Nontrivial elements of Sha explained through K3 surfaces
Algebraic Geometry
2007-06-06 v1 Number Theory
Abstract
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.
Cite
@article{arxiv.0706.0541,
title = {Nontrivial elements of Sha explained through K3 surfaces},
author = {Adam Logan and Ronald van Luijk},
journal= {arXiv preprint arXiv:0706.0541},
year = {2007}
}