Milne’s rationality conjecture and geometry after reduction

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Family 001 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

A geometric object defined by polynomial equations can be studied over the complex numbers or reduced modulo a prime. These two settings have different tools. Complex geometry uses topology and differential forms; geometry in characteristic $p$ uses arithmetic substitutes that retain information invisible to ordinary point counting.

Suppose both descriptions produce an intersection number. Must they produce the same rational number? For actual algebraic subvarieties, the geometry supplies a common answer. The difficulty is that topology predicts classes which are not already known to come from algebraic subvarieties, while reduction can create additional divisors that did not exist upstairs.

Family 001 concerns this gap for abelian varieties. Its manuscript, M1, claims Milne’s rationality conjecture: a rational Hodge class, after good reduction, has the same rational pairing with every complementary product of divisors in all the specified cohomology theories. A stronger consequence supplies an actual algebraic cycle on the reduction, using a separate theorem about complex multiplication. Keeping these two assertions distinct is essential to understanding the result.

2. Background and essential definitions

Abelian varieties and reduction

An abelian variety is a smooth projective algebraic variety with a commutative algebraic group law. An elliptic curve is the one-dimensional example. Over the complex numbers, an abelian variety of dimension $d$ looks analytically like $\mathbb C^d/\Lambda$, where $\Lambda$ is a lattice of rank $2d$, with an additional condition ensuring projectivity.

The coefficients of the variety here belong to $\overline{\mathbb Q}$, the field of algebraic numbers. They can be placed in a number field. Choosing a prime above a rational prime $p$ then permits reduction of an integral model. Good reduction means that the resulting special fiber is again a smooth abelian variety. Write the original variety as $A$ and its reduction, over an algebraic closure of the field with $p$ elements, as $A_0$.

Good reduction preserves much structure, but it need not preserve the collection of available algebraic subvarieties. In particular, $A_0$ may have divisors that do not lift to $A$. The question is therefore not settled simply by reducing intersections already performed in characteristic zero.

Cycles, Hodge classes, and realizations

An algebraic cycle of codimension $r$ is a finite formal combination of subvarieties of dimension $d-r$. Rational coefficients are allowed throughout this digest. A divisor has codimension one. Intersecting a codimension-$r$ cycle with $d-r$ divisors gives a zero-dimensional cycle, whose degree is a rational number.

Cohomology assigns a linear-algebraic class to a cycle. Over $\mathbb C$, the relevant group is $H^{2r}(A(\mathbb C),\mathbb Q)$. Complex differential forms split cohomology into types $(a,b)$, according to the numbers of holomorphic and antiholomorphic directions. A rational Hodge class in degree $2r$ is a rational cohomology class of type $(r,r)$. Algebraic cycles give such classes. The rational Hodge conjecture asks for the converse.

A realization is one of the compatible cohomological descriptions used in different arithmetic settings. For each prime $\ell\ne p$, étale cohomology uses the field $\mathbb Q_\ell$ of $\ell$-adic numbers. At $p$, crystalline cohomology plays the corresponding role. A $p$-adic completion of $\overline{\mathbb Q}$, denoted $\mathbb C_w$, is used for the comparison in M1.

The notation $\mathbb Q(r)$, and its arithmetic analogues, denotes a Tate twist. Here its role is normalization: it makes cycle classes and the trace on top-degree cohomology agree with geometric degree. Understanding the claim does not require constructing the twist explicitly.

For a divisor $D$, write $c_1(D)$ for its cohomology class. Multiplication of classes is the cup product, the cohomological counterpart of intersection. The trace $\operatorname{Tr}$ turns a top-degree class into its numerical intersection value. A Lefschetz class is a rational linear combination of products of divisor classes.

3. Development history and previous partial results

The broad background is the relationship between Hodge classes in characteristic zero and Tate classes in positive characteristic. A Tate class is, roughly, an appropriately normalized arithmetic cohomology class fixed by a power of Frobenius, the symmetry induced by raising coordinates to the field characteristic. These classes are candidates for algebraic cycle classes, just as Hodge classes are over the complex numbers.

