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, relate 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.