Tate-Shafarevich groups and K3 surfaces

TL;DR

This paper demonstrates explicit examples of genus-2 curves with Jacobians that have nontrivial 2-torsion in their Tate-Shafarevich groups, advancing understanding of these groups in algebraic geometry.

Contribution

It provides explicit constructions of genus-2 curves with Jacobians exhibiting nontrivial 2-torsion in their Tate-Shafarevich groups, building on previous methods.

Findings

Explicit genus-2 curves with nontrivial 2-torsion in Tate-Shafarevich groups

Extension of Logan and van Luijk's method

New examples in algebraic geometry

Abstract

Following (and elaborating on) a method of Logan and van Luijk, we exhibit explicit genus-2 curves whose Jacobians have nontrivial 2-torsion in their Tate-Shafarevich groups.