Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian

Abstract

We describe a method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve. This can be used to improve the upper bound coming from 2-descent for the rank of the group of rational points on the Jacobian. Our method remains practical regardless of the Galois action on the Weierstrass points of the genus 2 curve. It does however depend on being able to find a rational point on a certain twisted Kummer surface. The latter does not appear to be a severe restriction in practice. In particular, we have used our method to unconditionally determine the ranks of all genus 2 Jacobians in the L-functions and modular forms database (LMFDB).