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Local-global principles for 1-motives

David Harari·Tamas 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 $\Be (X)$ 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-Shaferevich 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.

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David Harari, Tamas Szamuely. 2007-09-28. Local-global principles for 1-motives. https://arxiv.org/abs/math/0703845

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