A divisibility related to the Birch and Swinnerton-Dyer conjecture

Mentzelos Melistas

doi:10.1016/j.jnt.2022.10.004

A divisibility related to the Birch and Swinnerton-Dyer conjecture

Mentzelos Melistas^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

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

Original language	English
Pages (from-to)	150-168
Number of pages	19
Journal	Journal of Number Theory
Volume	245
Early online date	24 Nov 2022
DOIs	https://doi.org/10.1016/j.jnt.2022.10.004
Publication status	Published - Apr 2023
Externally published	Yes

Keywords

Elliptic curve
Shafarevich-Tate group
Tamagawa number
Torsion point
BSD conjecture
n/a OA procedure

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10.1016/j.jnt.2022.10.004

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title = "A divisibility related to the Birch and Swinnerton-Dyer conjecture",

abstract = "Let E/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/Q divides the product of the order of the Shafarevich–Tate group of E/Q, the (global) Tamagawa number of E/Q, and the Tamagawa number of E/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/Q is semi-stable.",

keywords = "Elliptic curve, Shafarevich-Tate group, Tamagawa number, Torsion point, BSD conjecture, n/a OA procedure",

author = "Mentzelos Melistas",

note = "Funding Information: The author would like to thank Dino Lorenzini for pointing out that some of the author's techniques in [28] could be applied to Conjecture 1.2 . The author would also like to thank Paul Voutier for some useful comments on an earlier version of this manuscript. I would also like to thank the anonymous referee for many insightful comments and many useful suggestions that greatly improved the exposition of this manuscript. This work was performed at the Steklov International Mathematical Center, while the author was a member there, and supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-15-2019-1614 ). Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2023",

month = apr,

doi = "10.1016/j.jnt.2022.10.004",

language = "English",

volume = "245",

pages = "150--168",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

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T1 - A divisibility related to the Birch and Swinnerton-Dyer conjecture

AU - Melistas, Mentzelos

N1 - Funding Information: The author would like to thank Dino Lorenzini for pointing out that some of the author's techniques in [28] could be applied to Conjecture 1.2 . The author would also like to thank Paul Voutier for some useful comments on an earlier version of this manuscript. I would also like to thank the anonymous referee for many insightful comments and many useful suggestions that greatly improved the exposition of this manuscript. This work was performed at the Steklov International Mathematical Center, while the author was a member there, and supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-15-2019-1614 ). Publisher Copyright: © 2022 Elsevier Inc.

PY - 2023/4

Y1 - 2023/4

N2 - Let E/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/Q divides the product of the order of the Shafarevich–Tate group of E/Q, the (global) Tamagawa number of E/Q, and the Tamagawa number of E/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/Q is semi-stable.

AB - Let E/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/Q divides the product of the order of the Shafarevich–Tate group of E/Q, the (global) Tamagawa number of E/Q, and the Tamagawa number of E/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/Q is semi-stable.

KW - Elliptic curve

KW - Shafarevich-Tate group

KW - Tamagawa number

KW - Torsion point

KW - BSD conjecture

KW - n/a OA procedure

U2 - 10.1016/j.jnt.2022.10.004

DO - 10.1016/j.jnt.2022.10.004

M3 - Article

SN - 0022-314X

VL - 245

SP - 150

EP - 168

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

A divisibility related to the Birch and Swinnerton-Dyer conjecture

Abstract

Keywords

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