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Arithmetic geometry Millennium Prize Problem Status: Open Status last verified

Birch and Swinnerton-Dyer Conjecture

The number of independent rational points on an elliptic curve (its rank) equals the order of vanishing of its L-function at s = 1.

Problem statement

For an elliptic curve E over Q, the rank of the group E(Q) equals the order of vanishing of L(E, s) at s = 1. The full statement (including the leading-coefficient formula) is in Wiles's official problem description.

E : y² = x³ + ax + b
E(Q) ≅ Z^r ⊕ (finite group)

Conjecture: r = ord_{s=1} L(E, s)

Source: Andrew Wiles — The Birch and Swinnerton-Dyer Conjecture (official problem description)

Introduction

An elliptic curve over the rationals is a curve y² = x³ + ax + b (with a non-zero discriminant). Its rational points form a group, and by Mordell's theorem that group is generated by finitely many points. The number of independent points of infinite order is the rank.

Birch and Swinnerton-Dyer, using early computer experiments in the 1960s, conjectured that the rank can be read off from an analytic object: the L-function L(E, s), which is built from counting points on the curve modulo each prime.

It is one of the Clay Millennium Prize Problems.

Definitions

Rank r
The number of independent rational points of infinite order on E.
a_p
p + 1 − #E(F_p), from counting the points of E modulo a prime p; the L-function is built from these numbers.
Analytic rank
The order of vanishing of L(E, s) at s = 1.

What is known

Each statement says how certain it is: an established (proved) result, an experimental observation, an open conjecture, or an unverified claim.

  • Established result If the analytic rank is 0 or 1, the rank equals the analytic rank (Gross–Zagier and Kolyvagin, combined with the modularity theorem). [Andrew Wiles]
  • Established result When curves are ordered by height, a majority of elliptic curves over Q satisfy the rank part of the conjecture (Bhargava, Skinner and Zhang). [Manjul Bhargava, Christopher Skinner, Wei Zhang]
  • Experimental observation Extensive tables of curves, ranks and L-function values (for example in the LMFDB) agree with the conjecture wherever both sides have been computed. [LMFDB]
  • Open conjecture The rank equals the analytic rank for every elliptic curve over Q (unproven in general). [Andrew Wiles]

Important limitations

  • Numerical evidence for many curves does not prove the conjecture for all curves.
  • For analytic rank 2 or more, even computing the rank of a single curve can be hard to certify.

Status history

  1. Status: Open Named one of the seven Millennium Prize Problems by the Clay Mathematics Institute.
  2. Status: Open Checked against the Clay Mathematics Institute page: still unsolved.

This status is maintained by the Zerethon IDE team from authoritative sources. Contributions, runs and AI answers never change it.