Nontrivial elements of Sha explained through K3 surfaces
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- by Adam Logan and Ronald van Luijk PDF
- Math. Comp. 78 (2009), 441-483 Request permission
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.References
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Additional Information
- Adam Logan
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
- Email: a5logan@math.uwaterloo.ca
- Ronald van Luijk
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6
- Email: rmluijk@gmail.com
- Received by editor(s): June 16, 2007
- Received by editor(s) in revised form: November 19, 2007
- Published electronically: May 2, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 78 (2009), 441-483
- MSC (2000): Primary 14H40, 11G10, 14J27, 14J28
- DOI: https://doi.org/10.1090/S0025-5718-08-02105-4
- MathSciNet review: 2448716