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Mathematics of Computation

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Nontrivial elements of Sha explained through K3 surfaces

Authors: Adam Logan and Ronald van Luijk
Journal: Math. Comp. 78 (2009), 441-483
MSC (2000): Primary 14H40, 11G10, 14J27, 14J28
Published electronically: May 2, 2008
MathSciNet review: 2448716
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Abstract: We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.

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Additional Information

Adam Logan
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

Ronald van Luijk
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6

Received by editor(s): June 16, 2007
Received by editor(s) in revised form: November 19, 2007
Published electronically: May 2, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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