Modular Curves

• Introduction
• Creation Functions
  • Creation of a Modular Curve
    • ModularCurve(X,t,N): Sch, MonStgElt, RngIntElt → CrvMod
    • ModularCurve(D, N): DB, RngIntElt → CrvMod
  • Creation of Points
    • ModuliPoints(X,E): CrvMod, CrvEll → SeqEnum
    • Example: Moduli Points
• Invariants
  • Level(X): CrvMod → RngIntElt
  • Genus(X): CrvMod → RngIntElt
  • ModelType(X): CrvMod → MonStgElt
  • Indices(X): CrvMod → SeqEnum
• Modular Polynomial Databases
  • AtkinModularPolynomial(N): RngIntElt → RngMPolElt
  • CanonicalModularPolynomial(N): RngIntElt → RngMPolElt
  • ClassicalModularPolynomial(N): RngIntElt → RngMPolElt
  • ModularCurveDatabase(t): MonStgElt → DB
  • ModularCurveDatabase(t,i): MonStgElt, RngIntElt → DB
  • N in D: RngIntElt, DB → BoolElt
  • ExistsModularCurveDatabase(t): MonStgElt → BoolElt
  • ExistsModularCurveDatabase(t,i): MonStgElt, RngIntElt → BoolElt
  • Example: Comparing Modular Polynomial Models
• Parametrized Structures
  • Isogeny(E,P): CrvEll, Pt → MapCrvEll
  • SubgroupScheme(E,P): CrvEll, Pt → CrvEllSubgroup
  • Example: Parametrized Subgroup Schemes
• Associated Structures
  • FunctionField(X): CrvMod → FldFun
  • jFunction(X): CrvMod → FldFunElt
  • BaseCurve(X): CrvMod → CrvMod, MapSch
  • Example: Base Curve of a Modular Curve
• Automorphisms
  • CanonicalInvolution(X): CrvMod → MapSch
  • AtkinLehnerInvolution(X,N): CrvMod, RngIntElt → MapSch
• Class Polynomials
  • HilbertClassPolynomial(D): RngIntElt → RngUPolElt
  • WeberClassPolynomial(D): RngIntElt → RngUPolElt, FldFunRatUElt
  • WeberToHilbertClassPolynomial(f,D): RngUPolElt, RngIntElt → RngUPolElt
  • Example: Standard Class Polynomials
• Modular Curves and Quotients (Canonical Embeddings)
  • ModularCurveQuotient(N,A): RngIntElt, [RngIntElt] → Crv
  • X0NQuotient(N,A): RngIntElt, [RngIntElt] → Crv
  • X0Nplus(N): RngIntElt → Crv
  • X0Nstar(N): RngIntElt → Crv
  • GenusX0N(N): RngIntElt → RngIntElt
  • GenusX1N(N): RngIntElt → RngIntElt
  • GenusX0NQuotient(N, As): RngIntElt, [RngIntElt] → RngIntElt
  • AtkinLehnerNumberOfFixedPoints(N, A): RngIntElt, RngIntElt → RngIntElt
  • Example: Quotient of X0(N) by Atkin-Lehner Involutions
• Modular Curves of Given Level and Genus
  • SetVerbose("ModularCurve", v): MonStgElt, RngIntElt
  • NewModularHyperellipticCurves(N, g): RngIntElt, RngIntElt → SeqEnum
  • NewModularHyperellipticCurve(B): [ModSym] → BoolElt, RngUPol
  • NewModularHyperellipticCurve(F): [RngSerPowElt] → BoolElt, RngUPol
  • ModularHyperellipticCurve(B): [ModSym] → BoolElt, RngUPol
  • ModularHyperellipticCurve(F): [RngSerPowElt] → BoolElt, RngUPol
  • NewModularNonHyperellipticCurvesGenus3(N): RngIntElt → SeqEnum
  • NewModularNonHyperellipticCurveGenus3(B): [ModSym] → BoolElt, RngMPolElt
  • NewModularNonHyperellipticCurveGenus3(F): [RngSerPowElt] → BoolElt, RngMPolElt
  • ModularNonHyperellipticCurveGenus3(F): [RngSerPowElt] → BoolElt, RngMPolElt
  • Example: Hyperelliptic Modular Curve X1(13)
  • Example: Hyperelliptic Intermediate Modular Curve
  • Example: Non-Hyperelliptic Curve of Level 97
  • Example: Genus 3 Curves from X1(20)
• Bibliography