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

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.