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MODULAR ABELIAN VARIETIES

 
Acknowledgements
 
Introduction
      Categories
      Verbose Output
 
Creation and Basic Functions
      Creating the Modular Jacobian J0(N)
      Creating the Modular Jacobians J1(N) and JH(N)
      Abelian Varieties Attached to Modular Forms
      Abelian Varieties Attached to Modular Symbols
      Creation of Abelian Subvarieties
      Creation Using a Label
      Invariants
      Conductor
      Number of Points
      Inner Twists and Complex Multiplication
      Predicates
      Equality and Inclusion Testing
      Modular Embedding and Parameterization
      Coercion
      Modular Symbols to Homology
      Embeddings
      Base Change
      Additional Examples
 
Homology
      Creation
      Invariants
      Functors to Categories of Lattices and Vector Spaces
      Modular Structure
      Additional Examples
 
Homomorphisms
      Creation
      Restriction, Evaluation, and Other Manipulations
      Kernels
      Images
      Cokernels
      Matrix Structure
      Arithmetic
      Polynomials
      Invariants
      Predicates
 
Endomorphism Algebras and Hom Spaces
      Creation
      Subgroups and Subrings
      Pullback and Pushforward of Hom Spaces
      Arithmetic
      Quotients
      Invariants
      Structural Invariants
      Matrix and Module Structure
      Predicates
      Elements
 
Arithmetic of Abelian Varieties
      Direct Sum
      Sum in an Ambient Variety
      Intersections
      Quotients
 
Decomposing and Factoring Abelian Varieties
      Decomposition
      Factorization
      Decomposition with respect to an Endomorphism or a Commutative Ring
      Additional Examples
 
Building blocks
      Background and Notation
 
Orthogonal Complements
      Complements
      Dual Abelian Variety
      Intersection Pairing
      Projections
      Left and Right Inverses
      Congruence Computations
 
New and Old Subvarieties and Natural Maps
      Natural Maps
      New Subvarieties and Quotients
      Old Subvarieties and Quotients
 
Elements of Modular Abelian Varieties
      Arithmetic
      Invariants
      Predicates
      Homomorphisms
      Representation of Torsion Points
 
Subgroups of Modular Abelian Varieties
      Creation
      Elements
      Arithmetic
      Underlying Abelian Group and Lattice
      Invariants
      Predicates and Comparisons
 
Rational Torsion Subgroups
      Cuspidal Subgroup
      Upper and Lower Bounds
      Torsion Subgroup
 
Hecke and Atkin-Lehner Operators
      Creation
      Invariants
 
L-series
      Creation
      Invariants
      Characteristic Polynomials of Frobenius Elements
      Values at Integers in the Critical Strip
      Leading Coefficient
 
Complex Period Lattice
      Period Map
      Period Lattice
 
Tamagawa Numbers and Component Groups of Neron Models
      Component Groups
      Tamagawa Numbers
 
Elliptic Curves
      Creation
      Invariants
 
Bibliography







DETAILS

 
Introduction

      Categories
            Example ModAbVar_Categories (H136E1)

      Verbose Output
            Example ModAbVar_Verbosity (H136E2)

 
Creation and Basic Functions

      Creating the Modular Jacobian J0(N)
            JZero(N : parameters) : RngIntElt -> ModAbVar
            Example ModAbVar_Modabvar-Creating_the_Modular_Jacobian_$J_0(N)$ (H136E3)

      Creating the Modular Jacobians J1(N) and JH(N)
            JOne(N : parameters) : RngIntElt -> ModAbVar
            Js(N : parameters) : RngIntElt -> ModAbVar
            JH(N, d : parameters) : RngIntElt, RngIntElt, RngIntElt -> ModAbVar
            JH(N, gens : parameters) : RngIntElt, [RngIntElt] -> ModAbVar
            Example ModAbVar_Modabvar-Creating_the_Modular_Jacobians_$J_1(N)$_and_$J_H(N)$ (H136E4)

      Abelian Varieties Attached to Modular Forms
            ModularAbelianVariety(M : parameters) : ModFrm -> ModAbVar
            ModularAbelianVariety(X : parameters) : [ModFrm] -> ModAbVar
            ModularAbelianVariety(eps : parameters) : GrpDrchElt -> ModAbVar
            ModularAbelianVariety(f) : ModFrmElt -> ModAbVar
            Newform(A) : ModAbVar -> ModFrmElt
            Example ModAbVar_Modabvar-Abelian_Varieties_Attached_to_Modular_Forms (H136E5)
            Example ModAbVar_Modabvar-Abelian_Varieties_Attached_to_Modular_Forms3 (H136E6)
            Example ModAbVar_Modabvar-Abelian_Varieties_Attached_to_Modular_Forms5 (H136E7)

