Large Tate--Shafarevich orders from good $abc$ triples

TL;DR

This paper reports the discovery of large Tate--Shafarevich group orders for elliptic curves derived from good abc triples, significantly exceeding previous records and providing new insights into their distribution and properties.

Contribution

The authors compute record values of || for elliptic curves linked to good abc triples, revealing new large Tate--Shafarevich group orders and ratios, with efficient methods surpassing prior computational efforts.

Findings

Largest ||^2 exceeds 3.75 trillion

Discovered curves with G > 150 and || > 250000^2

Identified prime divisors of || values

Abstract

Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank zero and are isogenous to quadratic twists of Frey curves constructed from coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the radical $r$ is the product of the primes dividing $abc$. Curves with $|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have $G>150$. The largest value of $|\Sha|$ is $1937832^2>3.755\times10^{12}$. This is more than 3.5 times the previous record, which had been computed at a cost about 600 times greater than that for the new record. The primes 25913, 27457, 36929 and 49253 are identified as divisors of $|\Sha|$ values.