The numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension

Abstract

We prove the numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, and also in some related cases.

1 Introduction

Recently, Ancona proved the numerical Hodge standard conjecture for abelian fourfolds [1]. In fact, he proved a general theorem for certain rank 2 pure motives in mixed characteristic [1, 8.1], and showed that this general theorem is applicable to abelian fourfolds over finite fields. In this paper, we point out some other cases where the general theorem can be applied. (See [1, A.9] for another example.)

In Ancona’s work, the main cases are

• (1) an absolutely simple abelian fourfold, and

• (2) the product of a simple abelian threefold and an elliptic curve.

(See [1, A.8] for some discussion.) In this paper, we generalize the second case as follows:

Theorem 1.1

Let A be a simple abelian variety over a field k. Assume either

• $\dim A$ is prime, or

• some specialization of A to a finite field is absolutely simple and almost ordinary.¹

Let E be an elliptic curve. The numerical Hodge standard conjecture holds for $A\times A$ and $A\times E$.

As in [1, 1.6], this combined with [2] implies

Corollary 1.2

The numerical equivalence on $A\times A, A\times E$ coincides with the $\ell$-adic homological equivalence on $A\times A, A\times E$ for infinitely many $\ell$.

Remark 1.3

Assume that k is a finite field. In both cases, the Tate conjecture for A is known, and, all algebraic classes come from the intersection of divisors [6, 10, 11]. Therefore, the numerical Hodge standard conjecture holds for A itself and the numerical equivalence on A coincides with the $\ell$-adic homological equivalence on A for every $\ell$ [8, 3.7], [1, Section 5]. However, if $\dim A\ge 3$, $A^2$ and $A \times E$ may have an exotic Tate class in the middle degree, i.e., a class that cannot be written using Tate classes of degree 2, and the Tate conjecture is not known except the case of the product of a simple threefold and an ordinary elliptic curve [9].²

We prove a slightly more general statement. Let A be an absolutely simple abelian variety of dimension g over a finite field $\textbf{F}_q$. Let $\alpha _1, \dots , \alpha _{2g}$ be the Frobenius eigenvalues of the first cohomology so that $\overline{\alpha }_i=\alpha _{i+g}$. Set $\beta _{i}:=q/ \alpha _i^2, 1\le i\le g$. Let $\Gamma '$ denote the multiplicative group generated by $\beta _i, 1\le i \le g$ inside $\textbf{Q}(\alpha _1, \dots , \alpha _{2\,g})$. The rank of $\Gamma '$ has been studied, e.g., [12, 13]. Following [3], we call it the angle rank of A. The angle rank is always less than or equal to g. If the angle rank is g or A is a supersingular elliptic curve,³ all the Tate classes on $A^n$ for a positive integer n can be written using Tate classes of degree 2, and the Tate conjecture holds for $A^n$. (This is the case for all abelian surfaces and elliptic curves.) The converse is also true. Recall that such a Tate class is called Lefschetz and a Tate class is exotic if it is orthogonal to all Lefschetz classes. We are interested in the easiest case with possible exotic Tate classes:

Theorem 1.4

If the angle rank of A is $g-1$ or g and $\dim A >1$ is odd, then the numerical Hodge standard conjecture holds for $A\times A$ and $A\times E$, where E is an elliptic curve.

Remark 1.5

Tankeev [10, p.332] showed that the angle rank is $g-1$ or g if $g=\dim A$ is an odd prime⁴ and the endomorphism ring is commutative. (If the endomorphism ring is not commutative, then the Frobenius generates an imaginary quadratic field and A, $A\times E$ are neat.) Lenstra and Zarhin [6] showed that that if A is almost ordinary, the angle rank is $g-1$ when g is odd and g when g is even; see [6, 6.7] (and [3, 1.5]) for a slightly more general case.

Remark 1.6

If the angle rank of A is $g-1$ and E is ordinary, then $A\times E$ has no exotic Tate classes; see Corollary 3.4. The same holds trivially if the angle rank of A is g and E is supersingular.

Now, Theorem 1.4 clearly implies Theorem 1.1, so we will focus on Theorem 1.4. We shall show that $A\times A$ may have an exotic Tate class only in the middle degree, and they form a 2-dimensional space so that we can apply [1, 8.1]. The case of $A\times E$ is similar.

Finally, let us mention that we study the Tate conjecture and the Hodge standard conjecture for self-products of K3 surfaces in [4] and its sequel.

