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Some conjectures about ranks of elliptic curves over ℚ: goldfeld_conjecture

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Statement

Goldfeld's conjecture ([Goldfeld1979], Conjecture B), in the form stated in the introduction of [Smith2025]: when the quadratic twists EdE^d of an elliptic curve EE over Q\mathbb{Q} are ordered by ∣d∣|d|, 50% of them have rank 00 and 50% have rank 11. See goldfeld_conjecture.variants.two_le for the remaining case 2≤r2 \leq r.

Goldfeld states the conjecture for the analytic rank of EdE^d, which the Birch and Swinnerton-Dyer conjecture predicts is equal to the Mordell–Weil rank used here. Corollary 1.2 of [Smith2025] proves that the Birch and Swinnerton-Dyer conjecture for the quadratic twist family of EE implies Goldfeld's conjecture for EE.

Lean statement

theorem goldfeld_conjecture (E : RatEllipticCurve) (r : ℕ) (hr : r = 0 ∨ r = 1) :
    atTop.Tendsto
      (fun H ↦ ({d ∈ twistIndexLE H | E.twistRank d = r}.ncard / (twistIndexLE H).ncard : ℝ))
      (𝓝 (1 / 2)) := by
  sorry

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Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Wikipedia/EllipticCurveRank.lean.

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