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We describe a method for computing the Cassels-Tate pairing on the 2-Selmergroup of the Jacobian of a genus 2 curve. This can be used to improve the upperbound coming from 2-descent for the rank of the group of rational points on theJacobian. Our method remains practical regardless of the Galois action on theWeierstrass points of the genus 2 curve. It does however depend on being ableto find a rational point on a certain twisted Kummer surface. The latter doesnot appear to be a severe restriction in practice. In particular, we have usedour method to unconditionally determine the ranks of all genus 2 Jacobians inthe L-functions and modular forms database (LMFDB).

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