On the Hasse principle for finite group schemes over global function fields

TL;DR

This paper investigates the Hasse principle for finite flat group schemes over global function fields, showing the kernel of the localization map can be understood via Galois cohomology, with applications to abelian varieties.

Contribution

It establishes a method to compute the kernel of the localization map in flat cohomology using Galois cohomology, extending understanding of the Hasse principle in this setting.

Findings

Kernel of localization map computed via Galois cohomology

Applications to kernels of multiplication by p^m on abelian varieties

Enhanced understanding of the Hasse principle for finite group schemes

Abstract

Let K be a global function field of positive characteristic p and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization map H^{1}(K,M)\to\prod_{all v}H^{1}(K_{v},M) in flat (fppf) cohomology can be computed solely in terms of Galois cohomology. We then give applications to the case where M is the kernel of multiplication by p^{m} on an abelian variety defined over K.