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

Jossen, Peter

doi:10.2140/ant.2013.7.2511

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

Peter Jossen

Vol. 7 (2013), No. 10, 2511–2544

DOI: 10.2140/ant.2013.7.2511

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

Milestones

Received: 27 September 2012

Revised: 4 March 2013

Accepted: 11 April 2013

Published: 18 January 2014

Authors

Peter Jossen
CNRS, UMR 8628 Mathématiques Université Paris-Sud Bâtiment 425 91450 Orsay France

Vol. 7, No. 10, 2013