Cycles in the de Rham cohomology of abelian varieties over number fields

TL;DR

This paper investigates the relationship between crystalline de Rham cycles and Hodge cycles in abelian varieties over number fields, confirming Ogus' prediction for certain families using advanced cohomological techniques.

Contribution

It proves Ogus' conjecture that crystalline de Rham cycles coincide with Hodge cycles for specific families of abelian varieties, expanding understanding of their cohomological properties.

Findings

Confirmed Ogus' prediction for abelian varieties with prime dimension and nontrivial endomorphism ring.

Established a link between crystalline de Rham cycles and Hodge cycles in these families.

Utilized a crystalline analogue of Faltings' isogeny theorem and Mumford--Tate conjecture cases.

Abstract

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include abelian varieties that have both prime dimension and nontrivial endomorphism ring. The proof is based on a crystalline analogue of Faltings' isogeny theorem due to Bost and the known cases of the Mumford--Tate conjecture.