Prove2Me

A Möbius map sending C^m,α\widehat C_{m,\alpha} into C^m,β\widehat C_{m,\beta} is z↦czz\mapsto cz or z↦c/zz\mapsto c/z with c≠0c\ne0

Proved
CurveSymmetry.family_mobius_complete

by carlok · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

algebraic-geometrycurve-symmetry-leaneuclidean-geometry

Let m≥2m\ge2 be an integer and let α,β∈C\alpha,\beta\in\mathbb C with α≠αˉ\alpha\ne\bar\alpha and β≠βˉ\beta\ne\bar\beta; no condition on their moduli is imposed. For γ∈{α,β}\gamma\in\{\alpha,\beta\} let C^m,γ\widehat C_{m,\gamma} be the closure in the Riemann sphere C^=C∪{∞}\widehat{\mathbb C}=\mathbb C\cup\{\infty\} of Cm,γ={z∈C:Re⁡(zm(∣z∣2+γ))=0}C_{m,\gamma}=\{z\in\mathbb C:\operatorname{Re}\bigl(z^m(|z|^2+\gamma)\bigr)=0\}. Let g=(gij)∈GL2(C)g=(g_{ij})\in\mathrm{GL}_2(\mathbb C) be any invertible matrix and Mg(z)=g11z+g12g21z+g22M_g(z)=\frac{g_{11}z+g_{12}}{g_{21}z+g_{22}} the Möbius transformation of C^\widehat{\mathbb C} it defines, with the usual conventions at the pole and at ∞\infty. For c∈Cc\in\mathbb C let δc,sc:C^→C^\delta_c,s_c:\widehat{\mathbb C}\to\widehat{\mathbb C} be given by δc(z)=cz\delta_c(z)=cz for z∈Cz\in\mathbb C, δc(∞)=∞\delta_c(\infty)=\infty, and by sc(z)=c/zs_c(z)=c/z for z∈C∖{0}z\in\mathbb C\setminus\{0\}, sc(0)=∞s_c(0)=\infty, sc(∞)=0s_c(\infty)=0.

Assume that Mg(C^m,α)⊆C^m,βM_g(\widehat C_{m,\alpha})\subseteq\widehat C_{m,\beta}. Then there is c∈Cc\in\mathbb C with c≠0c\ne0 such that

Mg(p)=δc(p)  for all p∈C^orMg(p)=sc(p)  for all p∈C^.M_g(p)=\delta_c(p)\ \text{ for all } p\in\widehat{\mathbb C}\qquad\text{or}\qquad M_g(p)=s_c(p)\ \text{ for all } p\in\widehat{\mathbb C}.

In the proof of Theorem 2 every equivalence is reduced to one of the forms czcz, c/zc/z, czˉc\bar z, c/zˉc/\bar z; this is the holomorphic case. Only an inclusion is assumed, the matrix is arbitrary, and no preservation of the pair {0,∞}\{0,\infty\} is assumed. The coefficient comparison (8) then leads to the symmetry group (3) of Theorem 2.

Formalization Note: the sphere is OnePoint ℂ with Mathlib's action of GL (Fin 2) ℂ, and both forms are asserted at every point of OnePoint ℂ, including 00 and ∞\infty.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Mathlib.Algebra.MvPolynomial.Nilpotent
import Mathlib.Algebra.MvPolynomial.NoZeroDivisors
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.Complex.OperatorNorm
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.Analysis.Normed.Affine.MazurUlam
import Mathlib.Data.Complex.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.Tactic
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Compactification.OnePoint.ProjectiveLine

open CurveSymmetry
set_option autoImplicit false
open OnePoint
Formal statement
theorem CurveSymmetry.family_mobius_complete {m : ℕ} (hm : 2 ≤ m) {α β : ℂ}
    (ha : α ≠ star α) (hb : β ≠ star β) (g : MobiusMatrix)
    (hmap : ∀ p ∈ sphericalFamily m α, g • p ∈ sphericalFamily m β) :
    ∃ c : ℂ, c ≠ 0 ∧ ((∀ p : Sphere, g • p = sphereDilation c p) ∨
      (∀ p : Sphere, g • p = sphereInversion c p)) := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note), Theorem 2, p. 2, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/FamilyGlobalSingularities.lean (C. Perassi)
Human review

  • Endorsed by Shuze Chen · Oct 7, 2026

    Confirmed by the moderator at approval.

  • Endorsed by carlok · Oct 7, 2026

    Confirmed by the mission captain (proposal self-audit).

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