Conjecture Fieldbook

GlossaryStudy cards

Birch and Swinnerton-Dyer

Canonical problem dossier

The rank of an elliptic curve’s rational points is predicted by the order of vanishing of its L-function at s = 1.

Status
Open
Field
Arithmetic geometry
Verification
Reviewed dossier
Source grade
mixed
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

rank⁡E(Q)=ord⁡s=1L(E,s)\operatorname{rank}E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)

Why it matters

It links algebraic information about rational solutions to analytic behavior of an L-function and is central to modern arithmetic geometry.

Context and known approaches

The conjecture predicts that the algebraic rank equals the analytic rank and gives a refined leading-term formula.

Formal boundary

For an elliptic curve E over Q, rank E(Q) = ord(s=1) L(E,s).

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Clay Mathematics Institute — Official Millennium problem overview.
  2. Wikipedia — General overview and references.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.