On the Hodge, Tate, and standard conjectures for abelian varieties

To the memory of my teacher, John Tate, with regret that he did not see this support for his vision.

This is a report on two preprints released by OpenAI on October 6, 2026,

• Milne’s rationality conjecture for abelian varieties,

• The rational Hodge conjecture for CM abelian varieties.

and some of their consequences. In particular, I place the preprints in context.¹

The rationality conjecture

Statement of the rationality conjecture

Let $\mathbb{Q}^{\mathrm{al}}$ be an algebraic closure of $\mathbb{Q}$ and $w$ a place of $\mathbb{Q}^{\mathrm{al}}$ lying over $p$. The residue field $\mathbb{F}$ at $w$ is an algebraic closure of $\mathbb{F}_{p}$.

1.2. Let $A$ be an abelian variety of dimension $n$ over $\mathbb{Q}^{\mathrm{al}}$ with good reduction to an abelian variety $A_{0}$ over $\mathbb{F}_{p}$. Let $\gamma$ be an algebraic class on $A$ and $\delta$ a Lefschetz class on $A_{0}$ of complementary codimension,

Then

and we can ask whether this $\ell$-adic number is a rational number independent of $\ell$. In this case, the answer is obviously yes, because $\gamma$ reduces to an algebraic class $\gamma_{0}$ on $A_{0}$ and $\langle(\gamma_{\ell})_{0}\cdot\delta_{\ell}\rangle=\langle\gamma_{0}\cdot\delta\rangle\in\mathbb{Q}$.

The conjecture says that the same should be true if $\gamma$ is an absolute Hodge class.

1.4. What is an absolute Hodge class? Let $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$. Deligne [2] showed that there exists a $\mathbb{Q}$-subspace $\mathcal{B}^{r}(A)$ of $\prod_{\ell}H_{\mathrm{et}}^{2r}(A,\mathbb{Q}_{\ell}(r))$ mapping isomorphically onto $\mathcal{B}^{r}(A_{\mathbb{C}})\coloneqq H^{2r}(A_{\mathbb{C}},\mathbb{Q}(r))\cap H^{p,p}\subset\prod_{\ell}H_{\mathrm{et}}^{2r}(A,\mathbb{Q}_{\ell}(r))$ for every homomorphism $\mathbb{Q}^{\mathrm{al}}\to\mathbb{C}$,

The classes in $\mathcal{B}^{r}(A)$ are the absolute Hodge classes on $A$.

The condition for a class in $\prod_{\ell}H_{\mathrm{et}}^{2r}(A,\mathbb{Q}_{\ell}(r))$ to be absolutely Hodge is transcendental, and there is no rational $\gamma_{0}$. Whence the problem.

1.5. What was my motivation for the conjecture? Deligne’s theorem allows you to define a category of abelian motives over a field of characteristic zero with all the properties Grothendieck’s category would have if the Hodge conjecture were known. The conjecture allows you to extend this to mixed characteristic. There are many heuristic arguments proving things assuming a sufficiently good theory of motives. With the proof of the conjecture, and the existence of the categories of motives, some of these heuristic arguments become proofs.

Proof of the rationality conjecture.

1.7. There is an obvious approach to proving the conjecture. Let $A,\gamma,\delta$ be as in the statement of the conjecture. If we can lift $(A_{0},\gamma_{0},\delta)$ to $(A',\gamma',\delta')$ in characteristic zero with $\gamma'$ an absolute Hodge class and $\delta'$ a Lefschetz class,

then

because $\gamma'$ and $\delta'$ are both absolute Hodge classes. Sometimes it is possible to lift $(A_0,\gamma_0,\delta)$ and sometimes it isn’t. The problem is to show that it is possible to lift it often enough to force rationality.

1.8. First, two observations:

• an induction argument shows that it suffices to lift enough divisors (not Lefschetz classes);

• if $A$ is a polarized abelian variety, then

where $\dagger$ is the Rosati involution defined by the polarization, so lifting a divisor class is much the same as lifting an endomorphism.

