Vol. 312, No. 1, 2021

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Relating the Tate–Shafarevich group of an elliptic curve with the class group

Dipendra Prasad and Sudhanshu Shekhar

Vol. 312 (2021), No. 1, 203–218
Abstract

The paper formulates a precise relationship between the Tate–Shafarevich group of an elliptic curve E over with a quotient of the class group of (E[p]) on which Gal((E[p])) = GL2(p) operates by its standard 2-dimensional representation over p. We establish such a relationship in many cases.

Keywords
class group, Tate–Shafarevich group, Selmer group, elliptic curve, Galois cohomology
Mathematical Subject Classification 2010
Primary: 11G05
Milestones
Received: 30 December 2019
Revised: 16 December 2020
Accepted: 20 December 2020
Published: 4 August 2021
Authors
Dipendra Prasad
Department of Mathematics
Indian Institute of Technology Bombay
Mumbai 400076
India
Sudhanshu Shekhar
Department of Mathematics
Indian Institute of Technology
Kanpur 208016
India