenius element. Then no power ofA has an exotic Tate class if and only ifP (A) =L(A). Spiess (1999) proves this for products of elliptic curves,and Zarhin (1991) andLenstra and Zarhin (1993) prove it for certain abelian varieties. See also Milne 2001, A7.

REMARK 2.8. LetK be a CM subfield ofC, finite and Galois overQ, and letG =Gal(K/Q). We say that an abelian varietyA is split by K if End0(A) is split by K,i.e.,End0(A)⊗Q K is isomorphic to a product of matrix algebras overQ.

LetA be a simple abelian variety overF, and letπ be aq-Frobenius endomorphism forA. If A is split byK, then, for anyp-adic primew of K

fA(w)def=

ordw(π)

ordw(q)[Kw : Qp]

lies inZ (apply Tate 1966, p142). Clearly,fA(w) is independent of the choice ofπ, and theequalityπ · ιπ = q implies thatfA(w) + fA(ιw) = [Kw : Qp]. LetW be the set ofp-adicprimes ofK, and letd be the local degree[Kw : Qp]. The mapA 7→ fA defines a bijectionfrom the set of isogeny classes of simple abelian varieties overF split byK to the set

{f : W → Z | f + ιf = d, 0 ≤ f(w) ≤ d all w}. (5)

The character groups ofP (A) andL(A) have a simple description in terms offA, and socomputing the dimensions ofP (A) andL(A) is only an exercise in linear algebra (cf. Wei1993, Part I; Milne 2001, A7).

Abelian varieties with exotic Tate classes

Typically, some power of a simple abelian variety overF will have exotic Tate classes.

PROPOSITION2.9. Let K be a CM-subfield ofC, finite and Galois overQ, with GaloisgroupG.

3 K3 SURFACES 8

(a) There exists a CM-fieldK0 such that, ifK ⊃ K0, then

i) the decomposition groupsDw in G of thep-adic primesw of K are not normalin G,

ii) ι acts without fixed points on the setW of p-adic primes ofK.

(b) AssumeK ⊃ K0, so thatDw 6= Dw′ for somew,w′ ∈ W . Let A be the simpleabelian variety overF corresponding (as in 2.8) to a functionf : W → Z such thatf(w) 6= f(w′) and neitherf(w) norf(w′) lie in f(W r{w,w′}). Then some powerof A supports an exotic Tate class.

PROOF. (a) There exists a totally real fieldF with Galois groupS5 overQ having at leastthreep-adic primes (Wei 1993, 1.6.9), we can takeK0 = F · Q whereQ is any quadraticimaginary field in whichp splits.

(b) We havedimP (A) ≤ t+ 1, where|W | = 2t,

(Wei 1993, 1.4.4). Let

H = {g ∈ G | f(gw) = f(w) for all w ∈W}.

ThenC(A) ≈ KH , and so

dimL(A) = 12 (G : H) + 1.

The conditions onf imply thatH ⊂ Dw ∩ Dw′ , which is properly contained inDw. As2t = (G : Dw), we see thatdimL(A) > dimP (A), and soL(A) 6= P (A). ✷

The set (5) hastd elements, of whicht(t − 1)(t − 2)d−2 satisfy the conditions of (2.9b).As we letK grow, the ratiot(t − 1)(t − 2)d−2/td tends to1, which justifies the statementpreceding the proposition.

THEOREM 2.10. There exists a family of abelian varietiesA over F for which the TateconjectureT (A) is true andT ∗

ℓ(A) is not generated byT 1ℓ (A).

PROOF. See Milne 2001 (the proof makes use of Schoen 1988, 1998). ✷

3 K3 surfaces

The next theorem was proved by Artin and Swinnerton-Dyer (1973).

THEOREM 3.1. The Tate conjecture holds forK3 surfaces overF that admit a pencil ofelliptic curves.

SKETCH OF PROOF. Let X be an ellipticK3 surface, and letf : X → P1 be the pencilof elliptic curves. A transcendental Tate class onX gives rise to a sequence(pn)n≥1 ofelements of the Tate-Shafarevich group of the generic fibre of E = Xη of f such thatℓpn+1 = pn for all n. From these elements, we get a tower

· · · → Pn+1 → Pn → · · ·

of principle homogeneous spaces forE overF(P1). By studying the behaviour of certaininvariants attached to thePn, Artin and Swinnerton-Dyer were able to show that no suchtower can exist. ✷

4 ALGEBRAIC CLASSES HAVE THE TANNAKA PROPERTY 9

For later work on K3 surfaces, see Nygaard 1983; Nygaard and Ogus 1985; Zarhin 1996.

ASIDE 3.2. With the proof of the theorems of Tate (2.1) and of Artin and Swinnerton-Dyer (3.1),there was considerable optimism in the early 1970s that the Tate conjecture would soon be provedfor surfaces over finite fields — all one had to do was attach a sequence of algebro-geometric objectsto a transcendental Tate class, and then prove that the sequence couldn’t exist. However, the progresssince then has been meagre. For example, we still don’t know the Tate conjecture for allK3 surfacesoverF.

4 Algebraic classes have the Tannaka property

Let S be a class of algebraic varieties overF containing the projective spaces and closedunder disjoint unions and products and passage to a connected component.

THEOREM 4.1. Let HW be a Weil cohomology theory on the algebraic varieties overF

with coefficients in a fieldQ in the sense of Kleiman 1994,§3). Assume that for allX ∈ S the kernel of the cycle mapZ∗(X) → H2∗

W (X)(∗) consists exactly of the cy-cles numerically equivalent to zero. LetX ∈ S, and letGX be the algebraic subgroupof GL(H∗

W (X)) × GL(Q(1)) fixing all algebraic classes on all powers ofX. Then theQ-vector spaceH2∗

W (Xn)(∗)GX is spanned by algebraic classes for alln.

