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SMALL MODULAR CURVES

 
Acknowledgements
 
Introduction
 
Small Modular Curve Models
 
Projection Maps
 
Automorphisms
 
Cusps and Rational Points
 
Standard Functions and Forms
 
Parametrized Structures
 
Modular Generators and q-Expansions
 
Extended Example
 
Bibliography







DETAILS

 
Introduction

 
Small Modular Curve Models
      SmallModularCurve(N) : RngIntElt -> Crv
      IsInSmallModularCurveDatabase(N) : RngIntElt -> Boolelt
      Example SmallModCrv_sm_mod_crvs_basic_ex (H129E1)

 
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 SmallModCrv_sm_mod_crvs_proj_ex (H129E2)

 
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 SmallModCrv_sm_mod_crvs_auto_ex (H129E3)

 
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 SmallModCrv_sm_mod_crvs_cusps (H129E4)

 
Standard Functions and Forms
      jInvariant(CN,N) : Crv, RngIntElt -> FldFunRatMElt
      jInvariant(p,N) : Pt, RngIntElt -> RngElt
      jNInvariant(p,N) : Pt, 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
      Isogeny(p,N) : Pt, RngIntElt -> MapSch
      Example SmallModCrv_sm_mod_crvs_prms (H129E5)

 
Modular Generators and q-Expansions
      qExpansionExpressions(N) : RngIntElt ->
      qExpansionsOfGenerators(N,R,r) : RngIntElt, RngSerLaur, RngIntElt -> SeqEnum
      Example SmallModCrv_q_exps_ex_sm_mod_crvs (H129E6)

 
Extended Example
      Example SmallModCrv_big_ex (H129E7)

 
Bibliography

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