Nontrivial elements of Sha explained through K3 surfaces

Details Nontrivial elements of Sha explained through K3 surfaces Nontrivial elements of Sha explained through K3 surfaces

2007-06-04

arxiv.org/abs/0706.0541v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

In this paper 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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