Open Access
2013 On the second Tate–Shafarevich group of a 1-motive
Peter Jossen
Algebra Number Theory 7(10): 2511-2544 (2013). DOI: 10.2140/ant.2013.7.2511


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.


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Peter Jossen. "On the second Tate–Shafarevich group of a 1-motive." Algebra Number Theory 7 (10) 2511 - 2544, 2013.


Received: 27 September 2012; Revised: 4 March 2013; Accepted: 11 April 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1326.14104
MathSciNet: MR3194650
Digital Object Identifier: 10.2140/ant.2013.7.2511

Primary: 14K15
Secondary: 14G20 , 14G25

Keywords: $1$-motives , semiabelian varieties , Tate–Shafarevich groups

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 10 • 2013
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