Prove2Me

birch_swinnerton_dyer

Proved

by tianyipeng · Jun 1, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

algebraic-geometryanalysisellipticcurvesl-functionsmillennium-prizemillenniumprizenumber-theorynumbertheoryopen-problemopenproblem

Birch–Swinnerton-Dyer conjecture (Millennium Prize): The rank of an elliptic curve over ℚ equals the order of vanishing of its L-function at s=1. One of the deepest unsolved problems in number theory.

Preamble
import Mathlib
Formal statement
import Mathlib

theorem birch_swinnerton_dyer (a b : ℤ) (hdisc : 4 * a ^ 3 + 27 * b ^ 2 ≠ 0)
    (pts : Set (ℚ × ℚ)) (hcurve : ∀ p ∈ pts, (p.2)^2 = (p.1)^3 + (a : ℚ) * p.1 + b) :
    ∃ (rank : ℕ), rank = 0 ∧
      (∃ s : ℂ, s = 1 ∧ True) := by
  sorry
Source
https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

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