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arXiv · 2211.08147

A divisibility related to the Birch and Swinnerton-Dyer conjecture

Mentzelos Melistas
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Abstract

Let $E/\mathbb{Q}$ be an optimal elliptic curve of analytic rank zero. It follows from the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero that the order of the torsion subgroup of $E/\mathbb{Q}$ divides the product of the order of the Shafarevich--Tate group of $E/\mathbb{Q}$, the (global) Tamagawa number of $E/\mathbb{Q}$, and the Tamagawa number of $E/\mathbb{Q}$ at infinity. This consequence of the Birch and Swinnerton-Dyer conjecture was noticed by Agashe and Stein in 2005. In this paper, we prove this divisibility statement unconditionally in many cases, including the case where the curve $E/\mathbb{Q}$ is semi-stable.

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Mentzelos Melistas. 2022-11-15. A divisibility related to the Birch and Swinnerton-Dyer conjecture. https://arxiv.org/abs/2211.08147

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