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

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

PDF Access Denied

We have not been able to recognize your IP address 18.219.112.111 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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

Vol. 312, No. 1, 2021

Dipendra Prasad and Sudhanshu Shekhar

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors