Tate-Shafarevich groups and 𝐾3 surfaces

Corn, Patrick

doi:10.1090/S0025-5718-09-02264-9

Tate-Shafarevich groups and $K3$ surfaces
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Math. Comp. 79 (2010), 563-581 Request permission

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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