1.9. Lifting theorems for abelian varieties have a long history. In the 1930s, Deuring showed that it is always possible to lift an elliptic curve and an endomorphism from characteristic $p$ to characteristic 0. In the proof of his part of the Honda-Tate theorem, Honda showed that you can always lift an abelian variety to a CM abelian variety in characteristic zero.³ The conjecture of Langlands and Rapoport predicts that the reduction of any point on a Shimura variety lifts to a CM-point…The most recent lifting result is Theorem 2.2.7 of Kisin and Zhou [5].

1.10. It has long been known that it suffices prove the rationality conjecture for Weil classes on CM abelian varieties. This means that we have the following.

• $A$ an abelian variety of CM-type over $\mathbb{Q}^{\mathrm{al}}$ (of dimension $g$).

• An action $\nu:E \to\operatorname{End}^{0}(A)$ of a CM field $E$ such that $\bigwedge_E^{2r} H^1(A_{\mathbb{C}},\mathbb{Q}) \subset H^{2r}(A_{\mathbb{C}},\mathbb{Q})$ consists of Hodge classes (i.e., classes of type $(r,r)$. Here $2r = \dim_E H^1(A_{\mathbb{C}},\mathbb{Q})$. Correspondingly, we get a $\mathbb{Q}$-vector space $W_E^r(A)$ of absolute Hodge classes on $A$, which we call Weil classes.

• A polarization $\lambda$ of $A$ whose Rosati involution involution acts as complex conjugation on $E$.

Given such a triple, we have to prove that $\langle\gamma_0 \cdot\delta\rangle\in\mathbb{Q}$ for all Weil classes $\gamma$ on $A$ and Lefschetz classes of codimension $g-r$ on $A_0$.

1.11. We seek a set $S$ of Lefschetz classes on $A_0$ such that

(a) $S$ is big enough:

(b) $S$ is not too big: it is possible to lift $(A_0,\nu_0,\lambda_0,\delta)$ to characteristic zero.

OpenAI gives the following answer. Let

then

works. Here $\xi$ is the divisor class of the polarization.

Given $S$, it only takes a page to prove that it satisfies (a) and a couple of pages to deduce (b) from the lifting theorem of Kisin and Zhou. See Madapusi [9].

1.12. There are precedents for this argument. For example, there is an old theorem ([19 2.7]) stating that, under some conditions, it is possible to lift all semisimple subalgebras of $\operatorname{End}^{0}_{E}(A_{0})$, which allows you to prove rationality for some Weil classes.

I was not surprised that my conjecture was proved — indeed, for the last twenty years I’ve been trying to persuade everyone that it was accessible. However, as were most mathematicians, I was surprised by the next result.

The Hodge conjecture for CM abelian varieties

2.2. Why is this so important? After all, we know the Hodge conjecture for many abelian varieties. Indeed, we know the Hodge conjecture for any variety for which the “known” algebraic classes exhaust the $\mathbb{Q}$-algebra of Hodge classes. For example, since we know the Hodge conjecture for the $(1,1)$ Hodge classes, we know the conjecture for any variety for which these classes generate the $\mathbb{Q}$-algebra of Hodge classes. Results of this nature, while interesting, tell us nothing about the veracity of the Hodge conjecture. For that, we have to prove the conjecture for a class of algebraic variety for which the known classes do not exhaust the Hodge classes. The abelian varieties of CM type are such a class.

2.3. The Hodge conjecture for CM is not known to imply the Hodge conjecture for all abelian varieties, but it would be strange if this wasn’t true. The CM-points are dense in the moduli space of abelian varieties. If the $\mathbb{Q}$-algebra of algebraic classes differed from the $\mathbb{Q}$-algebra of Hodge classes at some point, it would be very odd if this doesn’t persist over a CM point.

2.4. Finally, as we shall see, once the Hodge conjecture is known for CM abelian varieties, it is possible to prove other very interesting results.

We now sketch the proof of the conjecture. Note that this all takes place over $\mathbb{C}$, so much of it is complex analysis.

