On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields

TL;DR

This paper establishes a finiteness theorem for the first flat cohomology of finite flat group schemes over higher dimensional varieties over finite fields, leading to invariance results for the $p$-torsion of Tate-Shafarevich groups of abelian schemes.

Contribution

It generalizes previous results on Tate-Shafarevich groups to higher dimensional bases and proves invariance of their $p$-torsion under isogenies and alterations.

Findings

Finiteness of the first flat cohomology group over higher dimensional bases.

Invariance of the $p$-part of Tate-Shafarevich groups under isogenies.

Generalization of Tate-Shafarevich group results to new geometric settings.

Abstract

We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the $p$-part. Along the way, we generalize previous results on the Tate-Shafarevich group in this situation.