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
Introduction
Categories
Example ModAbVar_Categories (H148E1)
Verbose Output
Example ModAbVar_Verbosity (H148E2)
Creation and Basic Functions
Creating the Modular Jacobian J0(N)
JZero(N : parameters) : RngIntElt -> ModAbVar
Example ModAbVar_Modabvar-Creating_the_Modular_Jacobian_$J_0(N)$ (H148E3)
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 -> ModAbVar
JH(N, gens : parameters) : RngIntElt, [RngIntElt] -> ModAbVar
Example ModAbVar_Modabvar-Creating_the_Modular_Jacobians_$J_1(N)$_and_$J_H(N)$ (H148E4)
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 (H148E5)
Example ModAbVar_Modabvar-Abelian_Varieties_Attached_to_Modular_Forms3 (H148E6)
Example ModAbVar_Modabvar-Abelian_Varieties_Attached_to_Modular_Forms5 (H148E7)
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 (H148E8)
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 (H148E9)
Creation Using a Label
ModularAbelianVariety(s : parameters) : MonStgElt -> ModAbVar
Example ModAbVar_Modabvar-Creation_Using_a_Label (H148E10)
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 (H148E11)
Example ModAbVar_Modabvar-Invariants2 (H148E12)
Example ModAbVar_Modabvar-Invariants3 (H148E13)
Example ModAbVar_Modabvar-Invariants3 (H148E14)
Example ModAbVar_Modabvar-Invariants4 (H148E15)
Conductor
Conductor(A) : ModAbVar -> RngIntElt
Example ModAbVar_Modabvar-Conductor (H148E16)
Number of Points
NumberOfRationalPoints(A) : ModAbVar -> RngIntElt, RngIntElt
#A : ModAbVar -> RngIntElt
Example ModAbVar_Modabvar-Number_of_Points (H148E17)
Inner Twists and Complex Multiplication
CMTwists(A : parameters) : ModAbVar -> SeqEnum
InnerTwists(A : parameters) : ModAbVar -> SeqEnum
Example ModAbVar_Modabvar-Inner_Twists_and_Complex_Multiplication (H148E18)
Example ModAbVar_Modabvar-Inner_Twists_and_Complex_Multiplication2 (H148E19)
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 (H148E20)
Example ModAbVar_Modabvar-Predicates2 (H148E21)
Example ModAbVar_Modabvar-Predicates3 (H148E22)
Example ModAbVar_Modabvar-Predicates4 (H148E23)
Example ModAbVar_Modabvar-Predicates5 (H148E24)
Example ModAbVar_Modabvar-Predicates6 (H148E25)
Example ModAbVar_Modabvar-Predicates7 (H148E26)
Example ModAbVar_Modabvar-Predicates8 (H148E27)
Equality and Inclusion Testing
A eq B : ModAbVar, ModAbVar -> BoolElt
A subset B : ModAbVar, ModAbVar -> BoolElt
Example ModAbVar_Modabvar-Equality_and_Inclusion_Testing (H148E28)
Example ModAbVar_Modabvar-Equality_and_Inclusion_Testing2 (H148E29)
Modular Embedding and Parameterization
CommonModularStructure(X) : [ModAbVar] -> List, List
ModularEmbedding(A) : ModAbVar -> MapModAbVar
ModularParameterization(A) : ModAbVar -> MapModAbVar
Example ModAbVar_Modabvar-Modular_Embedding_and_Parameterization (H148E30)
Example ModAbVar_Modabvar-Modular_Embedding_and_Parameterization2 (H148E31)
Coercion
A ! x : ModAbVar, . -> BoolElt, ModAbVarElt
Example ModAbVar_Modabvar-Coercion (H148E32)
Example ModAbVar_Modabvar-Coercion2 (H148E33)
Modular Symbols to Homology
ModularSymbolToIntegralHomology(A, x) : ModAbVar, SeqEnum -> ModTupFldElt
ModularSymbolToRationalHomology(A, x) : ModAbVar, ModSymElt -> ModTupFldElt
Example ModAbVar_Modabvar-Modular_Symbols_to_Homology (H148E34)
Example ModAbVar_Modabvar-Modular_Symbols_to_Homology2 (H148E35)
Embeddings
Embeddings(A) : ModAbVar -> List
AssertEmbedding(~A, phi) : ModAbVar, MapModAbVar ->
Example ModAbVar_Modabvar-Embeddings (H148E36)
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 (H148E37)
Additional Examples
Example ModAbVar_modabvar1 (H148E38)
Example ModAbVar_additional2 (H148E39)
Example ModAbVar_additional3 (H148E40)
Example ModAbVar_additional4 (H148E41)
Example ModAbVar_additional5 (H148E42)
Example ModAbVar_additional6 (H148E43)
Homology
Creation
Homology(A) : ModAbVar -> ModAbVarHomol
Example ModAbVar_Homology-Creation (H148E44)
Invariants
Dimension(H) : ModAbVarHomol -> RngIntElt
Example ModAbVar_Homology-Invariants (H148E45)
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 (H148E46)
Example ModAbVar_Homology-Functors_to_Categories_of_Lattices_and_Vector_Spaces (H148E47)
Example ModAbVar_Homology-Functors_to_Categories_of_Lattices_and_Vector_Spaces2 (H148E48)
Modular Structure
IsAttachedToModularSymbols(H) : ModAbVarHomol -> BoolElt