      Abelian Varieties Attached to Modular Symbols
            ModularAbelianVariety(M) : ModSym -> ModAbVar
            ModularAbelianVariety(X) : [ModSym] -> ModAbVar
            ModularSymbols(A) : ModAbVar -> SeqEnum
            Example ModAbVar_Modabvar-Abelian_Varieties_Attached_to_Modular_Symbols (H136E8)

      Creation of Abelian Subvarieties
            DefinesAbelianSubvariety(A, V) : ModAbVar, ModTupFld -> BoolElt, ModAbVar
            ZeroModularAbelianVariety() : -> ModAbVar
            ZeroModularAbelianVariety(k) : RngIntElt -> ModAbVar
            ZeroSubvariety(A) : ModAbVar -> ModAbVar
            Example ModAbVar_Modabvar-Creation_of_Abelian_Subvarieties (H136E9)

      Creation Using a Label
            ModularAbelianVariety(s : parameters) : MonStgElt -> ModAbVar
            Example ModAbVar_Modabvar-Creation_Using_a_Label (H136E10)

      Invariants
            BaseRing(A) : ModAbVar -> Rng
            Dimension(A) : ModAbVar -> RngIntElt
            DirichletCharacter(A) : ModAbVar -> GrpDrchElt
            DirichletCharacters(A) : ModAbVar -> List
            FieldOfDefinition(A) : ModAbVar -> Fld
            Level(A) : ModAbVar -> RngIntElt
            Sign(A) : ModAbVar -> RngIntElt
            Weights(A) : ModAbVar -> Set
            Example ModAbVar_Modabvar-Invariants (H136E11)
            Example ModAbVar_Modabvar-Invariants2 (H136E12)
            Example ModAbVar_Modabvar-Invariants3 (H136E13)
            Example ModAbVar_Modabvar-Invariants3 (H136E14)
            Example ModAbVar_Modabvar-Invariants4 (H136E15)

      Conductor
            Conductor(A) : ModAbVar -> RngIntElt
            Example ModAbVar_Modabvar-Conductor (H136E16)

      Number of Points
            NumberOfRationalPoints(A) : ModAbVar -> RngIntElt, RngIntElt
            #A : ModAbVar -> RngIntElt
            Example ModAbVar_Modabvar-Number_of_Points (H136E17)

      Inner Twists and Complex Multiplication
            CMTwists(A : parameters) : ModAbVar -> SeqEnum
            InnerTwists(A : parameters) : ModAbVar -> SeqEnum
            Example ModAbVar_Modabvar-Inner_Twists_and_Complex_Multiplication (H136E18)
            Example ModAbVar_Modabvar-Inner_Twists_and_Complex_Multiplication2 (H136E19)

      Predicates
            CanDetermineIsomorphism(A, B) : ModAbVar, ModAbVar -> BoolElt, BoolElt, MapModAbVar
            HasMultiplicityOne(A) : ModAbVar -> BoolElt
            IsAbelianVariety(A) : ModAbVar -> BoolElt
            IsAttachedToModularSymbols(A) : ModAbVar -> BoolElt
            IsAttachedToNewform(A) : ModAbVar -> BoolElt, ModAbVar, MapModAbVar
            IsIsogenous(A, B) : ModAbVar, ModAbVar -> BoolElt
            IsIsomorphic(A, B) : ModAbVar, ModAbVar -> BoolElt, MapModAbVar
            IsOnlyMotivic(A) : ModAbVar -> BoolElt
            IsQuaternionic(A) : ModAbVar -> BoolElt
            IsSelfDual(A) : ModAbVar -> BoolElt
            IsSimple(A) : ModAbVar -> BoolElt
            Example ModAbVar_Modabvar-Predicates (H136E20)
            Example ModAbVar_Modabvar-Predicates2 (H136E21)
            Example ModAbVar_Modabvar-Predicates3 (H136E22)
            Example ModAbVar_Modabvar-Predicates4 (H136E23)
            Example ModAbVar_Modabvar-Predicates5 (H136E24)
            Example ModAbVar_Modabvar-Predicates6 (H136E25)
            Example ModAbVar_Modabvar-Predicates7 (H136E26)
            Example ModAbVar_Modabvar-Predicates8 (H136E27)

      Equality and Inclusion Testing
            A eq B : ModAbVar, ModAbVar -> BoolElt
            A subset B : ModAbVar, ModAbVar -> BoolElt
            Example ModAbVar_Modabvar-Equality_and_Inclusion_Testing (H136E28)
            Example ModAbVar_Modabvar-Equality_and_Inclusion_Testing2 (H136E29)

      Modular Embedding and Parameterization
            CommonModularStructure(X) : [ModAbVar] -> List, List
            ModularEmbedding(A) : ModAbVar -> MapModAbVar
            ModularParameterization(A) : ModAbVar -> MapModAbVar
            Example ModAbVar_Modabvar-Modular_Embedding_and_Parameterization (H136E30)
            Example ModAbVar_Modabvar-Modular_Embedding_and_Parameterization2 (H136E31)

