At a point $(c,d)$ of $w^2=h(t)$ with $h(c)\ne0$ , $f\,dt=f\,du$ for some uniformizer $u$ of the local ring

Proved

CurveSymmetry.quad_unramified_coeff

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

curve-symmetry-leanfunction-fieldsgenus

Let $h\in\mathbb C[t]$ be a squarefree polynomial such that $W^2-h$ is irreducible in $\mathbb C(t)[W]$ . Let $K=\mathbb C(t)[W]/(W^2-h)$ be the function field of the double cover $w^2=h(t)$ , where $t$ and $w$ denote the classes of $t$ and $W$ , and let $A=\mathbb C[t][W]/(W^2-h)$ be its coordinate ring, viewed inside $K$ . For a point $(c,d)\in\mathbb C^2$ of the cover, $d^2=h(c)$ , let $\mathfrak m_{(c,d)}\subset A$ be the kernel of the evaluation $A\to\mathbb C$ , $t\mapsto c$ , $w\mapsto d$ , and let $\mathcal O_{(c,d)}=A_{\mathfrak m_{(c,d)}}\subset K$ be the local ring at $(c,d)$ , the valuation ring of the point place $(c,d)$ of $K$ ; it is a discrete valuation ring with maximal ideal $\mathfrak m_{(c,d)}\mathcal O_{(c,d)}$ . A uniformizer is an element of $\mathcal O_{(c,d)}$ generating this maximal ideal. Write $\Omega_{K/\mathbb C}$ for the $K$ -vector space of Kähler differentials of $K$ over $\mathbb C$ and $dx$ for the differential of $x\in K$ .

Let $(c,d)\in\mathbb C^2$ with $d^2=h(c)$ and $h(c)\ne0$ , and let $f\in K$ . Then there exists a uniformizer $u$ of $\mathcal O_{(c,d)}$ such that

f\,dt=f\,du\quad\text{in }\Omega_{K/\mathbb C}.

Thus the coefficient of $f\,dt$ against a uniformizer at an unramified point place can be taken to be $f$ itself: $dt$ has order zero there. It belongs to the local study of differentials on double covers $w^2=h(t)$ that the formalization carries out for the genus clause of Lemma 4 of the note (the normalization of $V_\alpha$ , a double cover of this kind, has genus $m$ ).

Formalization Note: the hypotheses that $h$ is squarefree and that $W^2-h$ is irreducible are Fact instances, and $d^2=h(c)$ is an explicit hypothesis. $\mathcal O_{(c,d)}$ is a subalgebra of $K$ : the localization of $A$ at the kernel of the evaluation map.

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 Definitions.Def_CurveSymmetry_08_Differentials
import Definitions.Def_CurveSymmetry_09_Genus
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.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Defs
import Mathlib.LinearAlgebra.TensorProduct.Basis
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DiscreteValuationRing.TFAE
import Mathlib.RingTheory.Etale.Field
import Mathlib.RingTheory.Etale.Kaehler
import Mathlib.RingTheory.Kaehler.Basic
import Mathlib.RingTheory.Kaehler.Polynomial
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.Module
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.RingTheory.MvPolynomial.Homogeneous
import Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic
import Mathlib.RingTheory.Nakayama
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
variable (h : ℂ[X]) [Fact (Squarefree h)] [Fact (Irreducible (quadRat h))]
  (c d : ℂ) (hd : d ^ 2 = h.eval c)

Formal statement

theorem CurveSymmetry.quad_unramified_coeff (hc : h.eval c ≠ 0) (f : QuadField h) :
    ∃ u : quadLocalRing h c d hd,
      IsLocalRing.maximalIdeal (quadLocalRing h c d hd) = Ideal.span {u} ∧
        f • KaehlerDifferential.D ℂ (QuadField h) (quadT h) =
          f • KaehlerDifferential.D ℂ (QuadField h) (u : QuadField h) := by sorry

Source

C. Perassi, Sharp symmetry bounds for real algebraic curves (note): lemma of the formalization, for Lemma 4, pp. 3-4, https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/sharp_symmetry_bounds.pdf. Lean: https://github.com/carlok/curve-symmetry-lean/blob/d99bc17a1c397956c05d7417de50f9beed56580f/lean/PlaceUniformizers.lean (C. Perassi)

View graph

At a point (c,d)(c,d)(c,d) of w2=h(t)w^2=h(t)w2=h(t) with h(c)≠0h(c)\ne0h(c)=0, f dt=f duf\,dt=f\,dufdt=fdu for some uniformizer uuu of the local ring

At a point $(c,d)$ of $w^2=h(t)$ with $h(c)\ne0$ , $f\,dt=f\,du$ for some uniformizer $u$ of the local ring