PROOF. Let Mot(F) be the category of motives overF based on the varieties inS andusing numerical equivalence classes of algebraic cycles ascorrespondences. Because theKunneth components of the diagonal are known to be algebraic, Mot(F) is a semisimpleTannakian category (Jannsen 1992). Our assumption on the cycle map implies thatHW

defines a fibre functorω onMot(F). It follows from the definition ofMot(F), that for anyvarietyX overF andn ≥ 0,

Z∗num(X

n)Q ≃ Hom(11, h2∗(Xn)(∗))

whereZ∗num(X

n) is the gradedZ-algebra of algebraic cycles modulo numerical equivalenceand the subscript means that we have tensored withQ. On applyingω to this isomorphism,we obtain an isomorphism

Z∗num(X

n)Q ≃ HomG(Q,H2∗W (Xn)(∗)) = H2∗

W (Xn)(∗)G

whereG = Aut⊗(ω). SinceGX is the image ofG in GL(H∗W (X)) × GL(Q(1)), this

implies the assertion. ✷

This is a powerful result: once we know that some cohomology classes are algebraic, itallows us to deduce that many more are (the group fixing the classes weknowto be algebraiccontains the group fixingall algebraic classes, and so any class that it fixes is in the spanof the algebraic classes). On applying the theorem to the smallest classS satisfying theconditions and containing a varietyX, we obtain the following criterion:

4.2 LetX be an algebraic variety overF such thatE(Xn, ℓ) holds for alln. In order to prove thatT (Xn, ℓ) holds for alln, it suffices to find enoughalgebraic classes on the powers ofX for some Frobenius map to be Zariskidense in the algebraic subgroup ofGL(H∗(X,Qℓ)) × GL(Qℓ(1)) fixing theclasses.

ASIDE 4.3. Theorem 4.1 holds also for almost-algebraic classes incharacteristic zero in the senseof Serre 1974, 5.2 and Tate 1994, p76.

5 ON THE EQUALITY OF EQUIVALENCE RELATIONS 10

5 On the equality of equivalence relations

Recall that an abelian varietyA has sufficiently many endomorphisms ifEnd0(A) containsan etaleQ-subalgebraE of degree2 dimA. It is known that such anE can be chosen to bea product of CM-fields. There then exists a unique involutionιE ofE such thatρ◦ιE = ι◦ρfor any homomorphismρ : E → C.

The next theorem (and its proof) is an abstract version of themain theorem of Clozel1999.

THEOREM 5.1. Let k be an algebraically closed field, and letH∗W be a Weil cohomology

theory with coefficient fieldQ . LetA be an abelian variety overk with sufficiently manyendomorphisms, and choose aQ-subalgebraE of End0(A) as above. Assume thatQ splitsE (i.e.,E ⊗Q Q ≈ Q

[E : Q]) and that there exists an involutionιQ of Q such that

⋄ σ ◦ ιE = ιQ ◦ σ for all σ : E → Q,

⋄ there exists a Weil cohomology theoryH∗W0⊂ H∗

W with coefficient fieldQ0def= Q〈ιQ〉

such thatH∗W = Q⊗Q0 H

∗W0

.

Then the kernel of the cycle class mapZ∗(X)→ H2∗W (X)(∗) consists exactly of the cycles

numerically equivalent to zero.

SKETCH OF PROOF. Choose a subsetΦ of Hom(E,Q) such that

Hom(E,Q) = Φ ⊔ ιQΦ

whereιQΦ = {ιQ ◦ ϕ | ϕ ∈ Φ} = ΦιE.

The spaceH1W (A) is free of rank one overE⊗QQ, and soH1

W (A) =⊕

σ∈Φ⊔ιΦH1W (A)σ

whereH1W (A)σ is the one-dimensionalQ-subspace on whichE acts throughσ. Similarly,

HrW (A) ≃

∧r

QH1

W (A) =⊕

I,J,|I|+|J |=r

HrW (A)I,J

whereI andJ are subsets ofΦ andιΦ respectively, andHrW (A)I,J is the one-dimensional

subspace on whicha ∈ E acts as∏

σ∈I⊔J σa. Let A∗W (A) be theQ-subalgebra of

H2∗W (A)(∗) generated by the algebraic classes. SinceA∗

W (A) ⊂ H2∗W0

(A)(∗), theQ-subspaceQ · A∗

W (A) of H2∗W (A)(∗) is stable under the involutionιH of H2∗

W (A)(∗) de-fined by theQ0-substructureH2∗

W0(A)(∗). Therefore, ifH2r

W (A)I⊔J (r) is contained inQ · A∗

W (A), then so also is

ιWH2rW (A)I⊔J (r) = H2r

W (A)ιQI⊔ιQJ(r).

Using this, Clozel constructs, for every nonzero algebraicclass, an algebraic class of com-plementary degree whose product with the first class is nonzero. ✷

COROLLARY 5.2. For an abelian varietyA overF, there is an infinite set of primesℓ 6= psuch thatE(An, ℓ) is true for alln.

6 THE HODGE CONJECTURE AND THE TATE CONJECTURE 11

PROOF. Choose aQ-subalgebraE of End0(A) as before, and letQ′ be the composite ofthe images ofE in C under homomorphismsE → C. ThenQ′ is a finite Galois extensionof Q that splitsE and it is a CM-field. LetS be the set of primesℓ 6= p such thatιlies in the decomposition group of someℓ-adic primev of Q′. For example, ifι is theFrobenius element of anℓ-adic prime ofQ′, thenℓ ∈ S, and soS has density> 0. TheWeil cohomology theoryHW

def= Hℓ ⊗ Q

′v satisfies the hypotheses of the theorem forA.

ForAn, we can choose theQ-algebra to beEn acting diagonally and use the same setS.✷

On combining (5.2) with (4.2) we obtain the following criterion:

5.3 LetA be an abelian variety overF. In order to prove thatT (An) holdsfor all n, it suffices to find enough algebraic classes on powers ofA for someFrobenius endomorphism to be Zariski dense in the algebraicsubgroup ofGL(H∗

ℓ (X)) ×GL(Qℓ(1)) fixing the classes for a suitableℓ.

6 The Hodge conjecture and the Tate conjecture

To go further, we shall need to consider the Hodge conjecture(following Deligne 1982).For a varietyX over an algebraically closed fieldk of characteristic zero, define

H∗A(X) = H∗

Af(X) ×H∗

dR(X) whereH∗Af

(X) =

(lim←−m

H∗(Xet,Z/mZ)

)⊗Z Q.