      Coercion
            A ! x : ModAbVar, . -> BoolElt, ModAbVarElt
            Example ModAbVar_Modabvar-Coercion (H136E32)
            Example ModAbVar_Modabvar-Coercion2 (H136E33)

      Modular Symbols to Homology
            ModularSymbolToIntegralHomology(A, x) : ModAbVar, SeqEnum -> ModTupFldElt
            ModularSymbolToRationalHomology(A, x) : ModAbVar, ModSymElt -> ModTupFldElt
            Example ModAbVar_Modabvar-Modular_Symbols_to_Homology (H136E34)
            Example ModAbVar_Modabvar-Modular_Symbols_to_Homology2 (H136E35)

      Embeddings
            Embeddings(A) : ModAbVar -> List
            AssertEmbedding(~A, phi) : ModAbVar, MapModAbVar ->
            Example ModAbVar_Modabvar-Embeddings (H136E36)

      Base Change
            CanChangeRing(A, R) : ModAbVar, Rng -> BoolElt, ModAbVar
            ChangeRing(A, R) : ModAbVar, Rng -> ModAbVar
            BaseExtend(A, R) : ModAbVar, Rng -> ModAbVar
            Example ModAbVar_Modabvar-Base_Change (H136E37)

      Additional Examples
            Example ModAbVar_modabvar1 (H136E38)
            Example ModAbVar_additional2 (H136E39)
            Example ModAbVar_additional3 (H136E40)
            Example ModAbVar_additional4 (H136E41)
            Example ModAbVar_additional5 (H136E42)
            Example ModAbVar_additional6 (H136E43)

 
Homology

      Creation
            Homology(A) : ModAbVar -> ModAbVarHomol
            Example ModAbVar_Homology-Creation (H136E44)

      Invariants
            Dimension(H) : ModAbVarHomol -> RngIntElt
            Example ModAbVar_Homology-Invariants (H136E45)

      Functors to Categories of Lattices and Vector Spaces
            IntegralHomology(A) : ModAbVar -> Lat
            Lattice(H) : ModAbVarHomol -> Lat
            RationalHomology(A) : ModAbVar -> ModTupFld
            RealHomology(A) : ModAbVar -> ModTupFld
            RealVectorSpace(H) : ModAbVarHomol -> ModTupFld
            VectorSpace(H) : ModAbVarHomol -> ModTupFld
            Example ModAbVar_additional8 (H136E46)
            Example ModAbVar_Homology-Functors_to_Categories_of_Lattices_and_Vector_Spaces (H136E47)
            Example ModAbVar_Homology-Functors_to_Categories_of_Lattices_and_Vector_Spaces2 (H136E48)

      Modular Structure
            IsAttachedToModularSymbols(H) : ModAbVarHomol -> BoolElt
            ModularSymbols(H) : ModAbVarHomol -> SeqEnum
            Example ModAbVar_Homology-Modular_Structure (H136E49)

      Additional Examples
            Example ModAbVar_homology1 (H136E50)
            Example ModAbVar_additional20 (H136E51)

 
Homomorphisms

      Creation
            IdentityMap(A) : ModAbVar -> MapModAbVar
            ZeroMap(A) : ModAbVar -> MapModAbVar
            nIsogeny(A, n) : ModAbVar, FldRatElt -> MapModAbVar
            Example ModAbVar_Morphisms-Creation (H136E52)

      Restriction, Evaluation, and Other Manipulations
            Restriction(phi, B) : MapModAbVar, ModAbVar -> MapModAbVar
            RestrictEndomorphism(phi, B) : MapModAbVar, ModAbVar -> MapModAbVar
            RestrictEndomorphism(phi, i) : MapModAbVar, MapModAbVar -> MapModAbVar
            RestrictionToImage(phi, i) : MapModAbVar, MapModAbVar -> MapModAbVar
            Evaluate(f, phi) : RngUPolElt, MapModAbVar -> MapModAbVar
            DivideOutIntegers(phi) : MapModAbVar -> MapModAbVar, RngIntElt
            SurjectivePart(phi) : MapModAbVar -> MapModAbVar
            UniversalPropertyOfCokernel(pi, f) : MapModAbVar, MapModAbVar -> MapModAbVar
            Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations (H136E53)
            Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations2 (H136E54)
            Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations3 (H136E55)
            Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations4 (H136E56)

      Kernels
            ComponentGroupOfKernel(phi) : MapModAbVar -> ModAbVarSubGrp
            ConnectedKernel(phi) : MapModAbVar -> ModAbVar, MapModAbVar
            Kernel(phi) : MapModAbVar -> ModAbVarSubGrp, ModAbVar, MapModAbVar
            Example ModAbVar_Morphisms-Kernels (H136E57)

      Images
            A @ phi : ModAbVar, MapModAbVar -> ModAbVar
            G @ phi : ModAbVarSubGrp, MapModAbVar -> ModAbVarSubGrp
            Image(phi) : MapModAbVar -> ModAbVar, MapModAbVar, MapModAbVar
            G @@ phi : ModAbVarSubGrp, MapModAbVar -> ModAbVarSubGrp
            Example ModAbVar_Morphisms-Images (H136E58)

