Algebra & Number Theory

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

Peter Jossen

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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.

Article information

Source
Algebra Number Theory, Volume 7, Number 10 (2013), 2511-2544.

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730115

Digital Object Identifier
doi:10.2140/ant.2013.7.2511

Mathematical Reviews number (MathSciNet)
MR3194650

Zentralblatt MATH identifier
1326.14104

Subjects
Primary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
Secondary: 14G25: Global ground fields 14G20: Local ground fields

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

Citation

Jossen, Peter. On the second Tate–Shafarevich group of a 1-motive. Algebra Number Theory 7 (2013), no. 10, 2511--2544. doi:10.2140/ant.2013.7.2511. https://projecteuclid.org/euclid.ant/1513730115


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