For any algebraically closed fieldK containingk,

H∗Af

(XK) ≃ H∗Af

(X) andH∗dR(XK) ≃ H∗

dR(X) ⊗k K,

and so there is a canonical homomorphismH∗A(X)→ H∗

A(XK).Let σ be a homomorphismk → C.5 An element ofH2∗

A (X)(∗) is Hodge relative toσif its image inH2∗

A (XC)(∗) is a Hodge class, i.e., lies inH2∗(XC,Q)(∗) ⊂ H2∗A (XC)(∗)

and is of type(0, 0).

CONJECTURE(DELIGNE). If an element ofH2∗A (X)(∗) is Hodge relative to one homo-

morphismσ : k → C, then it is Hodge relative to every such homomorphism.

An element ofH2∗A (X)(∗) is absolutely Hodgeif it is Hodge relative to everyσ. Let

B∗abs(X) be the set of absolutely Hodge classes onX. ThenB∗abs(X) is a gradedQ-subalgebra ofH2∗

A (X)(∗), and Deligne (1982) shows:

(a) for every regular mapf : X → Y , f∗ mapsB∗abs(Y ) into B∗abs(X) and f∗ mapsB∗abs(X) into B∗abs(Y );

(b) for everyX, B∗abs(X) contains the algebraic classes;

(c) for every homomorphismk → K of algebraically closed fields,B∗abs(X) ≃ B∗abs(XK);

(d) for any modelX1 of X over a subfieldk1 of k with k algebraic overk1, Gal(k/k1)acts onB∗abs(X) through a finite quotient.

5Throughout, I assume thatk is not too big to be embedded intoC. For fields that are “too big”, one canuse property (c) ofB∗

abs below as a definition ofB∗

abs(K).

6 THE HODGE CONJECTURE AND THE TATE CONJECTURE 12

Property (d) shows that the image ofB∗abs(X) in H2∗(X,Qℓ(∗)) consists of Tate classes.

THEOREM 6.2. If the Tate conjecture holds forX, then all absolutely Hodge classes onXare algebraic.

PROOF. LetA∗(X) be theQ-subspace ofH2∗A (X)(∗) spanned by the classes of algebraic

cycles, and consider the diagram defined by a homomorphismk → C,

B∗(XC)⊂

−−−−→ H2∗B (XC)(∗)

⊂−−−−→ H2∗

ℓ (XC)(∗)xS

x≃

A∗(X)⊂

−−−−→ B∗abs(X) −−−−→ T ∗ℓ (X)

⊂−−−−→ H2∗

ℓ (X)(∗).

The four groups at upper left are finite dimensionalQ-vector spaces, and the map at topright gives an isomorphismH2∗

B (XC)(∗) ⊗Q Qℓ ≃ H2∗ℓ (XC)(∗). Therefore, on tensoring

theQ-vector spaces in the above diagram withQℓ, we get injective maps

A∗(X) ⊗Q Qℓ → B∗abs(X)⊗Q Qℓ → T

∗ℓ(X).

If the Tate conjecture holds forX, then the composite of these maps is an isomorphism,and so the first is also an isomorphism. This implies thatA∗(X) = B∗abs(X). ✷

THEOREM 6.3. For varietiesX satisfying Deligne’s conjecture, the Tate conjecture forXimplies the Hodge conjecture forXC.

PROOF. For any homomorphismk → C, the homomorphismH2∗A (X)(∗) → H2∗

A (XC)(∗)mapsB∗abs(X) intoB∗(XC). When Deligne’s conjecture holds forX, B∗abs(X) ≃ B∗(XC).Therefore, ifB∗abs(X) consists of algebraic classes, so also doesB∗(XC). ✷

ASIDE 6.4. The Hodge conjecture is known for divisors, and the Tateconjecture is generally ex-pected to be true for divisors. However, there is little evidence for either conjecture in higher codi-mensions, and hence little reason to believe them. On the other hand, Deligne believes his conjectureto be true.6

ASIDE 6.5. As Tate pointed out at the workshop, one reason the Tate conjecture is harder than theHodge conjecture is that it doesn’t tell you which cohomology classes are algebraic; it only tells youtheQℓ-span of the algebraic classes.

Deligne’s theorem on abelian varieties

The following is an abstract version of the main theorem of Deligne 1982.

THEOREM 6.6. Let k be an algebraically closed subfield ofC. Suppose that for everyabelian varietyA overk, we have a gradedQ-subalgebraC∗(A) of B∗(AC) such that

(A1) for every regular mapf : A→ B of abelian varieties overk, f∗ mapsC∗(B) intoC∗(A) andf∗ mapsC∗(A) into C∗(B);

(A2) for every abelian varietyA, C1(A) contains the divisor classes; and

6When asked at the workshop, Tate said that he believes the Tate conjecture for divisors, but that in highercodimension he has no idea. Also it worth recalling that Hodge didn’t conjecture the Hodge conjecture: heraised it as a problem (Hodge 1952, p184). There is considerable evidence (and some proofs) that the classespredicted to be algebraic by the Hodge and Tate conjectures do behave as if they are algebraic, at least in somerespects, but there is little evidence that they are actually algebraic.

6 THE HODGE CONJECTURE AND THE TATE CONJECTURE 13

(A3) let f : A → S be an abelian scheme over a connected smooth (not necessarilycomplete)k-variety S, and letγ ∈ Γ (SC, R

2∗fC∗Q(∗)); if γt is a Hodge class for allt ∈ S(C) andγs lies inC∗(As) for ones ∈ S(k), then it lies inC∗(As) for all s ∈ S(k).ThenC∗(A) ≃ B∗(AC) for all abelian varieties overk.

For the proof, see the endnotes to Deligne 1982). I list threeapplications of this theorem.

THEOREM 6.7. In order to prove the Hodge conjecture for abelian varieties, it suffices toprove the variational Hodge conjecture.

