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

Abstract: In her 2021 thesis, my student Jiali Yan gave a practical method for computing the Cassels-Tate pairing on the 2-Selmer group of the Jacobian of a genus 2 curve all of whose Weierstrass points are rational. She also gave a second method without any assumption on the Weierstrass points, but instead assuming we can find a rational point on a certain twisted Kummer surface. The two methods can be thought of as generalising methods of Cassels and Donnelly in the elliptic curve case. I will describe a practical refinement of the second method which is now implemented in Magma, and has been used to unconditionally determine the ranks of all genus 2 Jacobians in the LMFDB.

number theory

Audience: researchers in the topic

London number theory seminar

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