A divisibility related to the Birch and Swinnerton-Dyer conjecture

TL;DR

This paper proves a divisibility property related to the Birch and Swinnerton-Dyer conjecture for optimal elliptic curves of rank zero, establishing it unconditionally in many cases, especially semi-stable curves.

Contribution

It unconditionally proves a divisibility statement connected to the BSD conjecture for certain elliptic curves, extending previous conditional observations.

Findings

Divisibility holds unconditionally for many elliptic curves.

Includes semi-stable elliptic curves as a special case.

Confirms a conjectured divisibility property in specific instances.

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.