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

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2021

Author dc:creator

• Yan, Jiali

Advisor dc:contributor.advisor

• Fisher, Tom

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

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/da1f8b48-8028-419b-9f6a-7562c548db32/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.72729

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/325273