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

DOI: 10.2140/pjm.2021.312.203

Abstract

The paper formulates a precise relationship between the Tate–Shafarevich group of an elliptic curve $E$ over $\mathbb{Q}$ with a quotient of the class group of $\mathbb{Q}\left(E\left[p\right]\right)$ on which $Gal\left(\mathbb{Q}\left(E\left[p\right]\right)∕\mathbb{Q}\right)={GL}_2\left(\mathbb{Z}∕p\right)$ operates by its standard 2-dimensional representation over $\mathbb{Z}∕p$. 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