Birch–Swinnerton-Dyer conjecture for elliptic curves

Prove the Birch–Swinnerton-Dyer conjecture for elliptic curves over \(\mathbb{Q}\), including that the order of vanishing of the associated \(L\)-function at \(s=1\) equals the rank of the elliptic curve and that its leading coefficient is given by the predicted product of arithmetic invariants.

Background

The paper introduces the rank of an elliptic curve as the rank of the free part of its Mordell–Weil group and emphasizes that this invariant remains poorly understood. It then describes the Birch–Swinnerton-Dyer conjecture, which relates the rank to the order of vanishing at s=1s=1 of the elliptic curve’s LL-function and predicts the leading coefficient of the Taylor expansion there in terms of arithmetic invariants.

Machine-learning experiments on elliptic-curve data are presented as a way to detect statistical structure associated with rank, but they do not resolve the Birch–Swinnerton-Dyer conjecture.

References

Elliptic curves have been studied for a long time, but the rank is still mysterious, with many questions still unanswered: in particular, the Birch and Swinnerton-Dyer conjecture which we will come to below.

— Mathematical Data Science  (2502.08620 - Douglas et al., 12 Feb 2025) in Section 2, subsection “Elliptic Curves”

\begin{conjecture}[The Birch and Swinnerton-Dyer] Let $E/F$ be an elliptic curve over a number field $F$.

— A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields  (2608.11969 - Li, 12 Aug 2026) in Conjecture 1.1, Section 1.1

Conversely, it is not known whether the modern formulation of the conjecture implies Conjecture \ref{OBSD}, even assuming the Riemann Hypothesis for $L(E, s)$.

— Products of point counts of higher genus curves over finite fields  (2608.18014 - Bucur et al., 18 Aug 2026) in Section 1, paragraph following the introduction of the modern Birch–Swinnerton-Dyer formulation

The original version of the Birch and Swinnerton-Dyer conjecture takes the following form.

— Products of point counts of higher genus curves over finite fields  (2608.18014 - Bucur et al., 18 Aug 2026) in Conjecture 1, Section 1