      Cokernels
            Cokernel(phi) : MapModAbVar -> ModAbVar, MapModAbVar
            Example ModAbVar_Morphisms-Cokernels (H136E59)

      Matrix Structure
            Matrix(phi) : MapModAbVar -> ModMatFldElt
            Eltseq(phi) : MapModAbVar -> SeqEnum
            Ncols(phi) : MapModAbVar -> RngIntElt
            Nrows(phi) : MapModAbVar -> RngIntElt
            Rows(phi) : MapModAbVar -> SeqEnum
            IntegralMatrix(phi) : MapModAbVar -> ModMatRngElt
            IntegralMatrixOverQ(phi) : MapModAbVar -> ModMatFldElt
            RealMatrix(phi) : MapModAbVar -> ModMatFldElt
            Example ModAbVar_Morphisms-Matrix_Structure (H136E60)

      Arithmetic
            Inverse(phi) : MapModAbVar -> MapModAbVar, RngIntElt
            phi * psi : MapModAbVar, MapModAbVar -> MapModAbVar
            a * phi : FldRatElt, MapModAbVar -> MapModAbVar
            phi * psi : MapModAbVar, AlgMatElt -> AlgMatElt
            psi * phi : AlgMatElt, MapModAbVar -> AlgMatElt
            phi ^ n : MapModAbVar, RngIntElt -> MapModAbVar
            phi + psi : MapModAbVar, MapModAbVar -> MapModAbVar
            n + phi : FldRatElt, MapModAbVar -> MapModAbVar
            phi + n : MapModAbVar, RngIntElt -> MapModAbVar
            phi + psi : MapModAbVar, AlgMatElt -> AlgMatElt
            psi + phi : AlgMatElt, MapModAbVar -> AlgMatElt
            phi - psi : MapModAbVar, MapModAbVar -> MapModAbVar
            n - phi : FldRatElt, MapModAbVar -> MapModAbVar
            phi - n : MapModAbVar, FldRatElt -> MapModAbVar
            phi - psi : MapModAbVar, AlgMatElt -> AlgMatElt
            psi - phi : AlgMatElt, MapModAbVar -> AlgMatElt
            Example ModAbVar_Morphisms-Arithmetic (H136E61)

      Polynomials
            CharacteristicPolynomial(phi) : MapModAbVar -> RngUPolElt
            FactoredCharacteristicPolynomial(phi) : MapModAbVar -> RngUPolElt
            MinimalPolynomial(phi) : MapModAbVar -> RngUPolElt
            Example ModAbVar_Morphisms-Polynomials (H136E62)

      Invariants
            Domain(phi) : MapModAbVar -> ModAbVar
            Codomain(phi) : MapModAbVar -> ModAbVar
            Degree(phi) : MapModAbVar -> RngIntElt
            Denominator(phi) : MapModAbVar -> RngIntElt
            ClearDenominator(phi) : MapModAbVar -> MapModAbVar
            FieldOfDefinition(phi) : MapModAbVar -> ModAbVar
            Nullity(phi) : MapModAbVar -> RngIntElt
            Rank(phi) : MapModAbVar -> RngIntElt
            Trace(phi) : MapModAbVar -> FldRatElt
            Example ModAbVar_Morphisms-Invariants (H136E63)

      Predicates
            IsMorphism(phi) : MapModAbVar -> BoolElt
            OnlyUpToIsogeny(phi) : MapModAbVar -> BoolElt
            HasFiniteKernel(phi) : MapModAbVar -> BoolElt
            IsInjective(phi) : MapModAbVar -> BoolElt
            IsSurjective(phi) : MapModAbVar -> BoolElt
            IsEndomorphism(phi) : MapModAbVar -> BoolElt
            IsInteger(phi) : MapModAbVar -> BoolElt, RngIntElt
            IsIsogeny(phi) : MapModAbVar -> BoolElt
            IsIsomorphism(phi) : MapModAbVar -> BoolElt
            IsOptimal(phi) : MapModAbVar -> BoolElt
            IsHeckeOperator(phi) : MapModAbVar -> BoolElt, RngIntElt
            IsZero(phi) : MapModAbVar -> BoolElt
            phi eq psi : MapModAbVar, MapModAbVar -> BoolElt
            n eq phi : RngIntElt, MapModAbVar -> BoolElt
            phi in X : MapModAbVar, List -> BoolElt
            Example ModAbVar_Morphisms-Predicates (H136E64)

 
Endomorphism Algebras and Hom Spaces

      Creation
            Hom(A, B) : ModAbVar, ModAbVar -> HomModAbVar
            End(A) : ModAbVar -> HomModAbVar
            BaseExtend(H, R) : HomModAbVar, Rng -> HomModAbVar
            HeckeAlgebra(A) : ModAbVar -> HomModAbVar
            Example ModAbVar_Homspace-Creation (H136E65)

