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Integral crystalline Tate classes of codimension two at supersingular points of Mumford's fourfolds

Primarily AI-generated textHuman understanding: no partsmath.AG — Algebraic Geometrymath.NT — Number Theory

Contributed by Antti Hippeläinen ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

For a supersingular abelian variety XX over Fˉp\bar{\mathbb{F}}_p, let T4(X)T^4(X) be the lattice of integral crystalline Tate classes of codimension 22 and S(X)S(X) the span of products of divisor classes. Let p≥5p\ge5 and Gp\mathscr{G}_p be the pp-divisible group of the supersingular points of Mumford's families of abelian fourfolds at primes inert in their cubic field, for the hyperspecial levels given by tensor-induced lattices. For every abelian fourfold XX with X[p∞]≅GpX[p^\infty]\cong\mathscr{G}_p we show that T4(X)/S(X)≅(Z/p)4T^4(X)/S(X)\cong(\mathbb{Z}/p)^4, and that the integral crystalline Tate conjecture holds in codimension 22 on XX if statements in a preprint of Markman hold. At these primes, Kisin's crystalline tensors make the Dieudonné module a tensor product of three rank-22 modules permuted cyclically by Frobenius, and at supersingular points Frobenius maps a tensor basis to multiples of basis vectors. For such monomial modules all lattices involved are spanned by multiples of basis vectors, with exponents given by min-plus formulas independent of pp. Under the endomorphisms, T4/ST^4/S is a non-split extension of (Z/p)2(\mathbb{Z}/p)^2 by (Z/p)2(\mathbb{Z}/p)^2, generated by one Weil class of an inert imaginary quadratic action. Assuming these statements, an algebraic such class comes from specializing a Weil cycle on a fourfold with complex multiplication. A projector that is a polynomial in an endomorphism, and an intersection matrix with unit determinant, fix its pp-adic normalization without a comparison theorem. At unramified non-inert primes with such levels, T4=ST^4=S at the supersingular points. Of the fourteen isomorphism classes of monomial supersingular Dieudonné modules of height 88, the conjecture holds unconditionally for nine, under the same assumption for that of Gp\mathscr{G}_p, and is open for four. The indices are computed exactly with integers.

Provenance statement

The proofs, the computations and the text of this note were produced by large language models (Claude models of Anthropic running as autonomous agents) in October 2026, and checked by other instances of the same models acting as referees, which re-derived the results and recomputed the numbers with programs of their own. The author directed the work, read through the note and verified the bibliographic data of its references. However, no human expert has checked the proofs. Part (b) of the main theorem is conditional on statements of a preprint of E. Markman (arXiv:2502.03415) whose proofs have not been verified. The computations of Appendix A can be reproduced by the code openly available at https://github.com/ahippelainen/integral-tate-supersingular-fourfolds-public.

Tools used

Anthropic
Claude OpusVersion 5.5
Python Software Foundation and others
Python

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Version history

  1. v1Submitted by Antti HippeläinenInitial depositCurrentOct 29, 2026