Milne’s rationality conjecture and algebraic specialization

The explanation

Take the elliptic curve y² = x³ − x, call it E, with its point at infinity O as zero. Form the surface E × E of all pairs of points. Inside it sit curves. One is the horizontal copy E × {O}. Another is the graph of doubling, all pairs (P, 2P). They meet where 2P = O. That happens at four points: O, (0, 0), (1, 0) and (−1, 0). So their intersection number is 4. Counts like this are whole numbers, because they count actual points.

Cohomology turns such shapes into algebra. Each curve gives a class, and intersections become products of classes. There are several cohomology theories. For each prime ℓ, ℓ-adic cohomology works with ℓ-adic numbers. Over a field of characteristic p, crystalline cohomology does the job at p itself. For actual curves and surfaces, every theory computes the same intersection numbers.

An abelian variety is a higher-dimensional version of E, a smooth shape whose points can be added. Some cohomology classes over the complex numbers look as if they come from subvarieties. These are Hodge classes. The Hodge conjecture predicts that every rational Hodge class comes from an algebraic cycle, a combination of subvarieties. For divisors, a 1924 theorem of Lefschetz settles this. In general it is open, even for abelian varieties.

Now reduce the variety modulo a prime p. Good reduction means the equations stay smooth. The reduced variety can gain divisors that do not lift back. For a typical E, the reduction of E × E gains the graph of Frobenius. That is the map that raises coordinates to a power of p. It does not come from any curve on the original surface.

Take a Hodge class and carry it down to the reduction. Pair it with a product of divisors there, which may include such new ones. Deligne proved in 1982 that Hodge classes on abelian varieties are absolute Hodge. That gives matching versions of the class in every theory. It does not show that its pairing with the new divisors is a rational number that all theories share.

Milne's rationality conjecture asks exactly that. It grew out of his work on Shimura varieties around 2000, and appears as a pairing statement in his 2009 paper. The paper claims it for every abelian variety over the algebraic numbers with good reduction at p. Every residue characteristic is included, even 2Source 1. Earlier, only special cases were known, such as CM varieties with simple ordinary reduction. A CM abelian variety is one with an unusually large set of symmetries.

The claim has consequences. Milne showed that the CM case is equivalent to a "good theory of rational Tate classes" on certain varieties over finite fields. The paper extends that theory to specializations of all Hodge classes. A companion paper claims the Hodge conjecture for CM abelian varieties. With it, this paper claims each reduced Hodge class comes from one rational algebraic cycleSource 2. The main theorem does not use that companionSource 3.

The obstacle is getting one rational number in every theory at once. Separate local arguments can give a different multiple in each theory. The proof first moves the class to a CM abelian variety. It then breaks CM classes into balanced Weil classes, a standard family of test cases. For these it claims a dichotomy. Either every pairing is zero, or the class is a rational polynomial in divisors in every theory simultaneouslySource 4. The second case lifts a divisor to an auxiliary variety and reads off a rational identity there.

The proof uses outside results, including recent work of Kisin and Zhou and of Gleason, Lim and Xu.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not claim that Hodge classes on the original variety over the algebraic numbers come from algebraic cycles.