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 $s=1$ of the elliptic curve’s $L$-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.