Nontrivial elements of Sha explained through K3 surfaces

TL;DR

This paper introduces a novel method using Brauer-Manin obstructions on K3 surfaces to demonstrate nontrivial elements in the Tate-Shafarevich group of Jacobians of genus two curves, revealing deep arithmetic properties.

Contribution

The paper presents a new approach to detect nontrivial elements of the Tate-Shafarevich group via Brauer-Manin obstructions on K3 surfaces, applied explicitly to genus two curves.

Findings

Identified infinitely many quadratic twists with nontrivial Tate-Shafarevich groups.

Developed a method linking K3 surfaces and Jacobian obstructions.

Provided explicit examples demonstrating the method's effectiveness.

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.