Prove2Me

Places of C(t)[W]/(W2−hα)\mathbb C(t)[W]/(W^2-h_\alpha) containing C[t]\mathbb C[t] are the points of w2=hα(t)w^2=h_\alpha(t), bijectively

Proved
CurveSymmetry.family_finite_place_classification

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

curve-symmetry-leanfunction-fieldsvaluation-theory

Let m≥1m\ge 1 be an integer and α∈C\alpha\in\mathbb C with α≠αˉ\alpha\ne\bar\alpha (that is, α\alpha is not real), and let hα(t)=−t (tm+1) (αtm+αˉ)∈C[t]h_\alpha(t)=-t\,(t^m+1)\,(\alpha t^m+\bar\alpha)\in\mathbb C[t]. Let Lα=C(t)[W]/(W2−hα(t))L_\alpha=\mathbb C(t)[W]/(W^2-h_\alpha(t)) and Rα=C[t][W]/(W2−hα(t))R_\alpha=\mathbb C[t][W]/(W^2-h_\alpha(t)), mapped into LαL_\alpha by sending WW to the class ww of WW. For c,d∈Cc,d\in\mathbb C with d2=hα(c)d^2=h_\alpha(c), let mc,d⊂Rα\mathfrak m_{c,d}\subset R_\alpha be the kernel of the evaluation t↦ct\mapsto c, W↦dW\mapsto d, and let Oc,d={a/s: a,s∈Rα, s∉mc,d}⊆Lα\mathcal O_{c,d}=\{a/s:\ a,s\in R_\alpha,\ s\notin\mathfrak m_{c,d}\}\subseteq L_\alpha be the local ring of the point (c,d)(c,d). Then:

  1. for all c,dc,d with d2=hα(c)d^2=h_\alpha(c), Oc,d≠Lα\mathcal O_{c,d}\ne L_\alpha and Oc,d\mathcal O_{c,d} contains C[t]\mathbb C[t];
  2. every valuation subring O\mathcal O of LαL_\alpha with O≠Lα\mathcal O\ne L_\alpha and C[t]⊆O\mathbb C[t]\subseteq\mathcal O equals Oc,d\mathcal O_{c,d} for some c,d∈Cc,d\in\mathbb C with d2=hα(c)d^2=h_\alpha(c);
  3. if d2=hα(c)d^2=h_\alpha(c), d′2=hα(c′)d'^2=h_\alpha(c') and Oc,d=Oc′,d′\mathcal O_{c,d}=\mathcal O_{c',d'}, then c=c′c=c' and d=d′d=d'.

Hence

{(c,d)∈C2: d2=hα(c)} → ∼  {O⊊Lα valuation subring: C[t]⊆O},(c,d)↦Oc,d.\{(c,d)\in\mathbb C^2:\ d^2=h_\alpha(c)\}\ \xrightarrow{\ \sim\ }\ \{\mathcal O\subsetneq L_\alpha\ \text{valuation subring}:\ \mathbb C[t]\subseteq\mathcal O\},\qquad (c,d)\mapsto\mathcal O_{c,d}.

LαL_\alpha is the function field of VαV_\alpha in the form of the double cover (7) from the proof of Lemma 4; the statement lists its places over the finite tt-line, one for each point of the affine curve w2=hα(t)w^2=h_\alpha(t).

Formalization Note: m>0m>0 and α≠αˉ\alpha\ne\bar\alpha are Fact instances, from which the squarefreeness of hαh_\alpha and the irreducibility of W2−hα(t)W^2-h_\alpha(t), needed to build Oc,d\mathcal O_{c,d}, are obtained.

Preamble
import Definitions.Def_CurveSymmetry_01_RealLoci
import Definitions.Def_CurveSymmetry_02_FamilyBasics
import Definitions.Def_CurveSymmetry_03_IsometriesAndCharts
import Definitions.Def_CurveSymmetry_05_FamilyFunctionField
import Definitions.Def_CurveSymmetry_06_QuadraticRing
import Definitions.Def_CurveSymmetry_07_Places
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.Algebra.Polynomial.SpecificDegree
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.RatFunc.Basic
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DiscreteValuationRing.TFAE
import Mathlib.RingTheory.Localization.AsSubring
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.RingTheory.Valuation.ValuationSubring
import Mathlib.Tactic

open CurveSymmetry
set_option autoImplicit false
open Polynomial
Formal statement
theorem CurveSymmetry.family_finite_place_classification {m : ℕ} {α : ℂ} [Fact (0 < m)]
    [Fact (α ≠ star α)] :
    (∀ (c d : ℂ) (hd : d ^ 2 = (familyH m α).eval c), quadPlace (familyH m α) c d hd ≠ ⊤ ∧
        ∀ p : ℂ[X], algebraMap ℂ[X] (QuadField (familyH m α)) p ∈
          quadPlace (familyH m α) c d hd) ∧
      (∀ O : ValuationSubring (QuadField (familyH m α)), O ≠ ⊤ →
        (∀ p : ℂ[X], algebraMap ℂ[X] (QuadField (familyH m α)) p ∈ O) →
          ∃ (c d : ℂ) (hd : d ^ 2 = (familyH m α).eval c), O = quadPlace (familyH m α) c d hd) ∧
      (∀ (c d c' d' : ℂ) (hd : d ^ 2 = (familyH m α).eval c)
        (hd' : d' ^ 2 = (familyH m α).eval c'),
        quadPlace (familyH m α) c d hd = quadPlace (familyH m α) c' d' hd' → c = c' ∧ d = d') := by sorry
Source
C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for proof of Lemma 4, equation (7), p. 3, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/QuadraticPlaces.lean (C. Perassi)

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