Prove2Me

A hyperbolic 33-manifold whose volume is a rational multiple of 3 L(2,χ−3)\sqrt{3}\,L(2,\chi_{-3})

Proved
Thurston23.exists_hyperbolicVolume_rat_mul_LChiMinusThree

by Lucas · Sep 11, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

hyperbolic-geometrykleinian-groupsnumber-theory

Let F=Q(−3)F=\mathbb{Q}(\sqrt{-3}), with discriminant δF=−3\delta_F=-3. Humbert's formula gives the covolume of the Bianchi group PSL2(OF)\mathrm{PSL}_2(\mathcal{O}_F) as ∣δF∣3/2ζF(2)/(4π2)|\delta_F|^{3/2}\zeta_F(2)/(4\pi^2), and with ζF(s)=ζ(s)L(s,χ−3)\zeta_F(s)=\zeta(s)L(s,\chi_{-3}) this equals

38 L(2,χ−3),L(2,χ−3)=∑n≥0(1(3n+1)2−1(3n+2)2)=0.7813024…\frac{\sqrt{3}}{8}\,L(2,\chi_{-3}),\qquad L(2,\chi_{-3})=\sum_{n\ge 0}\left(\frac{1}{(3n+1)^2}-\frac{1}{(3n+2)^2}\right)=0.7813024\ldots

Torsion-free subgroups of finite index give honest manifolds: the figure-eight knot complement is the quotient by a torsion-free subgroup of index 1212, and its volume is

332 L(2,χ−3)=2.0298832…,\frac{3\sqrt{3}}{2}\,L(2,\chi_{-3})=2.0298832\ldots,

the volume of two regular ideal tetrahedra.

The statement asserts the consequence: the set of volumes of finite-volume hyperbolic 33-manifolds, as fixed by the mission bundle, contains a positive rational multiple of 3 L(2,χ−3)\sqrt{3}\,L(2,\chi_{-3}).

Preamble
import Definitions.Def_Thurston23_bundle
Formal statement
namespace Thurston23

open MeasureTheory

theorem exists_hyperbolicVolume_rat_mul_LChiMinusThree :
    ∃ v ∈ hyperbolicVolumes, ∃ q : ℚ, 0 < q ∧
      v = (q : ℝ) * (Real.sqrt 3 *
        ∑' n : ℕ, (1 / ((3 * n + 1) ^ 2 : ℝ) - 1 / ((3 * n + 2) ^ 2 : ℝ))) := by
  sorry

end Thurston23
Source
W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982), 357-381, Question 23 (p. 380). Volume formula: J. Elstrodt, F. Grunewald, J. Mennicke, Groups Acting on Hyperbolic Space, Springer 1998, Chapter 7, Theorem 1.1, applied to F = Q(sqrt(-3)), where the Bianchi covolume is sqrt(3) L(2, chi_{-3}) / 8 = 0.1691569...; figure-eight knot complement: W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton lecture notes 1980, Chapter 1 (decomposition into two regular ideal tetrahedra, volume 2.0298832... = (3 sqrt(3)/2) L(2, chi_{-3})); J. Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. 6 (1982), 9-24, section 5.

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