ModularSymbols(H) : ModAbVarHomol -> SeqEnum
Example ModAbVar_Homology-Modular_Structure (H148E49)
Additional Examples
Example ModAbVar_homology1 (H148E50)
Example ModAbVar_additional20 (H148E51)
Homomorphisms
Creation
IdentityMap(A) : ModAbVar -> MapModAbVar
ZeroMap(A) : ModAbVar -> MapModAbVar
nIsogeny(A, n) : ModAbVar, FldRatElt -> MapModAbVar
Example ModAbVar_Morphisms-Creation (H148E52)
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 (H148E53)
Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations2 (H148E54)
Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations3 (H148E55)
Example ModAbVar_Morphisms-Restriction,_Evaluation,_and_Other_Manipulations4 (H148E56)
Kernels
ComponentGroupOfKernel(phi) : MapModAbVar -> ModAbVarSubGrp
ConnectedKernel(phi) : MapModAbVar -> ModAbVar, MapModAbVar
Kernel(phi) : MapModAbVar -> ModAbVarSubGrp, ModAbVar, MapModAbVar
Example ModAbVar_Morphisms-Kernels (H148E57)
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 (H148E58)
Cokernels
Cokernel(phi) : MapModAbVar -> ModAbVar, MapModAbVar
Example ModAbVar_Morphisms-Cokernels (H148E59)
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 (H148E60)
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 (H148E61)
Polynomials
CharacteristicPolynomial(phi) : MapModAbVar -> RngUPolElt
FactoredCharacteristicPolynomial(phi) : MapModAbVar -> RngUPolElt
MinimalPolynomial(phi) : MapModAbVar -> RngUPolElt
Example ModAbVar_Morphisms-Polynomials (H148E62)
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 (H148E63)
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 (H148E64)
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 (H148E65)
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 (H148E66)
Example ModAbVar_Homspace-Subgroups__and_Subrings2 (H148E67)
Example ModAbVar_Homspace-Subgroups__and_Subrings3 (H148E68)
Example ModAbVar_Homspace-Subgroups__and_Subrings4 (H148E69)
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 (H148E70)
Arithmetic
H1 + H2 : HomModAbVar, HomModAbVar -> HomModAbVar
H1 meet H2 : HomModAbVar, HomModAbVar -> HomModAbVar
Example ModAbVar_Homspace-Arithmetic (H148E71)
Quotients
Index(H2, H1) : HomModAbVar, HomModAbVar -> RngIntElt
Quotient(H2, H1) : HomModAbVar, HomModAbVar -> GrpAb, Map, Map
Example ModAbVar_Homspace-Quotients (H148E72)
Invariants
Domain(H) : HomModAbVar -> ModAbVar
Codomain(H) : HomModAbVar -> ModAbVar
FieldOfDefinition(H) : HomModAbVar -> ModAbVar
Discriminant(H) : HomModAbVar -> FldRatElt, AlgMatElt
Example ModAbVar_Homspace-Invariants (H148E73)
Structural Invariants
Basis(H) : HomModAbVar -> SeqEnum
Dimension(H) : HomModAbVar -> RngIntElt
Ngens(H) : HomModAbVar -> RngIntElt
H . i : HomModAbVar, RngIntElt -> MapModAbVar
Example ModAbVar_Homspace-Structural_Invariants (H148E74)
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 (H148E75)
Example ModAbVar_Homspace-Matrix_and_Module_Structure3 (H148E76)
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 (H148E77)
Elements
H ! x : HomModAbVar, . -> BoolElt, MapModAbVar
Example ModAbVar_Homspace-Random_Element (H148E78)
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 (H148E79)
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 (H148E80)
Example ModAbVar_Arithabvar-Intersections2 (H148E81)
Quotients
A / B : ModAbVar, ModAbVar -> ModAbVar, MapModAbVar
Cokernel(phi) : MapModAbVar -> ModAbVar, MapModAbVar
Example ModAbVar_Arithabvar-Quotients (H148E82)
Decomposing and Factoring Abelian Varieties
Decomposition
Decomposition(A) : ModAbVar -> SeqEnum
A(n) : RngIntElt, ModAbVar -> ModAbVar
Example ModAbVar_Decomp-Decomposition (H148E83)
Factorization
Factorisation(A) : ModAbVar -> List
Example ModAbVar_Decomp-Factorization (H148E84)
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 (H148E85)
Additional Examples
Example ModAbVar_homology1 (H148E86)
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 (H148E87)
Orthogonal Complements
Complements
Complement(A : parameters) : ModAbVar -> ModAbVar, MapModAbVar
ComplementOfImage(phi : parameters) : MapModAbVar -> ModAbVar, MapModAbVar
Example ModAbVar_Complements-Complements (H148E88)
Dual Abelian Variety
IsDualComputable(A) : ModAbVar -> BoolElt, ModAbVar
Dual(A) : ModAbVar -> ModAbVar
ModularPolarization(A) : ModAbVar -> MapModAbVar