PROOF. TakeC∗(A) to be theQ-subspace ofH2∗B (AC)(∗) spanned by the classes of alge-

braic cycles onA. Clearly (A1) and (A2) hold, and (A3) is (one form of) the variationalHodge conjecture. ✷

The following is the original version of the main theorem of Deligne 1982.

THEOREM 6.8. Deligne’s conjecture holds for all abelian varietiesA over k (hence theTate conjecture implies the Hodge conjecture for abelian varieties).

PROOF. TakeC∗(A) to beB∗abs(A). Clearly (A2) holds, and we have already noted that(A1) holds. That (A3) holds is proved in Deligne 1982. ✷

The theorem implies that, for an abelian varietyA over an algebraically closed fieldk of characteristic zero, any homomorphismk → C defines an isomorphismB∗abs(A) →B∗(AC). In view of this, I now writeB∗(A) for B∗abs(A) and call its elements theHodgeclassesonA.

Motivated classes (following Andre 1996)

Let k be an algebraically closed field, and letH∗W be a Weil cohomology theory on the

varieties overk with coefficient fieldQ. For a varietyX overk, letL andΛ be the operatorsdefined by a hyperplane section ofX, and define

E∗(X) = Q[L,Λ] · A∗W (X) ⊂ H2∗

W (X)(∗)

whereA∗W (X) is theQ-subspace ofH2∗

W (X)(∗) generated by algebraic classes. ThenE∗(X) is a gradedQ-subalgebra ofH2∗

W (X)(∗), but these subalgebras are not (obviously)stable under direct images. However, when we define

C∗(X) =⋃

Yp∗E

∗(X × Y ),

thenC∗(X) is a gradedQ-subalgebra ofH2∗W (X)(∗), and these algebras satisfy (A1). They

obviously satisfy (A2).

THEOREM 6.9. Let k be an algebraically closed subfield ofC, and letHW be the Weilcohomology theoryX 7→ H∗

B(XC). For every abelian varietyA, C∗(A) = B∗(AC).

PROOF. Clearly (A2) holds, and that (A3) holds is proved in Andre 1996, 0.5. ✷

The elements ofC∗(X) are calledmotivated classes.

ASIDE 6.10. As Ramakrishnan pointed out at the workshop, since proving the Hodge conjecture isworth a million dollars and the Tate conjecture is harder, itshould be worth more.

7 RATIONAL TATE CLASSES 14

7 Rational Tate classes

There are by now many papers proving that, if the Tate conjecture is true, then somethingelse even more wonderful is true. But what if we are never ableto decide whether the Tateconjecture is true? or worse, what if it turns out to be false?In this section, I suggest analternative to the Tate conjecture for varieties over finitefields, which appears to be muchmore accessible, and which has some of the same consequences.

An abelian variety with sufficiently many endomorphisms over an algebraically closedfield of characteristic zero will now be called aCM abelian variety. LetQal be the algebraicclosure ofQ in C. Then the functorA AC from CM abelian varieties overQal to CMabelian varieties overC is an equivalence of categories (see, for example, Milne 2006, §7).

Fix a p-adic primew of Qal, and letF be its residue field. It follows from the theoryof Neron models that there is a well-defined reduction functor A A0 from CM abelianvarieties overQal to abelian varieties overF, which the Honda-Tate theorem shows to besurjective on isogeny classes.

LetQal

w be the completion ofQ atw. For a varietyX overF, define

H∗A(X) = H∗

Af(X)×H∗

p (X) where

{H∗

Af(X) =

(lim←−m,p∤m

H∗(Xet,Z/mZ))⊗Z Q

H∗p (X) = H∗

crys(X) ⊗W (F) Qal

w.

For a CM abelian varietyA overQal,

H∗Af

(AK) (not-p) ≃ H∗Af

(A0)

H∗dR(A)⊗Qal Q

al

w ≃ H∗crys(A0)⊗W (F) Q

al

w,

and so there so there is a canonical (specialization) mapH∗A(A)→ H∗

A(A0).LetS be a class of smooth projective varieties overF satisfying the following condition:

(*) it contains the abelian varieties and projective spacesand is closed underdisjoint unions, products, and passage to a connected component.

DEFINITION 7.1. A family (R∗(X))X∈S with eachR∗(X) a gradedQ-subalgebra ofH2∗

A (X)(∗) is agood theory of rational Tate classesif(R1) for all regular mapsf : X → Y of varieties inS, f∗ mapsR∗(Y ) intoR∗(X) and

f∗ mapsR∗(X) intoR∗(Y );(R2) for all varietiesX in S,R1(X) contains the divisor classes;(R3) for all CM abelian varietiesA overQal, the specialization mapH2∗

A (A)(∗) →H2∗

A (A0)(∗) sends the Hodge classes onA to elements ofR∗(A0);(R4) for all varietiesX in S and all primesl (includingl = p), the projectionH2∗

A (X)(∗) →H2∗

l (X)(∗) defines an isomorphismR∗(X)⊗Q Ql → T∗l (X).

Thus, (R3) says that there is a diagram

B∗(A) ⊂ H2∗A (A)(∗)

↓ ↓R∗(A) ⊂ H2∗

A (A0)(∗),

and (R4) says thatR∗(X) is simultaneously aQ-structure on each of theQl-spacesT ∗l (X)

of Tate classes (including forl = p). The elements ofR∗(X) will be called therationalTate classes onX (for the theoryR).

The next theorem is an abstract version of the main theorem ofMilne 1999b.

8 REDUCTION TO THE CASE OF CODIMENSION 2 15

THEOREM 7.2. In the definition of a good theory of rational Tate classes, the condition(R4)can be weakened to:

(R4*) for all varietiesX in S and all primesl, the projection mapH2∗A (X) →

H2∗l (X)(∗) sendsR∗(X) into T ∗

ℓ(X).

In other words, if a family satisfies (R1-3), and (R4*), then it satisfies (R4). For the proof,see Milne 2007. I list three applications of it.

Any choice of a basis for aQl-vector space defines aQ-structure on the vector space.Thus, there are many choices ofQ-structures on theQl-spacesT ∗

l (X). The next theo-rem says, however, that there isexactlyone family of choices satisfying the compatibilityconditions (R1–4).