2.5 ([3, 4]). It suffices to prove the conjecture in codimension $2$.

Fix a (large) CM-field $K$ finite and Galois over $\mathbb{Q}$ with Galois group $G$. To give a CM abelian variety over $\mathbb{C}$ split by $K$, i.e., such that $\operatorname{End}^{0}(A) \otimes_{\mathbb{Q}} K$ is a product of matrix algebras over $K$, amounts to giving a finite $G$-set $S$ and a subset $S^{+}$ of $S$ such that $S = S^{+} \sqcup\iota S^{+}$. Here $\iota$ is complex conjugation. For example, if $A$ has complex multiplication by a CM-algebra $E$, then $S = \operatorname{Hom}(E,K)$ and $S^{+}$ is the CM type. If $A$ corresponds to $(S,S^{+})$, then it is possible to read off $H^{*}(A,\mathbb{Q})$ with its Hodge structure from $(S,S^{+})$, and even $H^{*}(A^{n},\mathbb{Q})$ because $A^{m}$ corresponds to the disjoint union of $n$ copies of $(S,S^{+})$. Let $[K:\mathbb{Q}] = 2m$. Let $H_{1},\ldots,H_{m}$ be the coordinate hyperplanes in $\mathbb{R}^{m}$, and take $S$ to be the set of connected regions of $\mathbb{R}^{m} \setminus\bigcup H_{i}$, and $S^{+}$ the subset on which $x_{1} > 0$. For a suitable action of $G$ on $\mathbb{R}^{m}$, hence on $(S,S^{+})$, there is a CM abelian variety $A(S,S^{+})$ over $\mathbb{C}$ and Hazama proves:

• (a) for every $n \ge0$, the $\mathbb{Q}$-algebra $\mathcal{B}^{*}(A^{n})$ of Hodge classes is generated by the classes of degree $\le2$;

• (b) every CM abelian variety split by $K$ is isogenous to an isogeny factor of $A^{n}$ for some $n$.

Thus, if the Hodge conjecture holds in codimension $\le2$ for abelian varieties split by $K$, then it holds in all codimensions for for $A(S,S^{+})^{n}$ by (a), and hence for every CM abelian variety split by $K$ by (b).

It suffices to prove the conjecture in dimension 2.

The standard conjecture of Lefschetz type is known for abelian varieties: the map on algebraic cycles

is an isomorphism. Here $L$ is the Lefschetz operator (cup-product with the class of an ample divisor).

Proof of the conjecture in dimension 2.

Thus, the problem comes down to embedding surfaces into $A$. Let $S$ be a surface. Suppose that the Hodge structure $H^{1}(A,\mathbb{Q})$ occurs in $H^{1}(S,\mathbb{Q})$. Then $A$ is an isogeny factor of the albanese of $S$, and so we get a map $f:S \to A$. For $S$ we take Picard surfaces. These are quotients of the 2-dimensional complex ball

by a discrete group attached to a three-dimensional hermitian space

over a CM-field $E$ such that $\psi$ has signature $(2,1)$ at one embedding of the totally real subfield of $E$ and signature $(3,0)$ at all other embeddings. From the Liu [7, 8], we find that the Hodge structure $H^{1}(A,\mathbb{Q})$ occurs in $H^{1}(S,\mathbb{Q})$ for $S$ a Picard surface. Now comes the hard part. We have to show that $f(S)$ is not numerically equivalent to zero. For this, we have to prove that $Z \cdot f(S) \ne0$ for some Hodge class $Z$ on $A$ of codimension 2. Here, we refer the reader to van Hoften and Sempliner [18], which, in addition to the original OpenAI manuscript draws on the manuscript Bergeron et al. [1], which proves the Hodge conjecture in small degrees for some arithmetic locally symmetric varieties.

The Hodge, Tate, and standard conjectures

The Hodge, Tate, and Grothendieck standard conjectures were stated in the 1950s and 1960s, and have haunted mathematicians ever since. Except for divisors on abelian varieties (Tate, Zarhin, Faltings), little progress has been made in proving the conjectures. Thus, the recent proofs for abelian fourfolds have attracted considerable attention.

Conjecture	Abelian fourfolds	General abelian varieties
Hodge 1950	Markman 20254	CM abelian varieties 2026 (OpenAI)
Tate 1963	Broe 2026	Divisors on AVs over finite fields 1966 (Tate)
		All abelian varieties over finite fields 1999* (Milne)
Grothendieck 1968	Ancona 2021	All abelian varieties 2002* (Milne)

Now we have the statements at right. The asterisk means that the results were conditional until 2026. Since the proofs of the asterisked results have suddenly become much more interesting (because unconditional), I shall explain them. Both are applications of the theory of tannakian categories.⁵

The Tate conjecture for abelian varieties over finite fields

For a smooth projective variety $X$ over $\mathbb{F}_q$, the full Tate conjecture says that

where $c(X) \ne0$ and $\rho_r$ is the is the rank of the group of numerical equivalence classes of algebraic cycles of codimension $r$. Let $H^i(X_{W\mathrm{et}},\mathbb{Z}(r))$ denote the Weil-étale cohomology of complex of sheaves defined by Bloch’s higher Chow groups, let $e\colon H^{2r}(X_{W\mathrm{et}},\mathbb{Z}(r)) \to H^{2r+1}(X_{W\mathrm{et}},\mathbb{Z}(r))$ denote cup-product with the canonical element of $H^1(k_{W\mathrm{et}},\mathbb{Z})$, and let

The conjecture of Lichtenbaum and Milne asserts that

where $\chi(X,\mathcal{O}_X,r)=\sum_{i\le r,j}(-1)^{i+j}(r-i)\dim H^j(X,\Omega^i_{X/k})$. This is now a theorem when $X$ is an abelian variety. See Milne [15], Theorem 11 and Remark 12.

In addition to Theorem 2.1, the proof of Theorem 3.1 relies on the magical properties of Lefschetz classes on abelian varieties and a serendipitous equality.

Lefschetz classes

Let $k$ be an algebraically closed field. The Lefschetz classes on abelian varieties have remarkable properties. For example, the Lefschetz standard conjecture holds for them

For any morphism $f:A\to B$ of abelian varieties, $f_*$ maps Lefschetz classes to Lefschetz classes.⁶ Numerical and homological equivalence agree on Lefschetz classes for every Weil cohomology theory. Using Lefschetz classes and abelian varieties, it is possible to define a category $\operatorname{LMot}(k)$ of “Lefschetz motives”. This is a canonically polarized tannakian category over $\mathbb{Q}$ with the expected fibre functors, and it is possible to describe its fundamental group.

A serendipitous equality

Let $\mathbb{Q}^{\mathrm{al}}$ denote the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$, let $w$ be a prime of $\mathbb{Q}^{\mathrm{al}}$ lying over $p$, and let $\mathbb{F}$ be the residue field at $w$. We have a diagram of tannakian categories at left, and the corresponding diagram of tannakian fundamental groups at right,

where

Category	Based on	Correspondences
$\operatorname{CM}(\mathbb{Q}^{\mathrm{al}})$	CM abelian varieties over $\mathbb{Q}^{\mathrm{al}}$	algebraic classes
$\operatorname{LCM}(\mathbb{Q}^{\mathrm{al}})$	CM abelian varieties over $\mathbb{Q}^{\mathrm{al}}$	Lefschetz classes
$\operatorname{Mot}(\mathbb{F})$	abelian varieties over $\mathbb{F}$	algebraic classes
$\operatorname{LMot}(\mathbb{F})$	abelian varieties over $\mathbb{F}$	Lefschetz classes.

The groups at right are affine group schemes of multiplicative type over $\mathbb{Q}$. We know nothing about $G$ except that it is contained in $S$ (because of 2.1) and in $L$ (because Lefschetz classes are algebraic).

