Nontrivial elements of Sha explained through K3 surfaces

Logan, Adam; van Luijk, Ronald

doi:10.1090/S0025-5718-08-02105-4

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.

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Electronic resources, available at http://oldweb.cecm.sfu.ca/~rluijk/ShaK3/.