When the BSD Real Period Remembers Elliptic Curves over ℚ
Abstract
We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question. For example, the real period ΩE =Γ(1/4)2/(2√π 171/4) and the Tate--Shafarevich group $ \Sha(E/\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ remember the elliptic curve E: y2=x3+17x. Let K be a number field admitting a real embedding, and fix one such embedding σ: K↪ℝ. We first prove that, for an elliptic curve E over K, the real period of E attached to σ determines the K-isogeny class of E. We then prove that an elliptic curve E over ℚ is determined up to isomorphism over ℚ by its real period, the number of connected components of E(ℝ), and the real period of the quadratic twist ED by a negative square-free integer D≡ 1 bmod 4 coprime to the minimal discriminant of E. Moreover, over ℚ, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on E (and D).
Related
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BibTeX
@misc{shiga2026when,
title = {{When the BSD Real Period Remembers Elliptic Curves over $\mathbbQ$}},
author = {Asuka Shiga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06015},
eprint = {2610.06015},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06015},
}
RIS
TY - UNPB
TI - When the BSD Real Period Remembers Elliptic Curves over ℚ
AU - Asuka Shiga
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06015
UR - https://arxiv.org/abs/2610.06015
AB - We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question. For example, the real period Ω_E =Γ(1/4)^2/(2√π 17^1/4) and the Tate--Shafarevich group $ \Sha(E/\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ remember the elliptic curve E: y^2=x^3+17x. Let K be a number field admitting a real embedding, and fix one such embedding σ: K↪ℝ. We first prove that, for an elliptic curve E over K, the real period of E attached to σ determines the K-isogeny class of E. We then prove that an elliptic curve E over ℚ is determined up to isomorphism over ℚ by its real period, the number of connected components of E(ℝ), and the real period of the quadratic twist E^D by a negative square-free integer D≡ 1 bmod 4 coprime to the minimal discriminant of E. Moreover, over ℚ, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on E (and D).
ER -