1 · The essentials
This family of results studies what happens to certain geometric data when the arithmetic setting in which they are viewed changes. The objects involved are abelian varieties defined over , under a “good reduction” assumption that allows this passage. The same data can then be examined using several mathematical tools. The manuscript shows that certain measurements made with these tools all give the same rational number, meaning a ratio of integers. This also covers the case where the new setting has characteristic . It additionally claims that the transferred data can be represented by one geometric object with rational coefficients, valid for all the tools under consideration. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine an object measured by several instruments using different readout systems. The first result says that the particular measurements under consideration agree exactly; the second says that one object can account for all those readings. This image does not describe the actual geometric objects or the good reduction assumption.
Why it matters
Rationality and agreement between theories are separate requirements: the values must be rational numbers, and those numbers must coincide. Representation by a single algebraic cycle adds geometric compatibility to this numerical agreement. The text presents these conclusions as resolving Milne’s rationality conjecture in the stated setting.
2 · Going further
The problem concerns an abelian variety defined over with good reduction in residue characteristic . A rational Hodge class is specialized and then paired with a product of divisor classes of complementary degree on the reduced variety. Milne’s rationality conjecture, as described here, asks that this pairing be the same rational number in every prime-to- realization and in crystalline cohomology.
The manuscript shows this assertion, including for . The abstract does not explain the method of the rationality argument. For the additional geometric conclusion, however, it identifies the use of the Hodge theorem for abelian varieties with complex multiplication: every such specialized Hodge class is represented by a single algebraic cycle with rational coefficients in all the realizations under consideration.
The repository summary attributes this simultaneous representation to the combination with result 032, for every residue characteristic . The main abstract also announces this conclusion, citing the Hodge theorem for abelian varieties with complex multiplication. Since no text of result 032 is supplied, the division of the arguments between the manuscripts cannot be specified further.
Useful notions
- Rational numbers and algebraic closure
- Abelian varieties
- Good reduction and residue characteristic
- Hodge classes and cohomology
- Divisors and algebraic cycles
What was known before / what is new
Before
The text identifies the rationality problem as Milne’s conjecture. It names the Hodge theorem for abelian varieties with complex multiplication as a tool for the conclusion about cycles. It supplies neither a detailed history nor an account of earlier partial results.
New
The manuscript shows rationality and independence of the cohomology theory for the stated pairings, including in residue characteristic . It also announces representation of each specialized class by a single rational algebraic cycle, simultaneously in the prime-to- and crystalline realizations. The repository summary connects this second conclusion with result 032.
Limits and assumptions
The supplied setting is abelian varieties over with good reduction. The numerical assertion concerns specialized rational Hodge classes paired with products of divisor classes of complementary degree, not arbitrary classes. Simultaneous representation concerns prime-to- realizations and crystalline cohomology, for every residue characteristic . The main result is not formalized in Lean and has not been formally verified. The supplied texts do not detail the rationality argument, provide the contents of result 032, or specify remaining open questions.
Mini glossary
- Good reduction —
- The assumption allowing passage to a reduced variety in residue characteristic in the setting of the result.
- Rational Hodge class —
- A type of cohomological data with rational coefficients whose specialization is studied in the manuscript.
- Specialization —
- The passage from a class on the original variety to a class on its reduction.
- Pairing —
- An operation that here assigns a number to a specialized class and a product of divisor classes of complementary degree.
- Realization —
- An expression of geometric data in a particular cohomology theory.
- Rational algebraic cycle —
- A combination of algebraic subvarieties with rational coefficients, used here to represent a specialized class.
Keywords : Milne’s rationality conjecture · Good reduction · Algebraic cycles
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves Milne's rationality conjecture for abelian varieties over with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Milne's rationality conjecture for abelian varieties
07/10/2026We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.