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Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves


Type

Thesis

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Authors

Yan, Jiali 

Abstract

Let J be the Jacobian variety of a genus two curve defined over a number field K. The main focus of this thesis is on computing the Cassels-Tate pairing on the 2-Selmer group of J.

We start by studying the Cassels-Tate pairing when J admits a Richelot 􏰑 isogeny φ : J → J. Suppose all points in J[2] are defined over K. We compute 􏰑􏰑 the Cassels-Tate pairing ⟨ , ⟩CT on Selφ􏰑(J)×Selφ􏰑(J) following the Weil pairing definition of the Cassels-Tate pairing.

We then study the pairing ⟨ , ⟩CT on Sel2(J) × Sel2(J) following the homogeneous space definition of the Cassels-Tate pairing. For ε,η ∈ Sel2(J), we compute ⟨ε,η⟩CT both in the case where all points in J[2] are defined over K and in the case where the twisted Kummer surface Kη has a K-rational point. In both cases, we give a computable formula for ⟨ε,η⟩CT and a practical algorithm for computation when K = Q.

In all cases, we calculate examples for which computing the Cassels-Tate pairing improves the rank bound of J obtained by carrying out standard de- scent calculations. We also give techniques to reduce the degree of the number field needed in the algorithm for computation.

Description

Date

2021-03-31

Advisors

Fisher, Tom

Keywords

Number Theory, Cassels-Tate pairing, rank bound, genus two curves

Qualification

Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge