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

### Volume 166 / 2021

Colloquium Mathematicum 166 (2021), 217-225
MSC: Primary 11G05, 11G40, 11Y50.
DOI: 10.4064/cm8008-9-2020
Published online: 22 March 2021

#### Abstract

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