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MODULAR SYMBOLS

 
Acknowledgements
 
Introduction
      Modular Symbols
 
Basics
      Verbose Output
      Categories
 
Creation Functions
      Ambient Spaces
      Labels
      Creation of Elements
 
Bases
 
Associated Vector Space
 
Degeneracy Maps
 
Decomposition
 
Subspaces
 
Twists
 
Operators
 
The Hecke Algebra
 
The Intersection Pairing
 
q-Expansions
 
Special Values of L-functions
      Winding Elements
 
The Associated Complex Torus
      The Period Map
      Projection Mappings
 
Modular Abelian Varieties
      Modular Degree and Torsion
      Tamagawa Numbers and Orders of Component Groups
 
Elliptic Curves
 
Dimension Formulas
 
Bibliography







DETAILS

 
Introduction

      Modular Symbols

 
Basics

      Verbose Output

      Categories
            Example ModSym_Creation (H133E1)

 
Creation Functions

      Ambient Spaces
            ModularSymbols(N) : RngIntElt -> ModSym
            ModularSymbols(N, k) : RngIntElt, RngIntElt -> ModSym
            ModularSymbols(N, k, F) : RngIntElt, RngIntElt, Fld -> ModSym
            ModularSymbols(N, k, sign) : RngIntElt, RngIntElt, RngIntElt -> ModSym
            ModularSymbols(N, k, F, sign) : RngIntElt, RngIntElt, Fld, RngIntElt -> ModSym
            ModularSymbols(eps, k) : GrpDrchElt, RngIntElt -> ModSym
            ModularSymbols(eps, k, sign) : GrpDrchElt, RngIntElt, RngIntElt -> ModSym
            Example ModSym_Creation-Ambient (H133E2)

      Labels
            ModularSymbols(s, sign) : MonStgElt, RngIntElt -> ModSym
            Example ModSym_Creation-Spaces (H133E3)

      Creation of Elements
            Example ModSym_Creation-Elements-1 (H133E4)
            Example ModSym_RepresentationConversion (H133E5)
            M ! x : ModSym, . -> ModSymElt
            ConvertFromManinSymbol(M, x) : ModSym, Tup -> ModSymElt
            ManinSymbol(x) : ModSymElt -> SeqEnum
            Example ModSym_Creation-Elements-2 (H133E6)

 
Bases
      Basis(M) : ModSym -> SeqEnum
      IntegralBasis(M) : ModSym -> Lat
      Example ModSym_IntegralBasis (H133E7)

 
Associated Vector Space
      VectorSpace(M) : ModSym -> ModTupFld, Map, Map
      DualVectorSpace(M) : ModSym -> ModTupFld
      Lattice(M) : ModSym -> Lat
      Example ModSym_Representation (H133E8)

 
Degeneracy Maps
      DegeneracyMap(M1, M2, d) : ModSym, ModSym, RngIntElt -> Map
      DegeneracyMatrix(M1, M2, d) : ModSym, ModSym, RngIntElt -> AlgMatElt
      ModularSymbols(M, N') : ModSym, RngIntElt -> ModSym
      M1 !! M2 : ModSym, ModSym -> ModSym
      Example ModSym_Coercion-spaces (H133E9)

 
Decomposition
      Decomposition(M, bound : parameters) : ModSym, RngIntElt -> SeqEnum
      NewformDecomposition(M : parameters) : ModSym -> SeqEnum
      AssociatedNewSpace(M) : ModSym -> ModSym
      SortDecomposition(D) : [ModSym] -> SeqEnum
      IsIrreducible(M) : ModSym -> BoolElt
      M1 lt M2 : ModSym, ModSym -> BoolElt
      Example ModSym_Decomposition (H133E10)

 
Subspaces
      CuspidalSubspace(M) : ModSym -> ModSym
      IsCuspidal(M) : ModSym -> BoolElt
      EisensteinSubspace(M) : ModSym -> ModSym
      IsEisenstein(M) : ModSym -> BoolElt
      NewSubspace(M) : ModSym-> ModSym
      IsNew(M) : ModSym -> BoolElt
      NewSubspace(M, p) : ModSym, RngIntElt -> ModSym
      Kernel(I, M) : [Tup], ModSym -> ModSym
      Complement(M) : ModSym -> ModSym
      BoundaryMap(M) : ModSym -> ModMatFldElt
      Example ModSym_Subspaces (H133E11)
      Example ModSym_BoundaryMap (H133E12)

 
Twists
      IsTwist(M1, M2, p) : ModSym, ModSym, RngIntElt -> BoolElt, GrpDrchElt
      IsMinimalTwist(M, p) : ModSym, RngIntElt -> BoolElt, ModSym, GrpDrchElt
      Example ModSym_example-twists (H133E13)

 
Operators
      Example ModSym_HeckeOperators (H133E14)
      HeckeOperator(M, n) : ModSym, RngIntElt -> AlgMatElt
      HeckePolynomial(M, n) : ModSym, RngIntElt -> RngUPolResElt
      IntegralHeckeOperator(M, n) : ModSym, RngIntElt -> AlgMatElt
      DualHeckeOperator(M, n) : ModSym, RngIntElt -> AlgMatElt
      AtkinLehner(M, q) : ModSym, RngIntElt -> AlgMatElt
      DualAtkinLehner(M, q) : ModSym, RngIntElt -> AlgMatElt
      StarInvolution(M) : ModSym -> AlgMatElt
      DualStarInvolution(M) : ModSym -> AlgMatElt
      ThetaOperator(M1, M2) : ModSym, ModSym -> Map
      Example ModSym_Operators (H133E15)
      Example ModSym_ThetaOperator (H133E16)

