Large Tate--Shafarevich orders from good abc triples

2021/11/15 by David Broadhurst, Broadhurst, David
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2111.07794

openalex publication_date 2021/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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|/√(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|>250000² and G>12 are found in 20 isogeny classes. Three curves have G>150. The largest value of |\Sha| is 1937832²>3.755×10¹². 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.

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