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Mathematics of Computation

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Tate-Shafarevich groups and $ K3$ surfaces

Author: Patrick Corn
Journal: Math. Comp. 79 (2010), 563-581
MSC (2000): Primary 14H40; Secondary 11G10
Published electronically: June 5, 2009
MathSciNet review: 2552241
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Abstract: This paper explores a topic taken up recently by Logan and van Luijk, finding nontrivial $ 2$-torsion elements of the Tate-Shafarevich group of the Jacobian of a genus-$ 2$ curve by exhibiting Brauer-Manin obstructions to rational points on certain quotients of principal homogeneous spaces of the Jacobian, whose desingularizations are explicit $ K3$ surfaces. The main difference between the methods used in this paper and those of Logan and van Luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras, rather than elliptic fibrations.

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Additional Information

Patrick Corn
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602-7403
Address at time of publication: Department of Mathematics and Computer Science, St. Mary’s College of Maryland, 18952 E. Fisher Road, St. Mary’s City, Maryland 20686-3001

Received by editor(s): March 27, 2008
Received by editor(s) in revised form: February 4, 2009
Published electronically: June 5, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.