The Tate conjecture for almost ordinary Abelian varieties over finite fields

The Täte Conjecture for Almost Ordinary

Abelian Varieties over Finite Fields

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin

University of California at Berkeley

and

Russian Academy of Sciences, Pushchino

l Introduction

Let K be a finite field of characteristic p consisting of q elements. Let K (o) denote the algebraic closure of K and G(K) := G&\(K(o)/K} the Galois group of K. Let Υ be a smooth projective variety over K and set Υ(α) := Υ χ Κ(α). Let £ be a rational prime different from p. The Galois group G(K) acts on the (twisted) £-adic etale cohomology groups F2 m(Y(a),Qi)(m). In [Tl] Täte conjectured that the subspace fixed under the Galois action is spanned by the cohomology classes of co-dimension m algebraic cycles on Y.

This conjecture has been proved in certain cases, e.g., Fermat hypersur-faces satisfying certain numerical conditions [Sh, Y], elliptic Ä'3 surhypersur-faces [AS], A'3 surfaces of finite height [N, NO] and powers of ordinary KZ sur-faces [Z3].

Now, let Y = X be a p-dimensional Abelian variety. In this case, Täte [T2] has proved his conjecture for m = 1. Notice that the well known Interpretation of ί-adic etale cohomology groups of X äs skew-symmetric multilinear forms on the Täte module Vt(X) allows us to identify the Ga-lois invariant subspace H2m(X(a),Qe)(m)G(-K^> with the space of all skew-symmetric 2m-linear forms E on Vt(X) such that

JS(Ft*i,··· , F r z2 m) = gmJ5(zi,··· , z2 m)

for all xj, · · · ,a;2m g V/(X). Here Fr is the Frobenius endomorphism of X acting on the Täte module. In view of the result of Täte mentioned above, in order to prove the Täte conjecture for arbitrary m, it suffices to check that each E can be presented äs a linear combination of exterior products of skew-symmetric bilinear forms ψ : Vt(X) x Vt(X) -*· Qt such that

,j/) for all x,y € Vt(X).

For example, this method makes it possible to verify easily the Täte con-jecture for a supersingular Abelian variety X (in fact, in this case, if K is sufficiently large, then all even-dimensional cohomology classes are alge-braic). Tankeev [T] verified the Täte conjecture for an absolutely simple Abelian variety X of prime dimension g. The papers [ZI, Z2] contain the proof of the Täte conjecture for the "almost supersingular" case (that is, for Abelian varieties with the set of slopes {0,1/2,1} and such that the slopes 0 and l have length 1).

The aim of this paper is to prove the Täte conjecture for the "almost ordinary" case. Namely, we consider a simple Abelian variety X having the same Newton polygon äs one of the product of a supersingular elliptic curve and an ordinary (g — l)-dirnensional Abelian variety [O]. This means that the set of slopes of X is {0,1/2,1} and the slope 1/2 has length 2. A general result of [LO] guarantees that such Abelian varieties exist in all dimensions g > l and all prime characteristics p.

Our proof is based on the study of multiplicative relations between the eigenvalues of Fr.

Acknowledgments. We are deeply grateful to Noriko Yui for her interest in this paper and -invaluable help during the preparation of the manuscript. The research of the first author was supported by the National Science Foundation under Grant No. DMS 9002939. Part of the research we report on was done while the second author was a Visiting Professor at Ohio State University and was finished while he was a Visiting Scholar at Harvard University; he would like to thank both universities for their hospitality.

2 Täte modules

Let X be a g>-dimensional Abelian variety defined over a commutative field K. Let us fix a separable algebraic closure K(s) of K and let G(K) :— G&\(K(s)/K) denote the Galois group of K. For a positive integer m, we denote by Xm the group of elements χ € X(K(s)) such that raz = 0. It is well known that if char(Ji) does not divide m, then Xm is a free TLImZ-module of rank Ig [M]. Let us fix a prime number t φ char(/i). Then one may define the Z^-Tate module Ti(X) äs the projective limit of the groups Xm where m runs through the set of all powers £' and the transition map is just multiplication by ί. It is well known that TI(X) is a free Z^-module of rank 2g [M]. Clearly, all Xm are finite Galois submodules of X(K(s)), and the Galois actions for m = £* glue together to yield a continuous homomorphism [S,R] pi - pt,x : G(K) -»· Aut Tt(X).

