Tate-Shafarevich groups and $K3$ surfaces

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.