Regularity of $f\,dt$ at a point $(c,d)$ of $w^2=h(t)$ with $h(c)=0$ : exactly when $fw$ is in the local ring

Proved

CurveSymmetry.isRegularAt_ramified

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 $x\in K$ . A differential $\omega\in\Omega_{K/\mathbb C}$ is regular at $(c,d)$ if for every uniformizer $u$ of $\mathcal O_{(c,d)}$ and every $g\in K$ with $\omega=g\,du$ one has $g\in\mathcal O_{(c,d)}$ .

Let $(c,d)\in\mathbb C^2$ with $d^2=h(c)$ and $h(c)=0$ , and let $f\in K$ . Then

f\,dt\text{ is regular at }(c,d)\iff f\,w\in\mathcal O_{(c,d)}.

This is the regularity criterion at the ramified point places; since $w$ is a uniformizer there, $f$ may have at most a simple pole. Applied to $h_\alpha(t)=-t(t^m+1)(\alpha t^m+\overline\alpha)$ , together with the criteria at the unramified point places and at the place over $t=\infty$ , it determines which differentials $f\,dt$ of the function field of $w^2=h_\alpha(t)$ are regular at every place; the formalization obtains from this the basis $t^i\,dt/w$ , $0\le i<m$ , of these differentials, hence the genus clause of Lemma 4 of the note (genus $m$ for the normalization of $V_\alpha$ ), which is used for $m=2$ in Remark 5.

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.isRegularAt_ramified (hc : h.eval c = 0) (f : QuadField h) :
    IsRegularAt h c d hd (f • KaehlerDifferential.D ℂ (QuadField h) (quadT h)) ↔
      f * AdjoinRoot.root (quadRat h) ∈ quadLocalRing h c d hd := 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/HolomorphicDifferentials.lean (C. Perassi)

View graph

Regularity of f dtf\,dtfdt 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)=0h(c)=0: exactly when fwfwfw is in the local ring

Regularity of $f\,dt$ at a point $(c,d)$ of $w^2=h(t)$ with $h(c)=0$ : exactly when $fw$ is in the local ring