Cohomology of groups and local-global problems

Wednesday, 9 August 2023

Abstract

Let $p$ be a prime and $n$ a positive integer. Let $\mathbb{F}_q$ be the finite field with $q=p^m$ elements and $G$ a subgroup of $\textrm{GL}_n(\mathbb{F}_q)$. We give sufficient conditions to the triviality of $H^1(G,\mathbb{F}_q^n)=0$. These result generalizes a  theorem of Nori published in 1987 and answer the long-standing problem, also considered by Serre,  of finding an effective version of it. As a consequence we refine a criterion, proved by \c{C}iperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels on the Tate-Shafarevich group, in the case of abelian varieties over number fields. We also describe the relation between Cassels' question and the local-global divisibility problem.

Joint work with Davide Lombardo (Universit\`a di Pisa