On Tate-Shafarevich groups of 1-motives over Galois extensions

TL;DR

This paper investigates the behavior of Tate-Shafarevich groups of 1-motives over Galois extensions of global fields, establishing finiteness and providing formulas for related group order quotients.

Contribution

It offers new results on the kernel and cokernel of restriction maps for Tate-Shafarevich groups of 1-motives, without assuming finiteness hypotheses.

Findings

Groups are finite for i=1,2

Formulas for quotient of group orders derived

Results hold independently of finiteness assumptions

Abstract

Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any finiteness hypotheses. We show that these groups are finite and obtain, in particular, formulas for the corresponding quotient of group orders.