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MODULAR CURVES

 
Acknowledgements
 
Introduction
 
Creation Functions
      Creation of a Modular Curve
      Creation of Points
 
Invariants
 
Modular Polynomial Databases
 
Parametrized Structures
 
Associated Structures
 
Automorphisms
 
Class Polynomials
 
Modular Curves and Quotients (Canonical Embeddings)
 
Modular Curves of Given Level and Genus
 
Bibliography







DETAILS

 
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 CrvMod_Moduli points (H128E1)

 
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
      N in D: RngIntElt, DB -> BoolElt
      ExistsModularCurveDatabase(t) : MonStgElt -> BoolElt
      Example CrvMod_Modular polynomials (H128E2)

 
Parametrized Structures
      Isogeny(E,P) : CrvEll, Pt -> MapCrvEll
      SubgroupScheme(E,P) : CrvEll, Pt -> CrvEllSubgroup
      Example CrvMod_Parametrized subgroup schemes (H128E3)

 
Associated Structures
      FunctionField(X) : CrvMod -> FldFun
      jFunction(X) : CrvMod -> FldFunElt
      BaseCurve(X) : CrvMod -> CrvMod, MapSch
      Example CrvMod_Modular base curve (H128E4)

 
Automorphisms
      CanonicalInvolution(X) : CrvMod -> MapSch

 
Class Polynomials
      HilbertClassPolynomial(D) : RngIntElt -> RngUPolElt
      WeberClassPolynomial(D) : RngIntElt -> RngUPolElt, FldFunRatUElt
      WeberToHilbertClassPolynomial(f,D) : RngUPolElt, RngIntElt -> RngUPolElt
      Example CrvMod_Standard class polynomials (H128E5)

 
Modular Curves and Quotients (Canonical Embeddings)
      ModularCurveQuotient(N,A) : RngIntElt, [RngIntElt] -> Crv
      Example CrvMod_x0n-quotient (H128E6)

 
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, RngIntElt -> SeqEnum
      NewModularNonHyperellipticCurveGenus3(B) : [ModSym] -> BoolElt, RngMPolElt
      NewModularNonHyperellipticCurveGenus3(F) : [RngSerPowElt] -> BoolElt, RngMPolElt
      ModularNonHyperellipticCurveGenus3(F) : [RngSerPowElt] -> BoolElt, RngMPolElt
      Example CrvMod_jim_ex_mod_crvs_hyp (H128E7)
      Example CrvMod_jim_ex_mod_crvs_jeon (H128E8)
      Example CrvMod_jim_ex_mod_crvs_jeon (H128E9)
      Example CrvMod_jim_ex_mod_crvs_gen3 (H128E10)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012