      Subgroups and Subrings
            Subgroup(X) : [MapModAbVar] -> HomModAbVar
            Subgroup(X, oQ : parameters) : [MapModAbVar], BoolElt -> HomModAbVar
            Subring(X) : [MapModAbVar] -> HomModAbVar
            Subring(phi) : MapModAbVar -> HomModAbVar
            Saturation(H) : HomModAbVar -> HomModAbVar
            RingGeneratedBy(H) : HomModAbVar -> HomModAbVar
            Example ModAbVar_Homspace-Subgroups__and_Subrings (H136E66)
            Example ModAbVar_Homspace-Subgroups__and_Subrings2 (H136E67)
            Example ModAbVar_Homspace-Subgroups__and_Subrings3 (H136E68)
            Example ModAbVar_Homspace-Subgroups__and_Subrings4 (H136E69)

      Pullback and Pushforward of Hom Spaces
            Pullback(H, phi) : HomModAbVar, MapModAbVar -> HomModAbVar
            Pullback(phi, H) : MapModAbVar, HomModAbVar -> HomModAbVar
            Pullback(phi, H, psi) : MapModAbVar, HomModAbVar, MapModAbVar -> HomModAbVar
            Example ModAbVar_Homspace-Pullback_and_Pushforward_of_Hom_Spaces (H136E70)

      Arithmetic
            H1 + H2 : HomModAbVar, HomModAbVar -> HomModAbVar
            H1 meet H2 : HomModAbVar, HomModAbVar -> HomModAbVar
            Example ModAbVar_Homspace-Arithmetic (H136E71)

      Quotients
            Index(H2, H1) : HomModAbVar, HomModAbVar -> RngIntElt
            Quotient(H2, H1) : HomModAbVar, HomModAbVar -> GrpAb, Map, Map
            Example ModAbVar_Homspace-Quotients (H136E72)

      Invariants
            Domain(H) : HomModAbVar -> ModAbVar
            Codomain(H) : HomModAbVar -> ModAbVar
            FieldOfDefinition(H) : HomModAbVar -> ModAbVar
            Discriminant(H) : HomModAbVar -> FldRatElt, AlgMatElt
            Example ModAbVar_Homspace-Invariants (H136E73)

      Structural Invariants
            Basis(H) : HomModAbVar -> SeqEnum
            Dimension(H) : HomModAbVar -> RngIntElt
            Ngens(H) : HomModAbVar -> RngIntElt
            H . i : HomModAbVar, RngIntElt -> MapModAbVar
            Example ModAbVar_Homspace-Structural_Invariants (H136E74)

      Matrix and Module Structure
            Lattice(H) : HomModAbVar -> Lat
            MatrixAlgebra(H) : HomModAbVar -> AlgMat
            RMatrixSpace(H) : HomModAbVar -> ModMatFld
            RModuleWithAction(H) : HomModAbVar -> ModED
            RModuleWithAction(H, p) : HomModAbVar, RngIntElt -> ModED
            Example ModAbVar_Homspace-Matrix_and_Module_Structure (H136E75)
            Example ModAbVar_Homspace-Matrix_and_Module_Structure3 (H136E76)

      Predicates
            IsRing(H) : HomModAbVar -> BoolElt
            IsField(H) : HomModAbVar -> BoolElt, Fld, Map, Map
            IsCommutative(H) : HomModAbVar -> BoolElt
            IsHeckeAlgebra(H) : HomModAbVar -> BoolElt
            IsOverQ(H) : HomModAbVar -> HomModAbVar
            IsSaturated(H) : HomModAbVar -> BoolElt
            H1 eq H2 : HomModAbVar, HomModAbVar -> BoolElt
            H1 subset H2 : HomModAbVar, HomModAbVar -> BoolElt
            Example ModAbVar_Homspace-Predicates (H136E77)

      Elements
            H ! x : HomModAbVar, . -> BoolElt, MapModAbVar
            Example ModAbVar_Homspace-Random_Element (H136E78)

 
Arithmetic of Abelian Varieties

      Direct Sum
            DirectSum(A, B) : ModAbVar, ModAbVar -> ModAbVar, List, List
            DirectSum(X) : [ModAbVar] -> ModAbVar, List, List
            A ^ n : ModAbVar, RngIntElt -> ModAbVar
            Example ModAbVar_direct-sum (H136E79)

      Sum in an Ambient Variety
            A + B : ModAbVar, ModAbVar -> ModAbVar
            SumOf(X) : [ModAbVar] -> ModAbVar
            SumOfImages(phi, psi) : MapModAbVar, MapModAbVar -> ModAbVar, MapModAbVar, List
            SumOfMorphismImages(X) : List -> ModAbVar, MapModAbVar, List
            FindCommonEmbeddings(X) : [ModAbVar] -> BoolElt, List