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin 181 Ig [M] and one may naturally identify Tt(X) with a certain Z^-lattice of rank 2dim(.X") in Vt(X). In particular, AutT^(X) becomes an open compact subgroup in A\itVt(X). This allows us to regard pt äs an £-adic representation [S, R],

Pt = Ρί,χ ·· G(K) ~* AutT/(X) C Aut Vt(X). We have

Gt C A.\ATt(X) C AxAVt(X).

Let End(Jf) denote the ring of all Jf-endomorphisrns of X. Clearly, all Xm are End(X)-invariant subgroups of X(K(s}), and their invariance gives rise to a canonical ring homomorphism

End(X) ® Z/mZ -> End(Xm)

which is an embedding if char^i) does not divide m. The image of the homomorphism lies in EndG(^)(Xm). For m running through powers of t, these homomorphisms glue together to the embedding [R, T2]

(2.1) EndpO<8>Z<-»End2i(X).

The image of this embedding lies in Έηάο(κ}Τί(Χ). Extending (2.1) by Qi -linearity, we obtain a canonical embedding [R, T2]

(2.2) EndpQ ® Q< -+ End Vt(X}. The image of this embedding lies in

We write \t : G(K) — »· Z*t for the cyclotomic character defining the Galois action on all the £-power roots of unity. Let £ be an invertible ample sheaf on X [M] . One may associate to £ a skew-symmetric non-degenerate bilinear form

K£:Vt(X)xVt(X)->Qt,

called the Riemann form [M] . This form is uniquely defined up to multi-plication by a constant in Z£, and enjoys the foliowing property:

for all σ € G(K) and x,y € Vt(X). It follows that G ι lies in the group Gp(Vt(X)) of symplectic similitudes of Vt(X) with respect to HC, and there is a continuous homomorphism x't : Gt — > Z*t such that

Hc(gx,gy) = XitäKcte, v) for all g € G£ and z, y € Vi(X). If CJ is another (ample) invertible sheaf on X, then there exists u E End(J3Q ® <Q>£ such that

On the other band, each skew-symmetric bilinear form of the type Vi(X) χ Vt(X) -> Qi, x, y >-+ Hc(ux, y) where u € End(X) can be presented äs a Qt -linear combination of the forms of type %c, (see [M], Section 20).

We write End°X for the tensor product End(X) ® Q. It is well known that End°X is a finite-dimensional semi-simple

Q-algebra. We have End(X) ®Qt = End°J\T <g><u Q*.

Notice that X is a simple Abelian variety if and only if End°X is a division algebra.

3 Abelian varieties over finite fields

Let us assume that K is a finite field of characteristic p with q elements. Clearly, q is a a power of p. The Galois group G(K) is pro-cyclic and its canonical topological generator is the Frobenius automorphism:

σ κ '· K (a) —* K (a), χ >—*· xq·

One may easily check that χι(σκ) — 1· If K1 is a finite algebraic extension of K, then K' is also finite and σκ> = <?% · Let X be an Abelian variety defined over K. We denote by

Fr = Fr# G End(X)

the Frobenius endomorphism Fr of Χ [Τ2, M, R]. It is known that the action of Fr on TAX] coincides with the action of σκ, i.e., we may write

Fr = FrK := κ(σκ) e Gt C AutT«(X) C Aut Vt(X). Clearly,

According to a well known result of A. Weil [M], the linear operator Fr : Vt(X) -»· Vt(X)

is semi-simple and its characteristic polynomial

P(<) = Px(0 := det(i id - Fr | Vt(X)) E Ze[t] lies in Z [i] and does not depend on the choice of t. (Here

id : Vt(X) - Vt(X)

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin 183

of the polynomial P(<) £ Z[t] C Q[t]. Then L is a finite Galois extension of Q. Let R = β χ denote the collection of roots of P(i) in L, i.e., the set of eigenvalues of Fr. By the very definition of L, we have R C L. Then the last statement of a theorem of Weil asserts the following: For each embedding

of L into the field C of complex numbers, we have \a\ = g1/2 for all α £ R. It follows easily that R c L* and the map α —* q/α is a permutation of

R [M] (because g/α is just the complex conjugate of o·) and furthermore,

the multiplicities of roots α and g/α coincide. If a = g/a G R, then its multiplicity is even, since the action of Fr multiplies by g a non-degenerate skew-symmetric bilinear (Riemann) form. This means that it is possible to rearrange the collection R of all roots (with multiplicities) of P(i) in such a way that it will be of the form

«1,0:2,···, otg-q/ai, g/o-2, ··· , q/ag.