THEOREM 7.3. There exists at most one good theory of rational Tate classesonS. In otherwords, ifR∗

1 andR∗2 are two such theories, then, for allX ∈ S, theQ-subalgebrasR∗

1(X)andR∗

2(X) of H2∗A (X)(∗) are equal.

PROOF. It follows from (R4) that ifR∗1 andR∗

2 are both good theories of rational Tateclasses andR∗

1 ⊂ R∗2, then they are equal. But ifR∗

1 andR∗2 satisfy (R1–4), thenR∗

1 ∩R∗2

satisfies (R1–3) and (R4*), and hence also (R4). ThereforeR∗1 = R

∗1 ∩R

∗2 = R

∗2. ✷

The following is the main theorem of Milne 1999b.

THEOREM 7.4. The Hodge conjecture for CM abelian varieties implies the Tate conjecturefor abelian varieties overF.

PROOF. Let S0 be the smallest class satisfying (*). ForX ∈ S0, let R∗(X) be theQ-subalgebra ofH2∗

A (X)(∗) spanned by the algebraic classes. The family(R∗(X))X∈S0

satisfies (R1), (R2), and (R4*), and the Hodge conjecture implies that it satisfies (R3).Therefore it satisfies (R4), which means that the Tate conjecture holds for abelian varietiesoverF. ✷

The following is the main theorem of Andre 2006.

THEOREM 7.5. All Tate classes on abelian varieties overF are motivated.

PROOF. LetS0 be the smallest class satisfying (*). Fix anℓ, and forX ∈ S0, letC∗(X) betheQℓ-algebra of motivated classes inH2∗

ℓ (X)(∗). The family(C∗(X))X∈S0 satisfies (R1),(R2), and (R4∗) with A replaced byℓ, and Andre shows that it satisfies (R3). Therefore, byTheorem 7.2 withA replaced byℓ, it satisfies (R4). ✷

ASIDE 7.6. Assume that there exists a good theory of rational Tate classes for abelian varieties overF. Then we expect that all Hodge classes on all abelian varieties overQal with good reduction atw(not necessarily CM) specialize to rational Tate classes. This will follow from knowing that everyF-point on a Shimura variety lifts to a special point, which isperhaps already known (Zink 1983,Vasiu 2003). Note that it implies the “particularly interesting” corollary of the Hodge conjecturenoted in Deligne 2006,§6.

8 Reduction to the case of codimension 2

Throughout this section,K is a CM subfield ofC, finite and Galois overQ, andG =Gal(K/Q). Recall that an abelian varietyA is said to be split byK if End0(A) ⊗Q K isisomorphic to a product of matrix algebras overK.

8 REDUCTION TO THE CASE OF CODIMENSION 2 16

CM abelian varieties

8.1 Let S be a finite leftG-set on whichι acts without fixed points, and letS+ be asubset ofS such thatS = S+ ⊔ ιS+. ThenE

def= Hom(S,K)G is aQ-algebra split by

K such thatHom(E,K) ≃ S. The condition onι implies thatE is a CM-algebra, andthe condition onS+ implies that it is a CM-type onE, and soS+ defines an isomorphismE ⊗Q C ≃ CS+

. The quotient ofCS+by any lattice inE is an abelian variety of CM-type

(E,S+). Thus, from such a pair(S, S+) we obtain a CM abelian varietyA(S, S+), well-defined up to isogeny, which is split byK, and every such abelian variety arises in this way(up to isogeny). For example, an abelian variety of CM-type(E,Φ) whereE is split byKis isogenous to(S,Φ) whereS = Hom(E,Q). Note that if(S, S+) =

⊔(Si, S

+i ), then, by

construction,A(S, S′) is isogenous to the product of the varietiesA(Si, S+i ).

PROPOSITION8.2. LetG act on the setS of CM-types onK by the rule

gΦ = Φ ◦ g−1 def= {ϕ ◦ g−1 | ϕ ∈ Φ}, g ∈ G, Φ ∈ S,

and letS+ be the subset of CM-types containing1G. The pair(S, S+) defines an abelianvarietyA(S, S+), and every simple CM abelian variety split byK occurs as an isogenyfactor ofA(S, S+).

PROOF. CertainlyS is a finite leftG-set on whichι acts without fixed points, and for anyCM-typeΦ onK, exactly one ofΦ or ιΦ contains1G, and soA(S, S′) is defined.

Let A be a simple CM abelian variety split byK, say, of CM-type(E,Φ). Fix ahomomorphismi : E → K, and let

Φ′ = {g ∈ G | g ◦ i ∈ Φ}

— it is a CM-type onK. An elementg of G fixes iE if and only if gΦ′ = Φ′ (see, forexample, Milne 2006, 1.10), and sog ◦ i 7→ gΦ′ is a bijection ofG-sets fromHom(E,K)

onto the orbitO of Φ′ in S. Moreover,1G ∈ gΦ′ def= Φ′ ◦ g−1 if and only if g ∈ Φ′, i.e.,

g ◦ i ∈ Φ. Thus,g ◦ i 7→ gΦ′ sendsΦ ontoO ∩ S+. This shows thatA is isogenous to thefactorA(O,O ∩ S+) of A(S, S+). ✷

The Hodge conjecture

Let 2m = [K : Q]. Let a(2m) be the hyperplane arrangement{H1, . . . ,Hm} in Rm withHi the coordinate hyperplanexi = 0. The hyperplanesHi divideRmr

⋃Hi into connected

regionsR(ε1, . . . , εm) = {(x1, . . . , xm) ∈ Rm r

⋃Hi | sign(xi) = εi}

indexed by the set{±}m. LetR(2m) be the set of connected regions, and letR(2m)+ bethe subset of those withε1 = +.

Let ρ be a faithful linear representation ofG onRm such thatG acts transitively on theseta(2m) of coordinate hyperplanes andρ(ι) acts as−1 onRm. The pair(R(2m), R(2m)+)then satisfies the conditions of (8.5), and so defines a CM abelian varietyA = A(G,K, ρ)of dimension2m−1 split byK.

The next statement is the main technical result of Hazama 2003.

