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

Abstract

We prove finiteness results for Tate-Shafarevich groups in degree 2 associated with 1-motives. We give a number theoretic interpretation of these groups, 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.