Let $P$ be the affine group scheme attached to the category of motives over $\mathbb{F}$ when we assume the Tate and standard conjectures (cf. [10]). It is called the Weil-number pro-torus, and fits into the above diagram as follows,

For any abelian variety $A$, there are natural actions of both $G$ and $P$ on the cohomology of $A$. The first fixes the algebraic classes and the second the Tate classes, so

Looking at the diagram, we see that we could prove that $P=G$ if $S$ and $L$ intersect in $P$. A priori, there is no reason to believe this should be true, but serendipitously⁷ it is,

This completes the sketch of the proof of Theorem 3.1.

This section summarizes parts of Milne [11].

The Hodge standard conjecture for abelian varieties

3.6. Let $A$ be a polarized abelian variety of dimension $g$. As the Lefschetz standard conjecture holds for abelian varieties, we have an isomorphism

of spaces of algebraic classes. Here $L$, the Lefschetz operator, is cup-product with the divisor class $\xi$ of the polarization. Let

The Hodge standard conjecture says that the pairing

is positive definite.

3.7. By a specialization argument, it suffices to prove the conjecture over finite fields.

3.8. The Hodge standard conjecture has a tannakian interpretation. It holds for a collection of algebraic varieties over a field $k$ if and only if the category of motives $\operatorname{Mot}(k)$ generated by the varieties admits a polarization for which the geometric pairings are positive.

3.9. Roughly speaking, a polarization on a tannakian category is a compatible family of positive hermitian forms on the objects. There is an elaborate, and deep, theory of polarizations on tannakian categories. For example, if $G$ is an algebraic group over a field, then the (tannakian) category of finite-dimensional representations of $G$ admits a polarization if and only if $G$ is an inner form of a compact group. See Chapter VI of Milne [16].

3.10. Hodge theory shows that the Hodge standard conjecture holds in characteristic zero. Therefore $\operatorname{CM}(\mathbb{Q}^{\mathrm{al}})$ admits a polarization for which the geometric pairings are positive. Using the diagram

from 3.4, it is possible to show that this polarization passes to the quotient tannakian category $\operatorname{Mot}(\mathbb{F})$. Therefore, the Hodge standard conjecture holds for all abelian varieties over $\mathbb{F}$. ⁸

This section summarizes parts of Milne [13].

Applications

Let $\operatorname{Mot}(\mathbb{Q}^{\mathrm{al}})$ be the category of abelian motives over $\mathbb{Q}^{\mathrm{al}}$ for absolute Hodge classes, and let $\operatorname{Mot}'(\mathbb{Q}^{\mathrm{al}})$ be the subcategory of those with good reduction (suitably defined). With the proof of the rationality conjecture, it is possible to show that there exists a tannakian category of motives $\operatorname{Mot}(\mathbb{F})$ and reduction functor $\operatorname{Mot}'(\mathbb{Q}^{\mathrm{al}}) \to\operatorname{Mot}(\mathbb{F})$ having the properties that Grothendieck's categories would have if the Hodge, Tate, and standard conjectures were known ([14]). With the proof of the Hodge conjecture for CM abelian varieties, we even know that $\operatorname{Mot}(\mathbb{F})$ is defined by algebraic cycles and $\operatorname{Mot}'(\mathbb{Q}^{\mathrm{al}})$ by almost-algebraic cycles. This allows us to turn some heuristic arguments into proofs.

3.11. Let $A$ be an abelian variety over a number field $k$, and let $G$ be the Mumford-Tate group of $A_{\mathbb{C}}$. It follows from Deligne's theorem on absolute Hodge classes that, after a finite extension of $k$, there are well-defined Frobenius conjugacy classes in $G(\mathbb{Q}_{\ell})$, $\ell$ a prime. It is an immediate consequence of the above theory that these are rational, i.e., arise from a single conjugacy class in $G(\mathbb{Q})$. See Serre [17], 12.6.

3.12. We leave it as a challenge to the reader to prove the conjecture of Langlands and Rapoport [6] following the heuristic argument they use to derive it.

References