THEOREM 8.3. LetA = A(G,K, ρ). For everyn ≥ 0, theQ-algebraB∗(An) is generatedby the classes of degree≤ 2.

8 REDUCTION TO THE CASE OF CODIMENSION 2 17

PROOF. See Hazama 2003,§7. ✷

LetF be the largest totally real subfield ofK. The choice of a CM-typeΦ = {ϕ1, . . . , ϕm}onK determines a commutative diagram:

1 −−−−→ 〈ι〉 −−−−→ G −−−−→ Gal(F/Q) −−−−→ 1yι 7→(−,...,−)

yρΦ

y

1 −−−−→ {±}m −−−−→ {±}m ⋊ Sm −−−−→ Sm −−−−→ 1

Here{±} denotes the multiplicative group of order2, and the symmetric groupSm acts on{±}m by permutating the factors,

(σε)i = εσ−1(i), σ ∈ Sm, ε = (εi)1≤i≤m ∈ {±}m.

Let ǫ be the unique isomorphism of groups{±} → {0, 1}. Forg ∈ G, write

g ◦ ϕi = ιǫ(εi)ϕσ−1(i).

ThenρΦ is the homomorphismg 7→ ((εi)i, σ).There is a natural action of{±}m ⋊ Sm onRm:

((εi)1≤i≤m, σ)(xi)1≤i≤m = (εixσ−1(i))1≤i≤m.

By composition, we get a linear representation ofG on Rm, also denotedρΦ. This actstransitively transitively ona(2m), andρΦ(ι) acts as−1.

The next statement is Hazama 2003, 6.2, but the proof there isincomplete.7

PROPOSITION8.4. Every simple CM abelian variety split byK is isogenous to a subvari-ety ofA(G,K, ρΦ).

PROOF. Because of Proposition 8.2, it suffices to show thatA(G,K, ρΦ) is isogenous tothe abelian varietyA(S, S+) of (8.1). For a CM-typeΦ′ onK, let εi(Φ′) equal+ or −according asϕi ∈ Φ′ or not. ThenΦ′ 7→ (εi(Φ

′))1≤i≤m : S → {±}m ≃ R(2m) isa bijection, sendingS+ ontoR(2m)+. This map isG-equivariant, and soA(S, S+) isisogenous toA(R(2m), R(2m)+)

def= A(G,K, ρΦ). ✷

The following is the main theorem of Hazama 2002, 2003.

THEOREM 8.5. In order to prove the Hodge conjecture for CM abelian varieties overC, itsuffices to prove it in codimension2.

PROOF. If the Hodge conjecture holds in codimension2, then Theorem 8.3 shows that itholds for the varietiesA(G,K, ρΦ)n, but Proposition 8.4 shows that every CM abelianAis isogenous to a subvariety ofA(G,K, ρΦ)n for someK andn. It is easy to see that if theHodge conjecture holds for an abelian varietyA, then it holds for any abelian subvariety(because it is an isogeny factor). ✷

7It applies only to the simple subvariety of the abelian variety of CM-type (K,Φ). Also the description(ibid. p632) of the divisor classes is incorrect, and so the statement in Theorem 7.14 thatAA(2n)(G;K) hasexotic Hodge classes whenn ≥ 3 should be treated with caution.

8 REDUCTION TO THE CASE OF CODIMENSION 2 18

The Tate conjecture

Theorems 8.5 and 7.4 show that, in order to prove the Tate conjecture for abelian varietiesoverF, it suffices to prove the Hodge conjecture in codimension2. The following is a morenatural statement.

THEOREM 8.6. In order to prove the Tate conjecture for abelian varieties overF, it sufficesto prove it in codimension2.

More precisely, we shall show that ifT 2(A, ℓ) holds for all abelian varietiesA overFand someℓ, thenT r(A, ℓ) andEr(A, ℓ) hold for all abelian varietiesA overF and allr andℓ. We fix ap-adic primew of Qal, and use the same notations as in§7.

LEMMA 8.7. Let A be an abelian variety overQal. If A is split byK, then so also isA0.Conversely, every abelian variety overF split byK is isogenous to an abelian varietyA0

with A split byK.

PROOF. LetE be an etale subalgebra ofEnd0(A) such that[E : Q] = 2dimA. ThenE isa maximal etale subalgebra ofEnd0(A0). BecauseE is split byK, so also isEnd0(A0).The converse follows from Tate 1968, Lemme 3. ✷

PROOF (OF THEOREM 8.6). LetA = A(S, S+) be the abelian variety in (8.2), which (see§7) we may regard as an abelian variety overQal. ThenA0 is an abelian variety overFsplit byK, and every simple abelian variety overF split byK is isogenous to an abeliansubvariety ofA0 (by 8.2, 8.7). The inclusionEnd0(A) → End0(A0) realizesC(A0) as aQ-subalgebra ofC(A), and hence defines an inclusionL(A0)→ L(A) of Lefschetz groups(see§2). Consider the diagram

MT(A) � �// L(A)

P (A0)

OO�

�

�

� �// L(A0)

?�

OO

in which MT (A) is the Mumford-Tate group ofA andP (A0) is the smallest algebraicsubgroup ofL(A0) containing a Frobenius endomorphism ofA0. Almost by definition,MT(A) is the largest algebraic subgroup ofL(A) fixing the Hodge classes inH2∗

B (AnC)(∗)

for all n, and so, for any primeℓ, MT(A)Qℓis the largest algebraic subgroup ofL(A)Qℓ

fixing the Hodge classes inH2∗ℓ (An)(∗) for all n. On the other hand, the classes in

H2∗ℓ (An

0 )(∗) fixed by P (A0)Qℓare exactly the Tate classes. The specialization isomor-

phismH2∗ℓ (A)(∗) → H2∗

ℓ (A0)(∗) is equivariant for the homomorphismL(A0) → L(A).As Hodge classes map to Tate classes inH2∗

ℓ (A)(∗) (see§6), they map to Tate classes inH2∗

ℓ (A0)(∗), and so they are fixed byP (A0)Qℓ. This shows thatP (A0)Qℓ

⊂ MT(A)Qℓ

(insideL(A)Qℓ), and soP (A0) ⊂ MT(A) (insideL(A)). The following is the main tech-

nical result of Milne 1999a (Theorem 6.1):

Assume thatK contains a quadratic imaginary field. Then the algebraic sub-groupMT(A) andL(A0) of L(A) intersect inP (A0).

