Modular Symbols

• Introduction
  • Modular Symbols
• Basics
  • Verbose Output
  • Categories
    • Example: Basis of Modular Symbols of Level 11
• 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: Creating Ambient Spaces of Modular Symbols
  • Labels
    • ModularSymbols(s, sign): MonStgElt, RngIntElt → ModSym
    • ModularSymbols(s): MonStgElt → ModSym
    • Example: Modular Symbols Spaces from Labels
  • Creation of Elements
    • Example: Entering Modular Symbols over a Finite Field
    • Example: Converting Between Manin and Modular Symbols
    • M ! x: ModSym, . → ModSymElt
    • ConvertFromManinSymbol(M, x): ModSym, Tup → ModSymElt
    • ManinSymbol(x): ModSymElt → SeqEnum
    • Example: Creating Modular Symbols of Level 14
• Bases
  • Basis(M): ModSym → SeqEnum
  • IntegralBasis(M): ModSym → SeqEnum
  • Example: Integral Basis of Modular Symbols
• Associated Vector Space
  • VectorSpace(M): ModSym → ModTupFld, Map, Map
  • DualVectorSpace(M): ModSym → ModTupFld
  • Lattice(M): ModSym → Lat
  • Example: Underlying Vector Space of Modular Symbols
• 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: Degeneracy Maps Between Levels 15 and 30
• 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: Decomposition of Modular Symbols of Level 37
• 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: Cuspidal Subspace of Modular Symbols
  • Example: Boundary Map on Modular Symbols
• Twists
  • IsTwist(M1, M2, p): ModSym, ModSym, RngIntElt → BoolElt, GrpDrchElt
  • IsMinimalTwist(M, p): ModSym, RngIntElt → BoolElt, ModSym, GrpDrchElt
  • Example: Newform That Is Its Own Twist
• Operators
  • Example: Hecke Operators as Matrices
  • 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: Integral Hecke Operators in Weight 4
  • Example: Theta Operator on Modular Symbols
• 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: Discriminant of a Hecke Algebra
• The Intersection Pairing
  • IntersectionPairing(x, y): ModSymElt, ModSymElt → FldRatElt
  • Example: Intersection Pairing on X0(37)
• \(q\)-Expansions
  • Eigenform(M, prec): ModSym, RngIntElt → RngSerPowElt
  • Eigenform(M): ModSym → RngSerPowElt
  • qEigenform(M, prec): ModSym, RngIntElt → RngSerPowElt
  • qEigenform(M): ModSym → RngSerPowElt
  • PowerSeries(M, prec): ModSym, RngIntElt → RngSerPowElt
  • PowerSeries(M): ModSym → RngSerPowElt
  • qExpansionBasis(M, prec : parameters): ModSym, RngIntElt → SeqEnum
  • qIntegralBasis(M): ModSym → SeqEnum
  • qIntegralBasis(M, prec): ModSym, RngIntElt → SeqEnum
  • qIntegralBasis(seq, prec): [ModSym], RngIntElt → SeqEnum
  • SystemOfEigenvalues(M, prec): ModSym, RngIntElt → SeqEnum
  • Example: q-Expansions from Modular Symbols
• 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: L-Series Values of Modular Symbols
  • 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: Cuspidal Subgroup of J0(N)
  • Example: Table of Cuspidal Subgroups of J0(N)
  • ModularKernel(M): ModSym → GrpAb
  • CongruenceGroup(M : parameters): ModSym → GrpAb
  • IntersectionGroup(M1, M2): ModSym, ModSym → GrpAb
  • IntersectionGroup(S): SeqEnum → GrpAb
  • Example: Numerical BSD Computation for an Abelian Surface
  • The Period Map
    • PeriodMapping(M, n): ModSym, RngIntElt → Map
    • PeriodMapping(M): ModSym → Map
    • Periods(M, prec): ModSym, RngIntElt → SeqEnum
    • Periods(M): ModSym → SeqEnum
    • ClassicalPeriod(M, j, prec): ModSym, RngIntElt, RngIntElt → FldPrElt
  • Projection Mappings
    • RationalMapping(M): ModSym → Map
    • IntegralMapping(M): ModSym → Map
    • Example: Rational Mapping of Modular Symbols
• Modular Abelian Varieties
  • Modular Degree and Torsion
    • ModularDegree(M): ModSym → RngIntElt
    • CongruenceModulus(M : parameters): ModSym → RngIntElt
    • TorsionBound(M, maxp): ModSym, RngIntElt → RngIntElt
    • Example: Congruence Modulus and Modular Degree
  • 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: Component Groups and Tamagawa Numbers
• Elliptic Curves
  • ModularSymbols(E): CrvEll → ModSym
  • ModularSymbols(E, sign): CrvEll, RngIntElt → ModSym
  • EllipticCurve(M): ModSym → CrvEll
  • pAdicLSeries(E, p): CrvEll, RngIntElt → RngSerPowElt
  • Example: Conjectural Order of Sha for Curve 389A
• 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: Dimensions of Spaces of Cusp Forms
• Bibliography