On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

TL;DR

This paper develops an explicit algorithm to identify elliptic curves with infinitely many quadratic twists satisfying the BSD conjecture, and applies it to a large database, providing evidence for related conjectures and observed biases.

Contribution

It encodes hypotheses into an algorithm to find elliptic curves with infinitely many BSD-satisfying twists and applies it to extensive data, advancing computational evidence in the field.

Findings

Identified all elliptic curves up to conductor 500,000 with infinitely many BSD-satisfying twists.

Provided numerical evidence supporting a conjecture on Gaussian behavior of the Shafarevich-Tate group.

Observed a positive bias in the distribution of BSD-satisfying twists.

Abstract

Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwi{\l}{\l} and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.