We now enlargeK so that it contains a quadratic imaginary field. Corollary 5.2 allowsus to chooseℓ so thatE(A0, ℓ) holds. LetG be the algebraic subgroup ofL(A0)Qℓ

fixingthe algebraic classes inH2∗

ℓ (An0 )(∗) for all n. If the Tate conjecture holds in codimension

9 THE HODGE STANDARD CONJECTURE 19

2, thenG fixes the Tate classes inH4ℓ (A

n0 )(2) (all n); therefore (by Theorem 8.3), it fixes

the Hodge classes inH2∗ℓ (An)(∗) (all n), and soG ⊂ MT(A)Qℓ

∩ L(A0)Qℓ= P (A0)Qℓ

;therefore,G fixes all Tate classes inH2∗

ℓ (An0 )(∗) (all n), which shows that the space of

Tate classes inH2∗ℓ (An

0 )(∗) (all n) is spanned by the algebraic classes (apply Theorem 4.1).Therefore, the equivalent statements in Theorem 1.5 hold for An

0 for all r andn. It followsthat the same is true of every abelian subvariety of some power A0, which includes allabelian varieties overF split byK (apply Lemma 8.7). Since every abelian variety overF

is split by some CM-field, this completes the proof. ✷

Optimists will now try to prove the Hodge conjecture in codimension2 for CM abelianvarieties, or, what may (or may not) be easier, the Tate conjecture in codimension2 forabelian varieties overF. Pessimists will try to prove the opposite. Others may prefer tolook at the questions in§10.

9 The Hodge standard conjecture

Let k be an algebraically closed field, and letHW be a Weil cohomology theory on thevarieties overk. For a varietyX over k, let Ar

W (X) be theQ-subspace ofH2rW (X)(r)

spanned by the classes of algebraic cycles. Letξ ∈ H2W (X)(1) be the class of a hyperplane

section ofX, and letL : H iW (X) → H i+2

W (X)(1) be the map∪ξ. The primitive partAr

W (X)prim of ArW (X) is defined to be

ArW (X)prim = {z ∈ Ar

W (X) | Ldim(X)−2r+1z = 0}.

CONJECTURE(HODGE STANDARD). Let d = dimX. For2r ≤ d, the symmetric bilinearform

(x, y) 7→ (−1)rx · y · ξd−2r : ArW (X)prim ×A

rW (X)prim → A

dW (X) ≃ Q

is positive definite (Grothendieck 1969,Hdg(X)).

The next theorem is an abstract version of the main theorem ofMilne 2002.

THEOREM 9.2. The Hodge standard conjecture holds for every good theory ofrationalTate classes.

In more detail, let(R∗(X))X∈S be a good theory of rational Tate classes. ForX ∈ S,the cohomology classξ of a hyperplane section ofX lies inR1(X), and we can defineRr(X)prim and the pairing onRr(X)prim by the above formulas. The theorem states thatthis pairing

Rr(X)prim ×Rr(X)prim → Q

is positive definite.For the proof, see Milne 2007. I list one application of this theorem.

THEOREM 9.3. If there exists a good theory of rational Tate classes for which all algebraicclasses are rational Tate classes, then the Hodge standard conjecture holds.

PROOF. The bilinear form onRr(X)prim restricts to the correct bilinear form onAr(X)prim.If the first is positive definite, then so is the second, which implies that the form onAr

W (X)prim

is positive definite for any Weil cohomology theoryHW . ✷

10 ON THE EXISTENCE OF A GOOD THEORY OF RATIONAL TATE CLASSES20

ASIDE 9.4. LetS0 be the smallest class satisfying (*) and letS be a second (possibly larger) class.If the Hodge conjecture holds for CM abelian varieties, thenthe family (Ar(X))X∈S0

is a goodtheory of rational Tate classes forS0; if moreover, the Tate conjecture holds for all varieties inS,then(Ar(X))X∈S is a good theory of rational Tate classes forS. However, the Tate conjecture alonedoes not imply that(Ar(X))X∈S is a good theory of rational Tate classes onS; in particular, wedon’t know that the Tate conjecture implies the Hodge standard conjecture. Thus, in some respects,the existence of a good theory of rational Tate classes is a stronger statement than the Tate conjecturefor varieties overF.

10 On the existence of a good theory of rational Tate classes

I consider this only for the smallest classS0 satisfying (*), which, I recall, contains theabelian varieties.

CONJECTURE(RATIONALITY CONJECTURE). Let A be a CM abelian variety overQal.The product of the specialization toA0 of any Hodge class onA with any Lefschetz classonA0 of complementary dimension lies inQ.

In more detail, a Hodge class onA is an element ofγ of H2∗A (A)(∗) and its specializa-

tion γ0 is an element ofH2∗A (A0)(∗). Thus the productγ0 · δ of γ0 with a Lefschetz class

of complementary dimensionδ lies in

H2dA (A0)(d) ≃ A

pf ×Qal

w, d = dim(A).

The conjecture says that it lies inQ ⊂ Apf ×Qal

w . Equivalently, it says that thel-componentof γ0 · δ is a rational number independent ofl.

REMARK 10.2. (a) The conjecture is true for a particularγ if γ0 is algebraic. Therefore,the conjecture is implied by the Hodge conjecture for CM abelian varieties (or even by theweaker statement that the Hodge classes specialize to algebraic classes).

(b) The conjecture is true for a particularδ if it lifts to a rational cohomology class onA. In particular, the conjecture is true ifA0 is ordinary andA is its canonical lift (becausethen all Lefschetz classes onA0 lift to Lefschetz classes onA).

For an abelian varietyA overF, letL∗(A) be theQ-subalgebra ofH2∗A (A)(∗) generated

by the divisor classes, and call its elements theLefschetz classesonA.

