On the second Tate-Shafarevich group of a 1-motive

TL;DR

This paper investigates the properties of the second Tate-Shafarevich group associated with 1-motives, proving finiteness under certain conditions, relating to Leopoldt's conjecture, and providing examples and duality results in arithmetic geometry.

Contribution

It establishes new finiteness results for degree 2 Tate-Shafarevich groups of 1-motives and introduces an arithmetic duality theorem over number fields.

Findings

Finiteness results for Tate-Shafarevich groups in degree 2 for 1-motives

Connection between Tate-Shafarevich groups and Leopoldt's conjecture

Example of a semiabelian variety with infinite Tate-Shafarevich group in degree 2

Abstract

We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction.