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On the $p$-torsion of the Tate-Shafarevich group of abelian varieties over higher dimensional bases over finite fields (2203.05798v1)
Published 11 Mar 2022 in math.NT and math.AG
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.