Vol. 7, No. 10, 2013

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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
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