Deligne’s theorem that Hodge classes on abelian varieties are absolute Hodge supplies strong compatibility between their realizations. Blasius’s work establishes the needed $p$-adic comparison. These results make it possible to transport the classes between theories, but compatibility alone does not say that pairing with a newly appearing divisor is rational. M1 reviews this distinction in its introduction; the original comparison result is Blasius’s A p-adic property of Hodge classes on abelian varieties.

Milne developed rationality formulations while studying reductions of Shimura varieties and the Langlands–Rapoport conjecture. His Rational Tate classes explains why one would like a common rational algebra inside the different Tate-class spaces. Such an algebra would support rational cohomological correspondences even before algebraic representatives were available.

One earlier positive case involves a CM abelian variety with simple ordinary reduction, together with its powers. In that setting the divisor classes on the reduction lift, removing the central obstruction. The general case must handle divisors that exist only after reduction.

A CM abelian variety has especially many commuting algebraic symmetries: for dimension $d$, its rational endomorphism algebra contains a commutative semisimple algebra of dimension $2d$. “CM” abbreviates complex multiplication. Earlier work of André reduces Hodge-class questions to special determinant-type classes associated with such symmetries. Milne’s 2025 account develops the lifting strategy and explains conditional criteria along this route.

M1 claims to complete the rationality argument through carefully controlled auxiliary lifts and a common rational normalization. The stronger algebraicity consequence uses the separately stated CM Hodge theorem. Neither the historical comparison theorems nor rationality by itself should be conflated with that additional input.

4. What the new manuscripts claim

One rational number in every realization

Fix a prime $p$ and a place $w$ above it. Let $A/\overline{\mathbb Q}$ have dimension $d$ and good reduction $A_0$, and let

$$\gamma\in H^{2r}(A(\mathbb C),\mathbb Q(r)),\qquad 0\leq r\leq d,$$

be a rational Hodge class. Write $\gamma_{0,\ell}$ and $\gamma_{0,p}$ for its specialized realizations. Given any Cartier divisors $D_1,\ldots,D_{d-r}$ on $A_0$, M1’s Theorem 1.1 states that one number $q\in\mathbb Q$ satisfies

$$\operatorname{Tr}_{\ell}\!\left( \gamma_{0,\ell}\prod_{j=1}^{d-r}c_{1,\ell}(D_j) \right)=q\qquad(\ell\ne p),$$

and

$$\operatorname{Tr}_{p}\!\left( \gamma_{0,p}\prod_{j=1}^{d-r}c_{1,\mathrm{cris}}(D_j) \right)=q.$$

The first equalities take place in $\mathbb Q_\ell$; the crystalline equality is interpreted after extending coefficients to $\mathbb C_w$. They identify the images of the same rational number in these different fields. When $r=d$, the empty divisor product means one.

The theorem includes $p=2$, arbitrary dimension, and divisors acquired only on reduction. It compares realizations at the fixed reduction place. It does not say that reductions at different places produce the same number.

A common class, then an algebraic representative

Using Milne’s equivalence between CM rationality and a good theory of rational Tate classes, M1’s Corollary 6.3 places every specialized Hodge class in one fixed such rational theory. The cohomological class is unique within that theory and has all the required realizations.

Corollary 6.4 adds the CM Hodge theorem from family 032. With this input, it claims one rational algebraic cycle $z_\gamma$ on $A_0$ whose cycle class equals $\gamma_{0,\ell}$ for every $\ell\ne p$, and equals $\gamma_{0,p}$ in the crystalline comparison. Pairing with any complementary algebraic cycle then gives its rational intersection number.

The paper explicitly separates this consequence from Theorem 1.1: its rationality proof does not use the CM Hodge theorem.

5. Why the advance matters

Importance: 4/5, provisional. The result addresses a structural obstacle in moving between complex geometry and arithmetic geometry. It would supply a common rational meaning to pairings computed in coefficient fields that otherwise look unrelated, while accommodating geometric classes that appear only after reduction.

For a simple analogy, several measurement systems may each report a compatible value without demonstrating that the value comes from a single rational geometric quantity. Rationality identifies that shared quantity. The algebraicity consequence goes further by supplying a geometric representative whose intersection degree explains it.

