Elliptic curves with exceptionally large analytic order of the Tate-Shafarevich groups

TL;DR

This paper presents 88 explicit examples of rank zero elliptic curves over rationals with exceptionally large Tate-Shafarevich groups, including a record value, and confirms some of these as the true orders using deep number theoretic results.

Contribution

The authors provide the largest known explicit examples of Tate-Shafarevich groups for rank zero elliptic curves and verify their true orders with advanced theoretical tools.

Findings

88 examples of large Tate-Shafarevich groups for elliptic curves

Record value of |(E)| = 1029212^2

Verification of some orders as the true Tate-Shafarevich group sizes

Abstract

We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|\sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\sza(E)| = 1029212^2 = 2^4\cdot 79^2 \cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $\sza$. For instance, $410536^2$ is the true order of $\sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.