3.0. Remark. By definition, L is a number field obtained by adjoining

to Q all elements of R. Let us fix an embedding L C C. Then the complex conjugation i : C —* C maps each root α into g/α. It follows that t,(L) C L, i.e., i(L) = L and one may view L äs an element of Gal(i/Q). Clearly, this

element does not depend on the choice of embedding of L into C. 3.0.1. Lemma, t lies in the center of G a l ( i / Q ) .

Proof. It suffices to check that ισ(α) = σι(α) for all roots α and

automorphisms σ of L.

We have σι(α) = <r(q/a) = g/cr(a). Since <r(a) € R, we have ισ(α) = g^(a),i.e, ι,σ(α) - σι(α).

3.1. Let us consider the multiplicative subgroup Γ of L* generated by

all the eigenvalues of Fr, i.e., by all elements of R. It is a finitely generated commutative group. Clearly, Γ is generated by g and α^,α^,· · · ,a3. It

follows that the rank of Γ is always less or equal to p H- 1; the equality holds if and only if all the roots of P(i) are simple and (in the notation above) the set {ΟΊ, a?, · · · , ag; q} consists of multiplicatively independent

elements. Incidentally, if the set {«1,0-2, · · · > <Xg\ l] consists of multiplica-tively independent elements, then the description of R given above implies that Γ is afree commutative group of rank g + 1. This means, in particular, that Γ does not contain non-trivial roots of unity, that is, roots of unity different from 1.

Example [T]. If X is an absolutely simple Abelian variety of prime

3.2. Remark. In our notation, the assertion "all roots of P(<) are simple" means that a,· ^ otj while i φ j and α» Φ q/ctj for all pairs (i,j), including i = j .

3.3. Remark. If X is simple, then W(t) is either irreducible over Q or a power of a Q-irreducible polynomial [T2, R].

3.4. Theorem of Täte [T2, R].

(a) End°A" is commutative if and only if all roots of P(i) are simple, and in this case, Έηά°Χ is generated by Fr äs Q-algebra.

(b) End°X «IQ Qi = EndG«Vi(X) = End^ Vt(X).

(c) Each skew-symmetric bilinear form ψ : Vt(X} x Vt(X) — > Q/ such that

y?(Fr x, Fry) = 9 <^>(z, y) for all x, y is a Qf-linear combination of forms of type He1·

3.5. Remark. Let us assume that 1P(£) is irreducible over Q, i.e., the endomorphism algebra End°X of X is a commutative field. Then the Galois group Gal(£/Q) acts transitively on R.

For each root a £ R the subfield Q(a) of L is canonically isomorphic to End°X; under this isomorphism α is mapped to Fr. Let us consider the commutative group U consisting of all functions / : R-^Z enjoying

/(a) = -/(g/a) for all a £ R.

Clearly, U is a free commutative group of rank g. (There are no square roots of q in R, because all elements of R are of (odd) multiplicity one.)

There is a natural Gal(L/Q)-action on U induced by the action on R. Namely, (tr/)(a) = /(σ"1«) for all σ G Gal(L/Q), α e R. Notice that under this action, complex conjugation L acts on U äs multiplication by — 1. Now assume that there exists an infinite cyclic Gal(L/Q) -invariant subgroup Δ of U. Since Aut(A) = {!,—!}, the Galois action on Δ is defined by quadratic character

ΚΔ : Gal(L/Q) - Aut(A) = {!,-!}.

This character gives rise to imaginary quadratic subfield B of L: notice that the complex conjugation acts äs multiplication by — l on Δ C U. The Galois group Gal(L/J5) coincides with the kernel of the character. The transitivity of the Gal(L/Q)-action on R implies that

/(a) = ±/(/J) for all a,ߣR.

