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

Cristian D. Gonzalez-Aviles, Ki-Seng Tan

Published 2011-09-09, updated 2012-01-17Version 5

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.