2 A lemma on the Hodge standard conjecture

Let A be an abelian variety of dimension g over a field with a polarization L. Let $\mathcal Z^{n}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ denote the space of algebraic cycles of codimension n modulo numerical equivalence. Recall that the Lefschetz standard conjecture holds for A and L, and we define the primitive part $\mathcal Z^{n, {{\,\textrm{prim}\,}}}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ of $\mathcal Z^{n}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ as follows

$$\begin{aligned} \mathcal Z^{n, {{\,\textrm{prim}\,}}}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}:=\left\{ \alpha \in \mathcal Z^{n}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}} \mid L^{g-2n+1}\cdot \alpha = 0 \right\} , \end{aligned}$$

assuming $n\le g/2$; see [1, 3.8, 3.9] for another description.

Conjecture 2.1

(The numerical Hodge standard conjecture) For a nonnegative integer $n\le g/2$, the pairing

$$\begin{aligned} \langle -, -\rangle _n :\mathcal Z^{n, {{\,\textrm{prim}\,}}}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}} \times \mathcal Z^{n, {{\,\textrm{prim}\,}}}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}} \rightarrow \textbf{Q}; (\alpha , \beta ) \mapsto (-1)^{n} \alpha \cdot \beta \cdot L^{g-2n} \end{aligned}$$

is positive definite.

Let us say that a class in $\mathcal Z^n_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ is exotic if it is orthogonal to any intersection of divisors.

Lemma 2.2

If A has exotic classes only in the middle degree, then the numerical Hodge standard conjecture is independent of L.

Proof

Let $\mathcal L^{n}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ denote the subspace of $\mathcal Z^n_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ spanned by the intersections of divisors. The numerical Hodge standard conjecture is known for $\mathcal L^{n}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ by specializing to a finite field [8, 3.7], [1, Section 5]. In particular, only the middle degree is a problem. There is an orthogonal decomposition with respect to $\langle -,- \rangle _n$

$$\begin{aligned} \mathcal Z^{g/2}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}=\mathcal L^{g/2}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}\oplus \mathcal E^{g/2}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}, \end{aligned}$$

where $\mathcal E^{g/2}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$ is the space of exotic classes. This decomposition is independent of L. The numerical Hodge conjecture holds for A and L if and only if $\langle -, - \rangle _{g/2}$ is positive definite on $\mathcal E^{g/2}_{{{\,\textrm{num}\,}}}(A)_{\textbf{Q}}$, and the latter statement is independent of L. $\square$

3 Exotic Tate classes

We assume that A is an absolutely simple abelian variety of dimension $g >2$ defined over a finite field $\textbf{F}_q$ of characteristic p. We use the notation $\alpha _i, \beta _i$ as in the introduction. Suppose first that the angle rank of A is $g-1$.

Lemma 3.1

The endomorphism ring ${{\,\textrm{End}\,}}(A)\otimes \textbf{Q}$ is a number filed of degree 2g generated by Frobenius.

Proof

By [11, Theorem 2 (c)], it suffices to show that the Frobenius eigenvalues $\alpha _1, \dots , \alpha _{2g}$ are all distinct. As A is simple, by [11, Theorem 2 (e)], the characteristic polynomial of Frobenius has the form of $P(t)^e$ for some irreducible polynomial P(t) in $\mathbb {Q}[t]$ and some integer e. If $e >1$, then the angle rank is less than or equal to g/2 and this cannot happen as $g-1 > g /2$ for $g >2$. $\square$

Lemma 3.2

Assume that the angle rank is $g-1$. After replacing $\alpha _i$ by $\alpha _{i+g}$ if necessary, the only relation among $\beta _1, \dots \beta _g$ has the form of

$$\begin{aligned} (\beta _1 \ldots \beta _g)^N = 1 \end{aligned}$$

for some N.

Proof

Let $\beta _1^{\textbf{Z}} \ldots \beta _g^{\textbf{Z}}$ denote the free abelian group of rank g with the basis $\beta _1, \dots , \beta _g$, and let $\Gamma _1$ be the kernel of the natural map

$$\begin{aligned} \beta _1^{\textbf{Z}} \ldots \beta _g^{\textbf{Z}} \rightarrow \textbf{Q}(\alpha _1, \dots , \alpha _{2g}){\setminus }\{0\}. \end{aligned}$$