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin 185 of L such that σβ - a. Then (σ/)(α) = /(/?). On the other band, since / € Δ, σ/ = ± / . This implies that /(a) = ±f(ß). Notice, that B lies in E :— Q(a) for each root a. Indeed, one has only to check that if σα = α for some σ e Gal(£/Q), then σ acts identically on Δ. In order to prove this, notice that (σ/)(α) = /(α) for all / 6 Δ (even for all / 6 U). Since σ f = ± / and /(a) does not vanish for non-zero /, we obtain that σ / = /. One may easily check that R consists of exactly two Gal(i/5)-orbits; a non-zero / is constant on each of these orbits and takes different values on different orbits. More precisely, /(a) = —/(/?) if roots a and β belong to different orbits. As above, / 6 Δ.

As a corollary, we obtain that ander our assumption (the existence of Galois invariant infinite cyclic subgroup of t/), End°X contains an imagi-nary quadratic subfield (namely, B).

Let us put Οί\ = «|/<? f°r *U i and denote by Γ' the multiplicative group generated by all a.\. One may easily check that the rank of Γ is equal to 1 + therankof Γ .

Consider the homomorphism

u:U->T',u(f) = Π <*/(α) = f[α'/(α'\

a€R t=l

Using the assumption f(a) = —/(?/<*)> °n e may easily check that u is a

surjective homomorphismfrom U to Γ'. It is also clear that u is Gal(£/Q)-equivariant.

3.5.1. Remark. Under the assumptions and notations of the previous Remark let β be a root and

NormB/B : E = Q(/?) -+ B

is the norm map. Clearly, g = [E : B]. Assume that / G Δ and /(/?) = m. Then one may easily check that

3.6. Theorem. Assume that P(<) κ irreducible over Q, i.e., the endo-morphism aJgebra End°X of X is a coinmutative field. Let us assume also that the rank of Γ is equa] to g.

Then

(a) The endomorphism algebra of X, End°X, contains an imaginary quadratic subGeld;

(c) Put Δ = {/ € U\u(f) is a root of unity}. Then Δ is infinite

cyclic Gal(L/Q) -invariant subgroup of U. If f is a generator of Δ, then

/(a) = ±1 for all α € R and Hf=1 a i/ ( a i ) = «(/) is a root of unity.

Proof of Theorem 3.6. Clearly, Γ" = u(U), and Δ is a commutative

free group of finite rank with rank equal to

g - rank(r') = ff - (rank(F) - 1) = 1.

So Δ is a Gal(L/Q) -invariant infinite cyclic group. Let / be its generator. Let us put n; = /(c*i)· We have /(?/<*») = —n,·. From the definition of a generator, if there are any integers mi , m%, · · · , mg such that

is a root of unity, then there exists an integer d such that mt- = dn,· for every i. Indeed, one has only to notice that the function

h : R —+ Z, h(üi) = m,·, h(q/ai) = — rat

-belongs to Δ. Clearly, the greatest common divisor of n ι,η^, ··· ,ng is

equal to 1.

Now it is time to apply results of Section 3.5. Recall that all /(a) have the same absolute value. Since

have the greatest common divisor equal to l, this absolute value is equal to l, i.e., all n, = ±1.

This yields the proof of the second and the third assertion. The first assertion also follows frorn the results of Section 3.5, because Δ is a cyclic Galois-invariant subgroup of U.

3.6.1. Remark. We keep the notations and assumptions of Theorem

3.6. Let β ζ. R. Let us identify canonically the endomorphism algebra of X with E — Q(ß) and let B be the imaginary quadratic subfield of E. Let /

be a generator of Δ normalized by the condition /(/?) = 1. Now, it follows

easily from Remark 3.5.1 and the definition of Δ, that Norm£/ß(/?2/g) is

a root of unity, i.e., the norm of β to the imaginary quadratic field B is a

root of unity times a power of the square root of q.

4 Newton polygons

We keep all notation and assumptions of the previous Section. Let ÖL be

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin . 187 a £ ÖL and q/a € ÖL for all a £ R. Since q is a power of p, we obtain that äf £J is a maximal ideal in ÖL not lying over p, then

orda(a) = 0 for all a 6 R

where orda : L* —»· Z is a discrete valuation attached to £J. It follows that ord£j(7) = 0 for all γ 6 Γ. This is because Γ is, by definition, generated by elements of R.

We denote by S the set of maximal ideals p in ÖL , that lie over p. For p € «S, we denote by ordp : L* -» Q the discrete valuation attached to p and normalized by the condition that ordp(g) = 1.