      Intersections
            A meet B : ModAbVar, ModAbVar -> ModAbVarSubGrp, ModAbVar, MapModAbVar
            IntersectionOfImages(X) : List -> ModAbVarSubGrp, ModAbVar, MapModAbVar
            ComponentGroupOfIntersection(A, B) : ModAbVar, ModAbVar -> ModAbVarSubGrp
            Example ModAbVar_Arithabvar-Intersections (H136E80)
            Example ModAbVar_Arithabvar-Intersections2 (H136E81)

      Quotients
            A / B : ModAbVar, ModAbVar -> ModAbVar, MapModAbVar
            Cokernel(phi) : MapModAbVar -> ModAbVar, MapModAbVar
            Example ModAbVar_Arithabvar-Quotients (H136E82)

 
Decomposing and Factoring Abelian Varieties

      Decomposition
            Decomposition(A) : ModAbVar -> SeqEnum
            A(n) : RngIntElt, ModAbVar -> ModAbVar
            Example ModAbVar_Decomp-Decomposition (H136E83)

      Factorization
            Factorisation(A) : ModAbVar -> List
            Example ModAbVar_Decomp-Factorization (H136E84)

      Decomposition with respect to an Endomorphism or a Commutative Ring
            DecomposeUsing(R) : HomModAbVar -> SeqEnum
            DecomposeUsing(phi) : MapModAbVar -> SeqEnum
            Example ModAbVar_Decomp-Decomposition_with_respect_to_an_Endomorphism_or_a_Commutative_Ring (H136E85)

      Additional Examples
            Example ModAbVar_homology1 (H136E86)

 
Building blocks

      Background and Notation
            BoundedFSubspace(epsilon, k, degrees) : GrpDrchElt, RngIntElt, [RngIntElt] -> [ ModSym ]
            HasCM(M : parameters) : ModSym -> BoolElt, RngIntElt
            InnerTwists(A : parameters) : ModAbVar -> [ GrpDrchElt ]
            DegreeMap(M : parameters) : ModSym -> [ Tup ], Fld
            BrauerClass(M) : ModSym -> SeqEnum
            ObstructionDescentBuildingBlock(M) : ModSym -> SeqEnum
            Example ModAbVar_Complements-Complements (H136E87)

 
Orthogonal Complements

      Complements
            Complement(A : parameters) : ModAbVar -> ModAbVar, MapModAbVar
            ComplementOfImage(phi : parameters) : MapModAbVar -> ModAbVar, MapModAbVar
            Example ModAbVar_Complements-Complements (H136E88)

      Dual Abelian Variety
            IsDualComputable(A) : ModAbVar -> BoolElt, ModAbVar
            Dual(A) : ModAbVar -> ModAbVar
            ModularPolarization(A) : ModAbVar -> MapModAbVar
            Example ModAbVar_Complements-Dual_Abelian_Variety (H136E89)

      Intersection Pairing
            IntersectionPairing(H) : ModAbVarHomol -> AlgMatElt
            IntersectionPairing(A) : ModAbVar -> AlgMatElt
            IntersectionPairingIntegral(A) : ModAbVar -> AlgMatElt
            Example ModAbVar_Complements-Intersection_Pairing (H136E90)

      Projections
            ProjectionOnto(A : parameters) : ModAbVar -> MapModAbVar
            Example ModAbVar_Complements-Projections (H136E91)
            Example ModAbVar_Complements-Projections2 (H136E92)

      Left and Right Inverses
            LeftInverse(phi : parameters) : MapModAbVar -> MapModAbVar, RngIntElt
            LeftInverseMorphism(phi : parameters) : MapModAbVar -> MapModAbVar
            RightInverse(phi : parameters) : MapModAbVar -> MapModAbVar, RngIntElt
            RightInverseMorphism(phi : parameters) : MapModAbVar -> MapModAbVar
            Example ModAbVar_Complements-Left_and_Right_Inverses (H136E93)

      Congruence Computations
            CongruenceModulus(A) : ModAbVar -> RngIntElt
            ModularDegree(A) : ModAbVar -> RngIntElt
            Example ModAbVar_Complements-Congruence_Computations (H136E94)

 
New and Old Subvarieties and Natural Maps

      Natural Maps
            NaturalMap(A, B, d) : ModAbVar, ModAbVar, RngIntElt -> MapModAbVar
            NaturalMap(A, B) : ModAbVar, ModAbVar -> MapModAbVar
            NaturalMaps(A, B) : ModAbVar, ModAbVar -> SeqEnum
            Example ModAbVar_New_old-Natural_Maps (H136E95)

      New Subvarieties and Quotients
            NewSubvariety(A, r) : ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
            NewSubvariety(A) : ModAbVar -> ModAbVar, MapModAbVar
            NewQuotient(A, r) : ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
            NewQuotient(A) : ModAbVar -> ModAbVar, MapModAbVar
            Example ModAbVar_New_old-New_Subvarieties_and_Quotients (H136E96)