By assumption, $\Gamma _1$ is a free abelian group of rank 1. So, the Galois group of $\textbf{Q}(\alpha _1, \dots , \alpha _{2g})$ acts naturally on $\Gamma _1$ via $\{\pm 1\}\subset {{\,\textrm{Aut}\,}}(\Gamma _1)$. Note that the Galois group acts on $\{\{\beta _1^{\pm 1}\}, \ldots , \{\beta _g^{\pm 1}\}\}$ by permutation and the action is transitive, and the Galois group contains the complex conjugation so that $\overline{\beta }_i=\beta _i^{-1}$. This implies that a generator of $\Gamma _1$ has the form of

$$\begin{aligned} \beta _1^{\pm N} \beta _2^{\pm N} \ldots \beta _g^{\pm N} \end{aligned}$$

for some N. $\square$

Corollary 3.3

Let $\ell$ be a prime different from p. If g is odd (resp. even), any exotic $\ell$-adic Tate class of $A\times A$, $A\times E$ (resp. A) is in the middle degree. If an exotic Tate class exists, then the space of exotic Tate classes is two-dimensional for $A\times A$ (resp. A) and four-dimensional for $A\times E$.

Proof

First note that, after extending the scalar, an exotic Tate class of degree 2d is the sum of products of 2d Frobenius eigenclasses of degree 1. Such a product expression gives rise to a nontrivial multiplicative relation

$$\begin{aligned} (\beta _{i_1}\ldots \beta _{i_{2d}})^N = 1 \end{aligned}$$

by the definition of exotic Tate classes.

Let us consider the case of A and $A\times A$. An exotic $\ell$-adic Tate class would give rise to a nontrivial multiplicative relation among $\beta _i$. By Lemma 3.2, the possible nontrivial relations are of the form of

$$\begin{aligned} (\beta _{1} \ldots \beta _g)^N= 1, \quad (\beta _{g+1} \ldots \beta _{2g})^N= 1. \end{aligned}$$

Moreover, note that Tate classes live in even degrees. Therefore, if g is odd, then any exotic Tate class of $A\times A$ is of degree 2g and N is even. Similarly, if g is even, any exotic Tate class of A is of degree g. The exotic Tate classes clearly span a 2-dimensional subspace.

The case of $A\times E$ is similar. Let $\alpha _{2\,g+1}, \alpha _{2+2\,g}$ denote the Frobenius eigenvalues of E with the associated $\beta _{2g+1}, \beta _{2g+2}$ as in the introduction. Again by Lemma 3.2, an exotic $\ell$-adic Tate class of $A\times E$ would give rise to a multiplicative relation

$$\begin{aligned} (\beta _{1} \ldots \beta _g \cdot \beta _{2g+i})^N = 1 \end{aligned}$$

for $i=1$ or 2 and the class is of degree $g+1$; recall that we are assuming g is odd. If E is ordinary, then neither of $\beta _{2g+1}, \beta _{2g+2}$ is a root of unity, so this cannot happen as $\beta _{1} \ldots \beta _g$ and $\beta _{g+1}\ldots \beta _{2\,g}$ are roots of unity by Lemma 3.2. If E is supersingular, $\beta _{2g+1}, \beta _{2g+2}$ are both roots of unity and the exotic Tate classes span a 4-dimensional subspace in the middle degree. $\square$

The following is also shown in the above proof.

Corollary 3.4

If g is odd and E is ordinary, then $A\times E$ has no exotic Tate classes.

A similar argument shows the following:

Lemma 3.5

Suppose the angle rank of A equals g and g is odd, then any exotic $\ell$-adic Tate class of $A\times E$ is in the middle degree. If an exotic Tate class exists, then E is ordinary and the space of exotic Tate classes is two-dimensional.

Proof

By assumption, there is no nontrivial multiplicative relation among $\beta _1, \dots , \beta _g$. Using Lemma 3.2 and the notation of the proof of Corollary 3.3, we see that any exotic Tate class of $A\times E$ is of degree $g+1$ and gives rise to a nontrivial multiplicative relation

$$\begin{aligned} (\beta _1 \ldots \beta _g \cdot \beta _{2g+i})^N =1 \end{aligned}$$

for $i=1$ or 2. Moreover, E must be ordinary and neither of $\beta _{2g+1}, \beta _{2g+2}$ is a root of unity. In this case, the exotic Tate classes span a 2-dimensional subspace. $\square$

