Duke Mathematical Journal

Local-global principles for $1$-motives

David Harari and Tamás Szamuely

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Abstract

Building upon our arithmetic duality theorems for $1$-motives, we prove that the Manin obstruction related to a finite subquotient ${\cyrille B} (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shafarevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate-type dual exact sequence for $1$-motives and give an application to weak approximation

Article information

Source
Duke Math. J. Volume 143, Number 3 (2008), 531-557.

Dates
First available in Project Euclid: 3 June 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1212500466

Digital Object Identifier
doi:10.1215/00127094-2008-028

Zentralblatt MATH identifier
1155.14020

Mathematical Reviews number (MathSciNet)
MR2423762

Subjects
Primary: 14G25: Global ground fields
Secondary: 14G05: Rational points

Citation

Harari, David; Szamuely, Tamás. Local-global principles for 1 -motives. Duke Math. J. 143 (2008), no. 3, 531--557. doi:10.1215/00127094-2008-028. http://projecteuclid.org/euclid.dmj/1212500466.


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