Now we are ready to recall the definition of the Newton polygon of X. The Newton polygon of X consists of a finite set Slpx C Q called the set of slopes and a positive integral-valued function length^ : Slpx —* Z+, which assigns to each slope its length. For each p € S, S\px is given by

Slp* = ordp(Ä), and for each c € Slpx, lengthA- (c) is given by

lengthx(c) = #{roots a 6 R (with multiplicities) such that ordp(a) = c}. Since L is a Galois extension of Q, the Galois group Gal(L/Q) acts transi-tively on S. This implies that the definition of S\px and lengthx does not depend on the choice of p, because R is the collection of roots of P(i), and the polynomialP(i) has rational coefficients (i.e., R is Gal(L/(Q>)-invariant).

Since the multiplicities of roots a and q/a coincide and ordp(g/a) = ordp(g) - ordp(a) = l - ordp(a),

the following assertion holds true: If c € Slpx, ihen l — c £ Slpx and lengthA-(c) = lengthA>-(l - c). If 1/2 is a slope then its length is even.

4.1. R e m a r k . Assume that the polynomial P(i) is a power of a Q-irreducible polynomial, i.e., there exists a polynomial P (t) and a positive integer i such that W (t) = P (t)*. Then, it is easily seen that lengthx(c) is divisible by i for all c G SIpx.

4.2. Definition. An Abelian variety X is called ordinary if Slpx = {0,1} and lengthx(0) = lengthx(l) -- g.

4.3. Definition. An Abelian variety X is called supersingular if SW = {1/2} and lengthx(l/2) = 2g.

4.4. Definition. ([ZI, Z2].) An Abelian variety X is of KZ type if

Slpx = {0, 1} oi {0, 1/2, 1} and lengthx(0) = lengthx(l) = 1.

If X is a simple Abelian variety of KZ type then the rank of Γ is equal

to g + 1 [ZI, 22].

4.5. Example. Let us assume that the endomorphisin algebra of X

is a commutative field. Let us define positive integer / by the formula

q = pf . Let e be the ramification index of L at p, i.e., the principal ideal

(p) is equal to (Ilpes?)*· ket ß £ R- ^et u s sP^t *n e principal ideal (/?)

into the product of maximal ideals in the ring of integers in L. Clearly,

for a certain non-negative integral valued function a on S. Now, using the transitivity of the Galois action on R and S, one may easily get the following description of the Newton polygon of X.

b \ .,

- J = - - 2 , for all slopes jj .

5 Almost ordinary Abelian varieties

Throughout this section we assume that X is a simple ^-dimensional Abelian variety, Slp^· = {0, 1/2, 1} and lengthx(l/2) = 2. These varieties were

in-troduced and studied by F. Oort [0]. In particular, he proved that the endomorphism algebra End°X of X is a commutative field, that is, the polynomial P(<) is irreducible over the field of rational numbers. This im-plies that all its roots have multiplicity 1.

5.1. Remark. It follows from the very definition of X that for each maximal ideal p 6 5 there exist exactly two roots α G R such that ordp(a)

are not integers. Clearly, if α is one of these roots, then q/a is other.

5.2. Corollary. In the notation above, for each maximal ideal p € S

there exists exactly one root a € {<*!, a2, · · · , ag] such that ordp(a) is not

an integer.

5.3. Corollary. Ifforsome/?€ {a1,a2,··· ,as} and p € <S, ordp(/?) =

}-Hendrik W. lenstra, Jr. and Yuri G. Zarhin 189 5.4. Proposition.. For e&ch a € {<*i, a-z, · · · , ag}, there exists a max-imal ideal p 6 S such that ord„(a) = 1/2. In particular, ordf(a) is not an integer.

Proof. Indeed, let us assume that there exists β G {α-^,α^,··· ,ots} such that ord„(/3) = 0 or l for all p e S. Since P(<) is irreducible over Q, the group Gal(L/Q) acts transitively. Since θΓΟσρ(/?) = οτάρ(σ~1β)) is an integer for all σ € Gal(Z-/Q), we obtain that ordp(a) is an integer for all

α e Ä. This contradicts Corollary 5.2.

5.5. Main Theorem. Assume i i a i there exist integers M and

mi)r r*2, · · · ,mg such that

Then all _{, · · · , mg are even.}

Proof. Assume that for some j the integer rrij is odd. According to the Proposition 5.4, there exists p € S such that ordp(o,·) = 1/2. It follows from the Corollary 5.3 that for all QH with i ji j we have that ordp(ai) is

an integer, Applying the homomorphism ordp to Y[ a™' = qM , we see that M = oidv(qM) = ordp( J J a"1' ) = some integer + (m, /2)

is not an integer, because m;· is odd. This leads us to a contradiction.

Therefore, m,j must be even for every j . This completes the proof. 5.5.1. Remark. The same arguments combined with the theorem of A. Weil prove the following generalization of the Theorem 5.5.

Assume that there exist a root of unity ε € L" and integers M and mi > "*2, · · · ,ηι such that

Then all m1 , m2, · · · , mg are even and M = Q3 mi)l^·

5.5.2. Remark. Assume that there exist a rooi of unity ε € L* ,

inte-gers M, mi, 77»a, · · · ,mg and a non-negative integer m such that (a) |mf | = m for all i, and

00 Π < · = * ? " ·

In this Situation, if </ is even, then m = m^ = M = 0 and e = 1.

9 s

M = ( \ J ?7^^)/2 = m(g/2) + 2_j(m' ~ m) / 2

» = 1 » = 1

is divisible by m So, if ??7 is not equal to zero, then we may assume, without loss of generahty, that m = l This imphes that

where all mt — ±1 and ε' is a icot of unity Recall the Remark 5 2 that for each p € «S, there exists exactly one root α = α, € {«1,03, )<*s} such that ordp (a) is not an integer Applymg the homomorphism ordp to Π<*["' = £'lM> w e obtam that

M = ordp(e'qM) = ordp(JJat m') = some integer+ mjordp(aj) = some integer ± oidp (o,)

and this is not an mtegei This is a contiadiction

5.5.3. Remark. Assume that theie exist a root of umty ε € L*, inte-gers mi, m2, , mg and a non-negative integer m such that

(a) |7nj| = m for all i, and

(b) ΠΧ"1' = ^ where <^ = a?Jq for all z

Under this Situation, if g is even, then m = mt = 0 and ε = l

Proof. We obtam Πα?™" = ε1Μ W l t n M = Y^ml One has only to apply the Remark 5 5 2

5.6. Corollary to Theorem 5.5. Assume that there exist mtegers M and mi,ni,m2,n2, ,mg,ng such that

Π «r·(«/«.)"· = «"

Then all mj — «i, ?n2 — n2, , mg — ng are even mtegers In particular, if all nt and m, are equal either to 0 or to l, tAen nt = m, for all ι

5.7. Theorem. TJbe set {ΛΙ, α2, , aä} consists of multiphcatively independent elements In particular, the rank of Γ js greater or equal than g, i e , it is equal either to g 01 to g -f l

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin 191 is a root of unity. Applying the Remark 5.5.1, we obtain that all mi/d are even integers. But this contradicts the definition of d.

5.8 Theorem. Assume that g is even. Then the sei {αϊ, »2, · · · , <xg, q} consists of multiplicatively independent elements, i.e., Γ is a free commu-tative group of rank g+l.

Proof of Theorem 5.8. We will use notation of Remark 3.5 and

Theorem 3.6. It follows from Theorem 3.6 combined with Remark 5.5.3 that Δ = {0}. In particular, u is an embedding. So u(Z3) = Γ' is the group generated by all a' . Since ω is an embedding, we obtain that Γ' is a commutative free group of rank g. On the other band, the rank of Γ is equal to 1 + the rank of Γ (3.5). Hence the rank of Γ is equal to 1 + g.

6 Skew-symmetric multiliiiear forms on the Täte

mod-ule

6.1. Theorem. Lei X be α α simple g-dimensional Abehan variety over K. Lei us assume that Slpx = {0, 1/2, 1} and lengthx(l/2) = 2. Then for each positive integer m and skew- Symmetrie '2m-hnear form

the following condition holds true: If

for all χι, · · · , z2 m € Vt(X), then E can be presented äs a linear combina-tion ofexterior products of Riemann forms

UC' : Vt(X) x Vt(X) -* Qi.

