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

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.