2026-09-23T20:22:52Zhttps://api.repository.cam.ac.uk/server/oai/request

oai:www.repository.cam.ac.uk:1810/3252732023-12-22T14:06:18Zcom_1810_213747com_1810_256064col_1810_213748

Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves 10.17863/CAM.72729 Yan, Jiali Fisher, Tom 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. University of Cambridge 2021-03-31 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/325273 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f7785500-5f49-4b73-aba6-58cddc64f6c1/download 7b72b0518cd287707a651e982db5210a https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/da1f8b48-8028-419b-9f6a-7562c548db32/download 353adac0d1ebdfd65ab16480263c3c87 https://www.rioxx.net/licenses/all-rights-reserved/ Number Theory Cassels-Tate pairing rank bound genus two curves