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arXiv · 2601.16044

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

Barinder S. Banwait·Xiaoyu Huang
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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.

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Barinder S. Banwait, Xiaoyu Huang. 2026-01-22. On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture. https://arxiv.org/abs/2601.16044

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