A divisibility related to the Birch and Swinnerton-Dyer conjecture

DOI10.1016/J.JNT.2022.10.004arXiv2211.08147WikidataQ122884311 ScholiaQ122884311MaRDI QIDQ6417336

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Publication date: 15 November 2022

Abstract: Let $E/mathbbQ$ 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 $E/mathbbQ$ divides the product of the order of the Shafarevich--Tate group of $E/mathbbQ$, the (global) Tamagawa number of $E/mathbbQ$, and the Tamagawa number of $E/mathbbQ$ 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 $E/mathbbQ$ is semi-stable.

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