half_rank_zero_and_half_rank_one

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Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1. See p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf.

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Lean 4 Statement

theorem half_rank_zero_and_half_rank_one (r : ℕ) (hr : r = 0 ∨ r = 1) :
    atTop.Tendsto
      (fun H ↦ ({E ∈ heightLE H | E.rank = r}.ncard / (heightLE H).ncard : ℝ)) (𝓝 (1 / 2))

ID: Wikipedia__EllipticCurveRank__half_rank_zero_and_half_rank_one

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