Categories
Example ModAbVar_Categories (H136E1)
Verbose Output
Example ModAbVar_Verbosity (H136E2)
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)
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)
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)
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)
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)
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)
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)
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)
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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Mon Dec 17 14:40:36 EST 2012