The distinction has consequences for motives, which organize cohomological information and correspondences in a common framework. Rational Tate classes provide cohomological correspondences; actual algebraic cycles provide stronger geometric information. The rating reflects the breadth of this connection and the arbitrary-dimensional scope, while recognizing that the strongest cycle statement uses a separate major theorem.

6. The main idea

Transport to a more symmetric setting

The first step moves the specialized class to an auxiliary CM abelian variety. An isogeny is a surjective group homomorphism with finite kernel; a quasi-isogeny allows rational inverses to such maps. Working up to isogeny permits the paper to use rational homomorphisms between special fibers and powers of varieties.

The transport must carry the original rational class in every realization. A collection of locally proportional classes would not suffice, since the proportionality factors might differ between coefficient fields. M1 normalizes these factors by ratios of rational divisor intersections, claiming one common rational scalar.

The maps at this stage concern reductions. The construction does not require an isogeny between the original characteristic-zero variety and its CM auxiliary.

Reduce to determinant-type classes

The CM symmetries decompose the relevant linear spaces into eigenspaces. Special Weil classes are built from their top alternating forms, analogous to determinant forms. A balanced signature means that conjugate eigenspaces occur with matching Hodge multiplicities. M1 reduces the general class to rational combinations of pullbacks of these balanced classes.

For such a class, the key claim is a dichotomy. Either all its complementary pairings with divisor products vanish, or the specialized class itself is one rational polynomial in divisor classes across all realizations. The first case supplies the rational answer zero. The second reduces the desired numerical comparison to ordinary intersections of divisors.

Lift the extra divisor with the right structure

In the second case, a suitable endomorphism produces a divisor with a nondegenerate pairing on the relevant eigenspaces. The construction chooses a filtration, a nested sequence of linear subspaces recording Hodge-type information, compatible with the divisor, the symmetry action, and the polarization.

Arithmetic deformation results are then used to realize this prescribed filtration on a second auxiliary abelian variety over a local field. This second auxiliary need not have complex multiplication and serves a different purpose from the first. Keeping its divisor and selected algebraic maps liftable is the point of the construction.

After passing to a complex realization of this lift, coefficient extraction in polynomial identities expresses the determinant-type class as a rational combination of divisor products. That rational identity specializes in every cohomology theory, and pullback returns it to the original reduction.

For the additional algebraicity statement, the CM Hodge theorem supplies an algebraic representative upstairs on the CM auxiliary. Specialization and rational pullback then give one representative on $A_0$. This explains precisely where the extra theorem enters.

7. Scope, limitations, and open questions

The theorem concerns abelian varieties over algebraic numbers with good reduction at a fixed place. It does not cover arbitrary varieties or bad reduction. Rational coefficients are essential; an integral cycle statement with controlled denominators is stronger than what is asserted here.

The rationality conclusion tests complementary products of divisors. Pairing with arbitrary complementary algebraic cycles is part of the stronger algebraicity consequence and uses its additional CM input.

The resulting cycle lies on the special fiber. M1 does not claim that it lifts to a cycle on the original variety, or that it proves the Hodge conjecture for arbitrary abelian varieties in characteristic zero. Nor is the representing algebraic cycle asserted to be unique modulo rational equivalence: uniqueness of its class in a fixed rational cohomological theory is a different assertion.

These boundaries leave natural further questions about integral refinements, behavior under bad reduction, and lifting algebraic representatives back to characteristic zero. They also indicate why a common intersection value, a common cohomology class, and a common algebraic cycle are three distinct levels of information.

8. Sources and further reading

M1’s Theorem 1.1 gives the rationality statement; Corollaries 6.3 and 6.4 distinguish rational Tate classes from algebraic specialization. Its introduction describes the history and the two different auxiliary lifts.

Milne’s Rational Tate classes and Abelian motives and Shimura varieties in nonzero characteristic supply the motivating framework. His notes on abelian varieties offer background on the geometric objects, divisors, and isogenies used here.

Source manuscripts

• M1. OpenAI, Milne’s rationality conjecture for abelian varieties (October 7, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.