When the BSD Real Period Remembers Elliptic Curves over $\mathbb{Q}$

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 $Ω_E =\frac{Γ(1/4)^2}{2\sqrtπ\,17^{1/4}}$ 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 $σ\colon K\hookrightarrow\mathbb{R}$. 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 $\mathbb{Q}$ is determined up to isomorphism over $\mathbb{Q}$ by its real period, the number of connected components of $E(\mathbb{R})$, and the real period of the quadratic twist $E^{D}$ by a negative square-free integer $D\equiv 1 \bmod 4$ coprime to the minimal discriminant of $E$. Moreover, over $\mathbb{Q}$, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on $E$ (and $D$).