      Old Subvarieties and Quotients
            OldSubvariety(A, r) : ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
            OldSubvariety(A) : ModAbVar -> ModAbVar, MapModAbVar
            OldQuotient(A, r) : ModAbVar, RngIntElt -> ModAbVar, MapModAbVar
            OldQuotient(A) : ModAbVar -> ModAbVar, MapModAbVar
            Example ModAbVar_New_old-Old_Subvarieties_and_Quotients (H136E97)

 
Elements of Modular Abelian Varieties

      Arithmetic
            a * x : FldReElt, ModAbVarElt -> ModAbVarElt
            x * a : ModAbVarElt, FldReElt -> ModAbVarElt
            x + y : ModAbVarElt, ModAbVarElt -> ModAbVarElt
            x - y : ModAbVarElt, ModAbVarElt -> ModAbVarElt
            Example ModAbVar_Elt-Arithmetic (H136E98)

      Invariants
            Order(x) : ModAbVarElt -> RngIntElt
            ApproximateOrder(x) : ModAbVarElt -> RngIntElt
            Degree(x) : ModAbVarElt -> RngIntElt
            FieldOfDefinition(x) : ModAbVarElt -> ModTupFldElt
            Example ModAbVar_Elt-Invariants (H136E99)

      Predicates
            x eq y : ModAbVarElt, ModAbVarElt -> BoolElt
            x in X : ModAbVarElt, List -> BoolElt
            IsExact(x) : ModAbVarElt -> BoolElt
            IsZero(x) : ModAbVarElt -> BoolElt
            Example ModAbVar_Elt-Predicates (H136E100)

      Homomorphisms
            x @ phi : ModAbVarElt, MapModAbVar -> ModAbVarElt
            x @@ phi : ModAbVarElt, MapModAbVar -> ModAbVarElt
            Example ModAbVar_Elt-Homomorphisms (H136E101)

      Representation of Torsion Points
            ApproximateByTorsionPoint(x : parameters) : ModAbVarElt -> ModAbVarElt
            Element(x) : ModAbVarElt -> ModTupFldElt
            LatticeCoordinates(x) : ModAbVarElt -> ModTupFldElt
            Eltseq(x) : ModAbVarElt -> SeqEnum
            Example ModAbVar_Elt-Representation_of_Torsion_Points (H136E102)

 
Subgroups of Modular Abelian Varieties

      Creation
            Subgroup(X) : [ModAbVarElt] -> ModAbVarSubGrp
            ZeroSubgroup(A) : ModAbVar -> ModAbVarSubGrp
            nTorsionSubgroup(A, n) : ModAbVar, RngIntElt -> ModAbVarSubGrp
            nTorsionSubgroup(G, n) : ModAbVarSubGrp, RngIntElt -> ModAbVarSubGrp
            ApproximateByTorsionGroup(G : parameters) : ModAbVarSubGrp -> ModAbVarSubGrp
            Example ModAbVar_Subgrp-Creation (H136E103)

      Elements
            Elements(G) : ModAbVarSubGrp -> SeqEnum
            Generators(G) : ModAbVarSubGrp -> SeqEnum
            Ngens(G) : ModAbVarSubGrp -> RngIntElt
            G . i : ModAbVarSubGrp, RngIntElt -> ModAbVarElt
            Example ModAbVar_Subgrp-Elements (H136E104)

      Arithmetic
            Quotient(A, G) : ModAbVar, ModAbVarSubGrp -> ModAbVar, MapModAbVar
            Quotient(G) : ModAbVarSubGrp -> ModAbVar, MapModAbVar, MapModAbVar
            A / G : ModAbVar, ModAbVarSubGrp -> ModAbVar, MapModAbVar, MapModAbVar
            A meet G : ModAbVar, ModAbVarSubGrp -> ModAbVarSubGrp
            G1 + G2 : ModAbVarSubGrp, ModAbVarSubGrp -> ModAbVarSubGrp
            G1 meet G2 : ModAbVarSubGrp, ModAbVarSubGrp -> ModAbVarSubGrp
            Example ModAbVar_Subgrp-Arithmetic (H136E105)

      Underlying Abelian Group and Lattice
            AbelianGroup(G) : ModAbVarSubGrp -> GrpAb, Map, Map
            Lattice(G) : ModAbVarSubGrp -> Lat
            Example ModAbVar_Subgrp-Underlying_Abelian_Group_and_Lattice (H136E106)

      Invariants
            AmbientVariety(G) : ModAbVarSubGrp -> ModAbVar
            Exponent(G) : ModAbVarSubGrp -> RngIntElt
            Invariants(G) : ModAbVarSubGrp -> SeqEnum
            Order(G) : ModAbVarSubGrp -> RngIntElt
            FieldOfDefinition(G) : ModAbVarSubGrp -> Fld
            Example ModAbVar_Subgrp-Invariants (H136E107)

