On Tate-Shafarevich groups of 1-motives

TL;DR

This paper proves the finiteness of certain kernel and cokernel groups related to 1-motives over global fields, extending understanding of Tate-Shafarevich groups in this context.

Contribution

It establishes the finiteness of the kernel and cokernel of restriction maps for 1-motives over global fields, a new result in the study of Tate-Shafarevich groups.

Findings

Finiteness of kernel and cokernel for i=1,2 in the restriction maps

Extension of Tate-Shafarevich group properties to 1-motives

Advancement in understanding Galois cohomology of 1-motives

Abstract

We establish the finiteness of the kernel and cokernel of the restriction map III^{i}(F,M) ---> III^{i}(K,M)^{G} for i=1 and 2, where M is a (Deligne) 1-motive over a global field F and K/F is a finite Galois extension of global fields with Galois group G.