A divisibility related to the Birch and Swinnerton-Dyer conjecture

Let $E\mathrm{/}\mathbb{Q}$ 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 $E\mathrm{/}\mathbb{Q}$ divides the product of the order of the Shafarevich–Tate group of $E\mathrm{/}\mathbb{Q}$, the (global) Tamagawa number of $E\mathrm{/}\mathbb{Q}$, and the Tamagawa number of $E\mathrm{/}\mathbb{Q}$ 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 $E\mathrm{/}\mathbb{Q}$ is semi-stable.