DEFINITION 10.3. LetA be an abelian variety overQal with good reduction to an abelianvarietyA0 overF. A Hodge classγ onA is locallyw-Lefschetzif its imageγ0 inH2∗

A (A0)(∗)is in theA-span of the Lefschetz classes, and it isw-Lefschetzif γ0 is itself Lefschetz.

CONJECTURE(WEAK RATIONALITY CONJECTURE). Let A be an abelian variety overQal with good reduction to an abelian varietyA0 overF. Every locallyw-Lefschetz Hodgeclass is itselfw-Lefschetz.

THEOREM 10.5. The following statements are equivalent:

(a) The rationality conjecture holds for all CM abelian varieties overQal.

(b) The weak rationality conjecture holds for all CM abelian varieties overQal.

(c) There exists a good theory of rational Tate classes on abelian varieties overF.

10 ON THE EXISTENCE OF A GOOD THEORY OF RATIONAL TATE CLASSES21

PROOF. (a) =⇒ (b): Choose aQ-basise1, . . . , et for the space of Lefschetz classes ofcodimensionr onA0, and letf1, . . . , ft be the dual basis for the space of Lefschetz classesof complementary dimension (here we use Milne 1999a, 5.2, 5.3). If γ is a locallyw-Lefschetz class of codimensionr, thenγ0 =

∑ciei for someci ∈ A. Now

〈γ0 · fj〉 = cj ,

which (a) implies lies inQ.(b) =⇒ (c): See Milne 2007.(c) =⇒ (a): If there exists a good theoryR of rational Tate classes, then certainly the

rationality conjecture is true, because thenγ0 · δ ∈ R2d ≃ Q. ✷

Two questions

QUESTION 10.6. LetA be a CM abelian variety overQal, letγ be a Hodge class onA, andlet δ be a divisor class onA0. Does(A0, γ0, δ) always lift to characteristic zero? That is,does there always exist a CM abelian varietyA′ overQal, a Hodge classγ′ onA′, a divisorclassδ′ onA′ and an isogenyA′

0 → A0 sendingγ′0 to γ0 andδ′0 to δ?

PROPOSITION10.7. If Question 10.6 has a positive answer, then the rationalityconjectureholds for all CM abelian varieties.

PROOF. Let γ be a Hodge class on a CM abelian varietyA of dimensiond overQal. If γhas dimension≤ 1, then it is algebraic and so satisfies the rationality conjecture. We shallproceed by induction on the codimension ofγ. Assumeγ has dimensionr ≥ 2, and letδ be a Lefschetz class of dimensiond − r. We may suppose thatδ = δ1 · δ2 · · · whereδ1, δ2, . . . are divisor classes. Apply (10.6) to(A, γ, δ). Thenγ′ · δ′1 is a Hodge class onA′

of codimensionr − 1, and

γ0 · δ ∈ (γ′ · δ′1)0 · δ2 · · · · · δd−rQ ⊂ Q.✷

A pair (A, ν) consisting of an abelian varietyA overC and a homomorphismν froma CM fieldE to End0(A) is said to be ofWeil typeif the tangent space toA at 0 is a freeE ⊗Q k-module. For such a pair(A, ν), the space

W d(A, ν)def=∧d

EH1(A,Q) ⊂ Hd(A,Q), whered = dimE H

1(A,Q),

consists of Hodge classes (Deligne 1982, 4.4). WhenE is quadratic overQ, the spacesW d

were studied by Weil (1977), and for this reason its elementsare calledWeil classes. A po-larization of an abelian variety(A, ν) of Weil type is a polarization ofAwhose Rosati invo-lution stabilizesE and induces complex conjugation on it. There then exists anE-hermitianform φ onH1(A,Q) and anf ∈ E× with f = −f such thatψ(x, y)

def= TrE/Q(fφ(x, y))

is a Riemann form forλ (ibid. 4.6). We say that the Weil classes on(A, ν) aresplit ifthere exists a polarization of(A, ν) for which theE-hermitian formφ is split (i.e., admits atotally isotropic subspace of dimensiondimE H1(A,Q)/2).

QUESTION 10.8. Is it possible to prove the weak rationality conjecture for split Weilclasses on CM abelian variety by considering the families ofabelian varieties considered inDeligne 1982, proof of 4.8, and Andre 2006,§3?

REFERENCES 22

A positive answer to this question implies the weak rationality conjecture because ofthe following result of Andre (1992) (or the results of Deligne 1982,§5).

Let A be a CM abelian variety overC. Then there exist CM abelian varietiesBi and homomorphismsA→ Bi such that every Hodge class onA is a linearcombination of the inverse images of split Weil classes on theBi.

In the spirit of Weil 1967, I leave the questions as exercisesfor the interested reader.

ASIDE 10.9. In the paper in which they state their conjecture concerning the structure of the pointson a Shimura variety over a finite field, Langlands and Rapoport prove the conjecture for some sim-ple Shimura varieties of PEL-type under the assumption of the Hodge conjecture for CM-varieties,the Tate conjecture for abelian varieties over finite fields,and the Hodge standard conjecture forabelian varieties over finite fields. I’ve proved that the first of these conjectures implies the othertwo (see 7.4 and 9.3), and so we have gone from needing three conjectures to needing only one. Aproof of the rationality conjecture would eliminate the need for the remaining conjecture. Probablywe can get by with much less, but having come so far I would liketo finish it off with no fudges.

ASIDE 10.10. Readers of the Wall Street Journal on August 1, 2007, were excited to find a headlineon the front page of Section B directing them to a column on “The Secret Life of Mathematicians”.The column was about the workshop, and included the following paragraph:

Progress, though, was made. V. Kumar Murty, of the University of Toronto, said thatas a result of the sessions, he’d be pursuing a new line of attack on Tate. It makesuse of ideas of the J.S. Milne of Michigan, who was also in attendance, and involvesAbelian varieties over finite fields, in case you want to get started yourself.

This becomes more-or-less correct when you replace “Tate” with the “weak rationality conjecture”.8

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Mathematics Department, University of Michigan, Ann Arbor, MI 48104, USAwww.jmilne.org/math/