6.2. Remark. In view of Täte's theorem 3.4 (c), it suffices to check that E can be presented äs a linear combination of exterior products of skew-symmetric bilinear forms φ : Vi(X) χ Vt(X) -+ Qi such that

V?(Fr x, Fr«/) = q<p(x, y) for all x,y € Vt(X).

this identification, Hc> becomes the cohomology class of any divisor whose linear equivalence class corresponds to £'). This proves the algebraicity of Galois-invariant cohomology classes for X satisfying the assumption of the Theorem 6.1. This means that the Täte conjecture [Tl] holds true for such X. We refer the reader to [Z2] for the details.

6.4. Elementary Lemma. Let r be a positive integer, V a 2r-dimensional vector space over a commutative field k of characteristic zero, φ : V x V — + k' a, skew-symmetric non-degenerate bilinear form, F : V — »· V an invertible semi-simple linear operator in V such that

<i>(Fx,Fy) = q<l}(x,y) for all x,y€V.

Here q is a non-zero element of k which is not a root of unity. Suppose that all the eigenvalues of F have multiplicity one. Let us present the spectrum of F in the form

oii, o>2, · · · , α r ; q/ai,q/a2, ··· , q/ar.

Further assume that the set {αϊ, α^, · · · , ar; q} enjoys the following prop-erties: Suppose .that there exist integers M and πΐι,ηι,πίζ,ηζ, · · · ,mg,ng such that

Π «r· (*/*.·)"· = «*·

Under this Situation, if all n; and m; are equal either to 0 or to l, then m = rat for all i.

Then for each positive integer M, the following conditions hold true: Assume that a skew-symmetric 2M-linear form

E : V χ · · · x V -<· k enjoys the following property:

E(Fxi,··· ,Fx2M) = qM E(XI,··· ,ΧΖΜ) for all χι,··· ,x2M e V. Then E can be presented äs a linear combination of exterior products of skew-symmetric bilinear forms ψ : V x V — >· k such that

tp(Fx, Fy) = qip(x, y) for all x,y$V.

6.5. Proof of Theorem 6.1. Using the Corollary 5.6, one has only

to apply the Elementary Lemma 6.4. to r = g, V = Vt(X), k = Q* and

6.6. Remark. One may deduce from the Theorem 5.8 (see [Z2]) the

Hendrik W. Lenstra, Jr. and Yuri G. Zarhin 193 even. If g js odd, then one may easily deduce from the Theorem 3-6 and the Theorem 5.7 that the Täte conjecture holds true for Xn if one restricts oneself by co-dimension m < g.

6.7. Remark. One may easily generalize results of this paper to the case of simple Abelian varieties X enjoying the following properties:

(a) The endomorphism algebra End°X of X is a commutative field, (b) 1/2 € Slpjy and lengthx(l/2) = 2, and

(c) each slope c different from 1/2 can be presented äs a rational fraction with odd denominator.

In particular, the Täte conjecture holds true for such an Abelian variety X.

7 Sketch of a proof of Elementary Lemma

Wemay assumethat k isalgebraically closed. Let {ei, · · · ,er; e_i, · · · , e _r} be a basis for V consisting of eigenvectors with respect to F.

Fet = a, e„ Fe„ = (g/at)e_,

for all i £ {1,2,··· , r}.Let e*lt · · · , e*; e*_lt · · · , e*_r be the basis for the dual space V. Now it is clear that if a skew-symmetric bilinear form is multiplied by q under the action of F, then it is a linear combination of ex-terior products e* Ae!^. For each subset D C {1,2,· · · ,r;-l, —2, ··· , —r} consisting of 2M elements, one may define (up to sign) a skew-symmetric multilinear form ED on V which is the exterior product of all e*, (ί Ε £>). Clearly, all ED constitute an eigenbasis with respect to F of the space of all skew-symmetric 2M-linear forms on V. One may easily check that ED is multiplied by qM under the action of F if and only if there exists &{+} C {1,2,···,r} such that D = D{ +} U - A + } = #{+} U { - i | i € -£*{+}}. Clearly, if D enjoys this property then £** is equal (up to sign) to the exterior product of bilinear forms £?{(''-')} = e? A e l , . The rest is plain.

References

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[Z3] Yu. Zarhin, The Täte conjecture for powers of ordinary K3 surfaces over finite fields. in preparation.

Hendrik W. Lenstra, Jr. Department of Mathematics University of California Berkeley, CA 94720 USA e-mail hwl@math.berkeley.edu and Yuri G. Zarhin

Institute for Mathematical Problems in Biology Russian Academy of Sciences

Pushchino, Moscow Region 142292 Russia