Example ModAbVar_Complements-Dual_Abelian_Variety (H148E89)
Intersection Pairing
IntersectionPairing(H) : ModAbVarHomol -> AlgMatElt
IntersectionPairing(A) : ModAbVar -> AlgMatElt
IntersectionPairingIntegral(A) : ModAbVar -> AlgMatElt
Example ModAbVar_Complements-Intersection_Pairing (H148E90)
Projections
ProjectionOnto(A : parameters) : ModAbVar -> MapModAbVar
Example ModAbVar_Complements-Projections (H148E91)
Example ModAbVar_Complements-Projections2 (H148E92)
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 (H148E93)
Congruence Computations
CongruenceModulus(A) : ModAbVar -> RngIntElt
ModularDegree(A) : ModAbVar -> RngIntElt
Example ModAbVar_Complements-Congruence_Computations (H148E94)
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 (H148E95)
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 (H148E96)
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 (H148E97)
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 (H148E98)
Invariants
Order(x) : ModAbVarElt -> RngIntElt
ApproximateOrder(x) : ModAbVarElt -> RngIntElt
Degree(x) : ModAbVarElt -> RngIntElt
FieldOfDefinition(x) : ModAbVarElt -> ModTupFldElt
Example ModAbVar_Elt-Invariants (H148E99)
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 (H148E100)
Homomorphisms
x @ phi : ModAbVarElt, MapModAbVar -> ModAbVarElt
x @@ phi : ModAbVarElt, MapModAbVar -> ModAbVarElt
Example ModAbVar_Elt-Homomorphisms (H148E101)
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 (H148E102)
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 (H148E103)
Elements
Elements(G) : ModAbVarSubGrp -> SeqEnum
Generators(G) : ModAbVarSubGrp -> SeqEnum
Ngens(G) : ModAbVarSubGrp -> RngIntElt
G . i : ModAbVarSubGrp, RngIntElt -> ModAbVarElt
Example ModAbVar_Subgrp-Elements (H148E104)
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 (H148E105)
Underlying Abelian Group and Lattice
AbelianGroup(G) : ModAbVarSubGrp -> GrpAb, Map, Map
Lattice(G) : ModAbVarSubGrp -> Lat
Example ModAbVar_Subgrp-Underlying_Abelian_Group_and_Lattice (H148E106)
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 (H148E107)
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 (H148E108)
Rational Torsion Subgroups
Cuspidal Subgroup
CuspidalSubgroup(A) : ModAbVar -> ModAbVarSubGrp
RationalCuspidalSubgroup(A) : ModAbVar -> ModAbVarSubGrp
Example ModAbVar_Torsion-Cuspidal_Subgroup (H148E109)
Upper and Lower Bounds
TorsionLowerBound(A) : ModAbVar -> RngIntElt
TorsionMultiple(A) : ModAbVar -> RngIntElt
Example ModAbVar_Torsion-Upper_and_Lower_Bounds (H148E110)
Torsion Subgroup
TorsionSubgroup(A) : ModAbVar -> BoolElt, ModAbVarSubGrp
Example ModAbVar_Torsion-Torsion_Subgroup (H148E111)
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 (H148E112)
Invariants
HeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt
FactoredHeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt
MinimalHeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt
Example ModAbVar_Operators-Invariants (H148E113)
L-series
Creation
LSeries(A) : ModAbVar -> ModAbVarLSer
Example ModAbVar_Lser-Creation (H148E114)
Invariants
CriticalStrip(L) : ModAbVarLSer -> RngIntElt, RngIntElt
ModularAbelianVariety(L) : ModAbVarLSer -> ModAbVar
Example ModAbVar_Lser-Invariants (H148E115)
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 (H148E116)
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 (H148E117)
Leading Coefficient
LeadingCoefficient(L, s, prec) : ModAbVarLSer, RngIntElt, RngIntElt -> FldReElt, RngIntElt
Example ModAbVar_Lser-Leading_Coefficient (H148E118)
Example ModAbVar_Lser-Leading_Coefficient2 (H148E119)
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 (H148E120)
Tamagawa Numbers
TamagawaNumber(A, p) : ModAbVar, RngIntElt -> RngIntElt, RngIntElt, BoolElt
TamagawaNumber(A) : ModAbVar -> RngIntElt, RngIntElt, BoolElt
Example ModAbVar_Compgrp-Tamagawa_Numbers (H148E121)
Elliptic Curves
Creation
EllipticCurve(A) : ModAbVar -> CrvEll
ModularAbelianVariety(E) : CrvEll -> ModAbVar
Example ModAbVar_Ellcrv-Creation (H148E122)
Invariants
EllipticInvariants(A, n) : ModAbVar, RngIntElt -> FldReElt, FldReElt, FldReElt, CrvEll
EllipticPeriods(A, n) : ModAbVar, RngIntElt -> FldReElt, FldReElt
Example ModAbVar_Ellcrv-Invariants (H148E123)
Bibliography
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