Himanshu Shukla

Computation of the Cassels-Tate pairing for hyperelliptic Jacobians

Cassels-Tate pairing (CTP) (first defined by Cassels for elliptic curves and generalized by Tate to principally polarized abelian varieties) on the n-Selmer group for an abelian variety A/k measures the image of n^2-Selmer group of A inside n-Selmer group, hence can be used to visualize elements in Sha(A/k)[n] or bound the algebraic rank of A(K). There are various definitions of CTP known. Cassels, Swinnerton-Dyer, Fisher-van Beek, Fisher-Newton, Fisher-Yan, Donelly-Fisher used the Weil-pairing definition or the homogeneous space definition or a combination of them to compute the pairing in various cases for Elliptic curves and on 2-Selmer group of genus 2 Jacobians in various cases. In this talk we will see how to make the Albanese-Albanese definition of CTP for Selmer groups corresponding to various isogenies on hyperelliptic Jacobians effective. Essentially we show that computing the global part of the Cassels-Tate pairing is computationally equivalent to trivializing matrix algebras over k given by 2-cocycles factoring through the field of definition of Selmer elements being paired in various cases.