Lefschetz classes of simple factors of Fermat Jacobian of prime degree over finite fields

TL;DR

This paper establishes a criterion for Tate classes to be Lefschetz in simple abelian varieties over finite fields and applies it to Fermat Jacobians of prime degree, showing all Tate classes are Lefschetz under certain conditions.

Contribution

It provides a necessary and sufficient matrix condition for Tate classes to be Lefschetz and applies this to Fermat Jacobians of prime degree.

Findings

All Tate classes are Lefschetz for simple factors of Fermat Jacobians under an assumption.

The paper introduces a matrix criterion for Lefschetz property of Tate classes.

It advances understanding of the structure of Tate classes in abelian varieties over finite fields.

Abstract

We give a necessary and sufficient condition in terms of a matrix for which all Tate classes are Lefschetz for simple abelian varieties over an algebraic closure of a finite field. As an application, we prove under an assumption that all Tate classes are Lefschetz for simple factors of Fermat Jacobian of prime degree.