      Predicates and Comparisons
            IsFinite(G) : ModAbVarSubGrp -> RngIntElt
            G1 subset G2 : ModAbVarSubGrp, ModAbVarSubGrp -> BoolElt
            G subset A : ModAbVarSubGrp, ModAbVar -> BoolElt
            A subset G : ModAbVar, ModAbVarSubGrp -> BoolElt
            G1 eq G2 : ModAbVarSubGrp, ModAbVarSubGrp -> BoolElt
            Example ModAbVar_Subgrp-Predicates_and_Comparisons (H136E108)

 
Rational Torsion Subgroups

      Cuspidal Subgroup
            CuspidalSubgroup(A) : ModAbVar -> ModAbVarSubGrp
            RationalCuspidalSubgroup(A) : ModAbVar -> ModAbVarSubGrp
            Example ModAbVar_Torsion-Cuspidal_Subgroup (H136E109)

      Upper and Lower Bounds
            TorsionLowerBound(A) : ModAbVar -> RngIntElt
            TorsionMultiple(A) : ModAbVar -> RngIntElt
            Example ModAbVar_Torsion-Upper_and_Lower_Bounds (H136E110)

      Torsion Subgroup
            TorsionSubgroup(A) : ModAbVar -> BoolElt, ModAbVarSubGrp
            Example ModAbVar_Torsion-Torsion_Subgroup (H136E111)

 
Hecke and Atkin-Lehner Operators

      Creation
            AtkinLehnerOperator(A, q) : ModAbVar, RngIntElt -> MapModAbVar, RngIntElt
            AtkinLehnerOperator(A) : ModAbVar -> MapModAbVar
            HeckeOperator(A, n) : ModAbVar, RngIntElt -> MapModAbVar
            Example ModAbVar_Operators-Creation (H136E112)

      Invariants
            HeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt
            FactoredHeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt
            MinimalHeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt
            Example ModAbVar_Operators-Invariants (H136E113)

 
L-series

      Creation
            LSeries(A) : ModAbVar -> ModAbVarLSer
            Example ModAbVar_Lser-Creation (H136E114)

      Invariants
            CriticalStrip(L) : ModAbVarLSer -> RngIntElt, RngIntElt
            ModularAbelianVariety(L) : ModAbVarLSer -> ModAbVar
            Example ModAbVar_Lser-Invariants (H136E115)

      Characteristic Polynomials of Frobenius Elements
            FrobeniusPolynomial(A : parameters) : ModAbVar -> RngUPolElt
            FrobeniusPolynomial(A, p : parameters) : ModAbVar, RngIntElt -> RngUPolElt
            FrobeniusPolynomial(A, P) : ModAbVar, RngOrdIdl -> RngUPolElt
            Example ModAbVar_Lser-Characteristic_Polynomials_of_Frobenius_Elements (H136E116)

      Values at Integers in the Critical Strip
            L(s) : RngIntElt, ModAbVarLSer -> RngElt
            LRatio(A, s) : ModAbVar, RngIntElt -> FldRatElt
            IsZeroAt(L, s) : ModAbVarLSer, RngIntElt -> BoolElt
            Example ModAbVar_Lser-Values_at_Integers_in_the_Critical_Strip (H136E117)

      Leading Coefficient
            LeadingCoefficient(L, s, prec) : ModAbVarLSer, RngIntElt, RngIntElt -> FldReElt, RngIntElt
            Example ModAbVar_Lser-Leading_Coefficient (H136E118)
            Example ModAbVar_Lser-Leading_Coefficient2 (H136E119)

 
Complex Period Lattice

      Period Map
            PeriodMapping(A, prec) : ModAbVar, RngIntElt -> Map

      Period Lattice
            Periods(A, n) : ModAbVar, RngIntElt -> SeqEnum

 
Tamagawa Numbers and Component Groups of Neron Models

      Component Groups
            ComponentGroupOrder(A, p) : ModAbVar, RngIntElt -> RngIntElt
            Example ModAbVar_Compgrp-Component_Groups (H136E120)

      Tamagawa Numbers
            TamagawaNumber(A, p) : ModAbVar, RngIntElt -> RngIntElt, RngIntElt, BoolElt
            TamagawaNumber(A) : ModAbVar -> RngIntElt, RngIntElt, BoolElt
            Example ModAbVar_Compgrp-Tamagawa_Numbers (H136E121)

 
Elliptic Curves

      Creation
            EllipticCurve(A) : ModAbVar -> CrvEll
            ModularAbelianVariety(E) : CrvEll -> ModAbVar
            Example ModAbVar_Ellcrv-Creation (H136E122)

      Invariants
            EllipticInvariants(A, n) : ModAbVar, RngIntElt -> FldReElt, FldReElt, FldReElt, CrvEll
            EllipticPeriods(A, n) : ModAbVar, RngIntElt -> FldReElt, FldReElt
            Example ModAbVar_Ellcrv-Invariants (H136E123)

 
Bibliography

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