Next, we construct a motivic counterpart of possible exotic Tate classes using complex multiplication. Let us first recall some facts about the motive of A [1, Section 4, Section 6]. Set $B:={{\,\textrm{End}\,}}(A)\otimes \textbf{Q}$ and write $L\subset \overline{\textbf{Q}}$ for the Galois closure of B with $\Sigma :={{\,\textrm{Hom}\,}}(B, L)$. As in [1, 6.6], there is the following decomposition in the category of Chow motives⁵ with coefficients in L:

$$\begin{aligned}{}[H^1 (A)]=\bigoplus _{\sigma \in \Sigma } M_{\sigma }, \end{aligned}$$

where $[H^1 (A)]$ is defined using Chow–Künneth decomposition [1, 4.1 (1)], $M_{\sigma }$ is of rank one, and B acts on $M_{\sigma }$ via $\sigma$ under the map $B\otimes _{\mathbb {Q}} L\rightarrow End ([H^1 (A)])$ supplied by [1, 4.1 (1), (3)]. By the product structure, it induces [1, 4.1 (2), 6.7 (1)]

$$\begin{aligned}{}[H^g (A)]=\bigoplus _{I\subset \Sigma , \# I =g} M_I, \end{aligned}$$

where $M_I=\otimes _{i \in I} M_i$ and $M_I$ is of rank one. This further induces the following decomposition [1, 6.7 (2)], in the category of Chow motives with coefficients in $\textbf{Q}$,

$$\begin{aligned}{}[H^g (A)]=\bigoplus M_{[I]}, \end{aligned}$$

where [I] denotes the Galois orbit of I and $M_{[I]}$ is the direct sum of $M_?$ over the Galois orbit. As stated in [1, 6.7(2)], this decomposition is orthogonal as numerical motives with respect to $\langle -, -\rangle ^{\otimes g}_{1, \text {mot}}$ defined in [1, 3.6]; the pairing $\langle -, -\rangle _{1, \text {mot}}$ is, up to sign, induced by $(g-1)$-th power of some polarization. Similarly, $[H^{2g}(A\times A)]$ has such a decomposition and we have summands like

$$\begin{aligned} M_{I^2}:=M_I\otimes M_{I}, \quad M_{[I^2]}. \end{aligned}$$

Proposition 3.6

Assume that the angle rank is $g-1$.

• (1) If g is even, there exists at most one [I] such that the $\ell$-adic realization of $M_{[I]}$ is exotic. The numerical algebraic classes in $M_{[I]}$ is zero or two-dimensional.

• (2) If g is odd and E is supersingular, there exists at most one [I] such that the $\ell$-adic realization of $M_{[I]}\otimes H^1 (E)$ is exotic. The numerical algebraic classes in $M_{[I]}\otimes H^1 (E)$ is zero, two, or four-dimensional.

• (3) If g is odd, there exists at most one $[I^2]$ such that the $\ell$-adic realization of $M_{[I^2]}$ is exotic. The numerical algebraic classes in $M_{[I^2]}$ is zero or two-dimensional.

Proof

(1) and (3) follow from the description of exotic Tate classes and [1, 6.8]. The key claim here is that the relevant Galois orbit only has two elements, and it controls the dimension of numerical algebraic classes. (2) is similar by finding a decomposition $M_{[I]}\otimes H^1(E)=M_1 \oplus M_2$ into rank 2 motives by choosing a CM structure on $A\times E$ as in the proof of [1, 7.16]. $\square$

We call $M_{[I]}, M_{[I]}\otimes H^1(E), M_{[I^2]}$ exotic if it has a nonzero numerical algebraic class. If it is the case, their $\ell$-adic realizations are the only exotic Tate classes. By [1, 5.3] and Lemma 2.2, the numerical Hodge standard conjecture for $A, A\times A, A\times E$ reduces to the corresponding problem on $M_{I}, M_{[I]}\otimes H^1(E), M_{[I^2]}$ respectively, with respect to $\langle -, -\rangle ^{\otimes g}_{1, \text {mot}}, \langle -, -\rangle ^{\otimes g+1}_{1, \text {mot}}. \langle -, -\rangle ^{\otimes 2\,g}_{1, \text {mot}}$ for some polarization.

A similar construction makes sense for $A \times E$ if g is odd, the angle rank is g, and E is ordinary.

Finally, when g is odd and E is supersingular, an exotic $M_{[I]}\otimes H^1(E)$ has a decomposition into rank 2 motives

$$\begin{aligned} M_{[I]}\otimes H^1(E) =M_1 \oplus M_2 \end{aligned}$$

orthogonal with respect to $\langle -, -\rangle ^{\otimes g+1}_{1, \text {mot}}$. More precisely, the Galois action on [I] gives rise to an imaginary quadratic field F inside B and there is an embedding $F\hookrightarrow {{\,\textrm{End}\,}}(E_{\overline{\textbf{F}}_q})\otimes \textbf{Q}$ by exactly the same argument as in the proof of [1, 7.16]. The actions of F on $M_{[I]}$ and $H^1(E)$ induce the above decomposition.

