#### Vol. 7, No. 10, 2013

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

### Peter Jossen

Vol. 7 (2013), No. 10, 2511–2544
##### 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.

##### Keywords
$1$-motives, semiabelian varieties, Tate–Shafarevich groups
##### Mathematical Subject Classification 2010
Primary: 14K15
Secondary: 14G25, 14G20