Small Modular Curves

• Introduction
• Small Modular Curve Models
  • SmallModularCurve(N): RngIntElt → Crv
  • SmallModularCurve(N,K): RngIntElt, Rng → Crv
  • IsInSmallModularCurveDatabase(N): RngIntElt → Boolelt
  • Example: Models from the Small Modular Curve Database
• Projection Maps
  • ProjectionMap(CN,N,CM,M): Crv, RngIntElt, Crv, RngIntElt → MapSch
  • ProjectionMap(CN,N,CM,M,r): Crv, RngIntElt, Crv, RngIntElt, RngIntElt → MapSch
  • Example: Projection Maps on Small Modular Curves
• Automorphisms
  • AtkinLehnerInvolution(CN,N,d): Crv, RngIntElt, RngIntElt → MapAutSch
  • SrAutomorphism(CN,N,r,u): Crv, RngIntElt, RngIntElt, RngElt → MapAutSch
  • ExtraAutomorphism(CN,N,u): Crv, RngIntElt, RngElt → MapAutSch
  • AutomorphismGroupOverQ(CN,N): Crv, RngIntElt → GrpAutCrv
  • AutomorphismGroupOverCyclotomicExtension(CN,N,n): Crv, RngIntElt, RngIntElt → GrpAutCrv
  • AutomorphismGroupOverExtension(CN,N,n,u): Crv, RngIntElt, RngIntElt, RngElt → GrpAutCrv
  • Example: Automorphisms of Small Modular Curves
• Cusps and Rational Points
  • Cusp(CN,N,d): Crv, RngIntElt, RngIntElt → Any
  • CuspIsSingular(N,d): RngIntElt, RngIntElt → BoolElt
  • CuspPlaces(CN,N,d): Crv, RngIntElt, RngIntElt → SeqEnum[PlcCrvElt]
  • NonCuspidalQRationalPoints(CN,N): Crv, RngIntElt → SeqEnum
  • Example: Cusps on Small Modular Curves
• Standard Functions and Forms
  • jInvariant(CN,N): Crv, RngIntElt → FldFunRatMElt
  • jFunction(CN,N): Crv, RngIntElt → FldFunFracSchElt
  • jInvariant(p,N): Pt, RngIntElt → RngElt
  • jInvariant(p,N): PlcCrvElt, RngIntElt → RngElt
  • jNInvariant(p,N): Pt, RngIntElt → RngElt
  • jNInvariant(p,N): PlcCrvElt, RngIntElt → RngElt
  • E2NForm(CN,N): Crv, RngIntElt → DiffCrvElt
  • E4Form(CN,N): Crv, RngIntElt → FldFunFracSchElt, DiffCrvElt
  • E6Form(CN,N): Crv, RngIntElt → FldFunFracSchElt, DiffCrvElt
• Parametrized Structures
  • SubgroupScheme(p,N): Pt, RngIntElt → SchGrpEll, CrvEll
  • SubgroupScheme(p,N): PlcCrvElt, RngIntElt → SchGrpEll, CrvEll
  • SubgroupScheme(p,N,E): Pt, RngIntElt, CrvEll → SchGrpEll
  • SubgroupScheme(p,N,E): PlcCrvElt, RngIntElt, CrvEll → SchGrpEll
  • Isogeny(p,N): Pt, RngIntElt → MapSch
  • Isogeny(p,N): PlcCrvElt, RngIntElt → MapSch
  • Isogeny(p,N,E): Pt, RngIntElt, CrvEll → MapSch
  • Isogeny(p,N,E): PlcCrvElt, RngIntElt, CrvEll → MapSch
  • Example: Rational Points and j-Invariants on X0(14)
• Modular Generators and \(q\)-Expansions
  • qExpansionExpressions(N): RngIntElt
  • qExpansionsOfGenerators(N,R,r): RngIntElt, RngSerLaur, RngIntElt → SeqEnum
  • Example: q-Expansions of Modular Generators
• Extended Example
  • Example: Rational Points on a Quotient of X0(45)
• Bibliography