 
The Hecke Algebra
      HeckeBound(M) : ModSym -> RngIntElt
      SetHeckeBound(M, n) : ModSym, RngIntElt -> RngIntElt
      HeckeAlgebra(M : Bound) : ModSym -> AlgMat
      DiscriminantOfHeckeAlgebra(M : Bound) : ModSym -> RngIntElt
      HeckeEigenvalueRing(M : parameters) : ModSym -> Rng, Map
      HeckeEigenvalueField(M) : ModSym -> Fld, Map
      Example ModSym_HeckeAlgebra (H133E17)

 
The Intersection Pairing
      IntersectionPairing(x, y) : ModSymElt, ModSymElt -> FldRatElt
      Example ModSym_IntersectionPairing (H133E18)

 
q-Expansions
      Eigenform(M, prec) : ModSym, RngIntElt -> RngSerPowElt
      qExpansionBasis(M, prec : parameters) : ModSym, RngIntElt -> SeqEnum
      qIntegralBasis(M) : ModSym -> SeqEnum
      SystemOfEigenvalues(M, prec) : ModSym, RngIntElt -> SeqEnum
      Example ModSym_qExpansions (H133E19)

 
Special Values of L-functions
      LSeries(M, j, prec) : ModSym, RngIntElt, RngIntElt -> FldPrElt
      LSeriesLeadingCoefficient(M, j, prec) : ModSym, RngIntElt, RngIntElt -> FldPrElt, RngIntElt
      RealVolume(M, prec) : ModSym, RngIntElt -> FldPrElt
      MinusVolume(M, prec) : ModSym, RngIntElt -> FldPrElt
      LRatio(M, j : parameters) : ModSym, RngIntElt -> FldRatElt
      LRatioOddPart(M, j) : ModSym, RngIntElt -> FldRatElt
      Example ModSym_LSeries (H133E20)

      Winding Elements
            WindingElement(M) : ModSym -> ModSymElt
            WindingElement(M, i) : ModSym, RngIntElt -> ModSymElt
            TwistedWindingElement(M, i, eps) : ModSym, RngIntElt, GrpDrchElt -> ModSymElt
            WindingLattice(M, j : parameters) : ModSym, RngIntElt -> Lat
            WindingSubmodule(M, j : parameters) : ModSym, RngIntElt -> ModTupFld
            TwistedWindingSubmodule(M, j, eps) : ModSym, RngIntElt, GrpDrchElt -> ModTupFld

 
The Associated Complex Torus
      SubgroupOfTorus(M, x) : ModSym, ModSymElt -> RngIntElt
      SubgroupOfTorus(M, s) : ModSym, SeqEnum -> GrpAb
      Example ModSym_CuspidalSubgroup (H133E21)
      Example ModSym_CuspidalSubgroupTable (H133E22)
      ModularKernel(M) : ModSym -> GrpAb
      CongruenceGroup(M : parameters) : ModSym -> GrpAb
      IntersectionGroup(M1, M2) : ModSym, ModSym -> GrpAb
      IntersectionGroup(S) : SeqEnum -> GrpAb
      Example ModSym_BSD (H133E23)

      The Period Map
            PeriodMapping(M, prec) : ModSym, RngIntElt -> Map
            Periods(M, prec) : ModSym, RngIntElt -> SeqEnum
            ClassicalPeriod(M, j, prec) : ModSym, RngIntElt, RngIntElt -> FldPrElt

      Projection Mappings
            RationalMapping(M) : ModSym -> Map
            IntegralMapping(M) : ModSym -> Map
            Example ModSym_ModularAbVarRational (H133E24)

 
Modular Abelian Varieties

      Modular Degree and Torsion
            ModularDegree(M) : ModSym -> RngIntElt
            CongruenceModulus(M : parameters) : ModSym -> RngIntElt
            TorsionBound(M, maxp) : ModSym, RngIntElt -> RngIntElt
            Example ModSym_ModularAbVarArithmetic (H133E25)

      Tamagawa Numbers and Orders of Component Groups
            ComponentGroupOrder(M, p) : ModSym, RngIntElt -> RngIntElt
            TamagawaNumber(M, p) : ModSym, RngIntElt -> RngIntElt
            RealTamagawaNumber(M) : ModSym -> RngIntElt
            MinusTamagawaNumber(M) : ModSym -> RngIntElt
            Example ModSym_ModularAbVarCompGrp (H133E26)

 
Elliptic Curves
      ModularSymbols(E) : CrvEll -> ModSym
      EllipticCurve(M) : ModSym -> CrvEll
      pAdicLSeries(E, p) : CrvEll, RngIntElt -> RngSerPowElt
      Example ModSym_BSD389A (H133E27)

 
Dimension Formulas
      DimensionCuspFormsGamma0(N, k) : RngIntElt, RngIntElt -> RngIntElt
      DimensionNewCuspFormsGamma0(N, k) : RngIntElt, RngIntElt -> RngIntElt
      DimensionCuspFormsGamma1(N, k) : RngIntElt, RngIntElt -> RngIntElt
      DimensionNewCuspFormsGamma1(N, k) : RngIntElt, RngIntElt -> RngIntElt
      DimensionCuspForms(eps, k) : GrpDrchElt, RngIntElt -> RngIntElt
      Example ModSym_DimensionFormulas (H133E28)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013