4 Ancona’s theorem for rank 2 motives

To conclude the proof of Theorem 1.4, we recall Ancona’s theorem and then use CM liftings to apply it.

Let K be a p-adic field with the ring of integers $O_K$ with residue field k. Fix an embedding $\sigma :K\hookrightarrow \textbf{C}$. We shall use the language of relative Chow motives over $O_K$, equipped with base changes to $\textbf{C}$ via $\sigma$ and to k via the specialization. For a relative Chow motive M over $O_K$, we write $V_B$ for the Betti realization of $M_{\textbf{C}}$. Let $V_Z$ denote the space of numerical algebraic cycles in $M_k$, i.e., homomorphisms from $\mathbbm {1}$ modulo numerical equivalences. Both $V_B$ and $V_Z$ are $\textbf{Q}$-vector spaces. If M has a quadratic form

$$\begin{aligned} q:{{\,\textrm{Sym}\,}}^2 (M) \rightarrow \mathbbm {1}, \end{aligned}$$

then it induces ($\textbf{Q}$-valued) quadratic forms $q_B, q_Z$ on $V_B, V_Z$ respectively.

Theorem 4.1

(Ancona [1, 8.1]) Let M be a relative Chow motive over $O_K$ with a quadratic form q. Assume that

• $\dim _{\textbf{Q}} V_B=\dim _{\textbf{Q}} V_Z=2$, and

• $q_B:V_B\times V_B \rightarrow \textbf{Q}$ is a polarization of Hodge structures.

Then, $q_Z$ is positive definite.

Remark 4.2

It is easier to show that $q_Z$ is positive definite or negative definite. Ancona uses the p-adic Hodge theory to determine the sign.

Following the proof of [1, 3.18], we use CM liftings to prove Theorem 1.4. The idea is that, under the assumption, the exotic classes gives rise to a Chow motive of rank 2 and it lifts along a CM lifting. We apply Theorem 4.1 to such lifting.

Proof of Theorem 1.4

Let A be as in Theorem 1.4. Set $B:={{\,\textrm{End}\,}}(A_{\textbf{F}_q})\otimes \textbf{Q}$. After enlarging $\textbf{F}_q$, we can find a finite extension $O_K$ of $W(\textbf{F}_q)$ and an abelian scheme $\mathcal A$ over $O_K$ with $B\rightarrow {{\,\textrm{End}\,}}(\mathcal A)$ such that the reduction $\mathcal A_{\textbf{F}_q}$ is B-isogenous to A. We may replace A by $\mathcal A_{\textbf{F}_q}$ and assume that some polarization on $\mathcal A_{\textbf{F}_q}$ lifts to a polarization on $\mathcal A$.

We will freely use notation from Sect. 3. If $A^2$ has no exotic classes, there is nothing to prove. So, assume some $M_{[I^2]}$ is exotic. By [1, 5.3], it suffices to show the the pairing $\langle -, - \rangle ^{\otimes 2g}_{1, \text {mot}}$, for a polarization that lifts to $\mathcal A$, is positive definite on the exotic $M_{[I^2]}$. By the construction of $M_{[I^2]}$ and the pairing explained in Sect. 3, it lifts to a relative Chow motive with a quadratic form over $O_K$ (cf. [1, 4.1, 4.2] and references therein, and the proof of [1, 3.18]). By the definition of the exotic $M_{[I^2]}$, this lift satisfies the assumption of Theorem 4.1. So, $\langle -, - \rangle ^{\otimes 2\,g}_{1, \text {mot}}$ is positive definite on $M_{[I^2]}$.

The case of $A\times E$ is similar as in [1]. Let us consider the case E is supersingular. The decomposition

$$\begin{aligned} M_{[I]}\otimes H^1(E) =M_1 \oplus M_2 \end{aligned}$$

is constructed using the action of the imaginary quadratic field $F\subset B$ on E, and it may also lifts by taking a lift of E with the action of F. (See the end of Sect. 3 for the notation, and we again refer to the proof of [1, 7.16] for details.) $\square$

Remark 4.3

Consider the case $A\times A$. The Hodge type of the Betti realization of the lifts of the exotic classes have the form of (2a, 2b), (2b.2a) with $a+b=g$. In particular, it is never (g, g) and any exotic class cannot be lifted to an algebraic class of $\mathcal A_{\textbf{C}} \times \mathcal A_{\textbf{C}}$. Therefore, Ancona’s theorem is essential.