Modular Abelian Varieties
- Introduction
- Creation and Basic Functions
- Creating the Modular Jacobian \(J_0(N)\)
- Creating the Modular Jacobians \(J_1(N)\) and \(J_H(N)\)
JOne(N : parameters): RngIntElt → ModAbVarJOne(N, k : parameters): RngIntElt, RngIntElt → ModAbVarJs(N : parameters): RngIntElt → ModAbVarJs(N, k : parameters): RngIntElt, RngIntElt → ModAbVarJH(N, d : parameters): RngIntElt, RngIntElt → ModAbVarJH(N, k, d : parameters): RngIntElt, RngIntElt, RngIntElt → ModAbVarJH(N, gens : parameters): RngIntElt, [RngIntElt] → ModAbVarJH(N, k, gens : parameters): RngIntElt, RngIntElt, [RngIntElt] → ModAbVar- Example: Creating the 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
CanDetermineIsomorphism(A, B): ModAbVar, ModAbVar → BoolElt, BoolElt, MapModAbVarHasMultiplicityOne(A): ModAbVar → BoolEltIsAbelianVariety(A): ModAbVar → BoolEltIsAttachedToModularSymbols(A): ModAbVar → BoolEltIsAttachedToNewform(A): ModAbVar → BoolElt, ModAbVar, MapModAbVarIsIsogenous(A, B): ModAbVar, ModAbVar → BoolEltIsIsomorphic(A, B): ModAbVar, ModAbVar → BoolElt, MapModAbVarIsOnlyMotivic(A): ModAbVar → BoolEltIsQuaternionic(A): ModAbVar → BoolEltIsSelfDual(A): ModAbVar → BoolEltIsSimple(A): ModAbVar → BoolElt- Example: Testing for Abelian Varieties
- Example: Modular Motives as Abelian Varieties
- Example: Attachment to Modular Symbols
- Example: Testing for Isogeny
- Example: Multiplicity One of Simple Factors
- Example: A Non-Quaternionic Surface
- Example: Self-Dual Abelian Varieties
- Example: Testing for Simplicity
- Equality and Inclusion Testing
- Modular Embedding and Parameterization
- Coercion
- Modular Symbols to Homology
ModularSymbolToIntegralHomology(A, x): ModAbVar, SeqEnum → ModTupFldEltModularSymbolToIntegralHomology(A, x): ModAbVar, Tup → ModTupFldEltModularSymbolToRationalHomology(A, x): ModAbVar, ModSymElt → ModTupFldEltModularSymbolToRationalHomology(A, x): ModAbVar, SeqEnum → ModTupFldEltModularSymbolToRationalHomology(A, x): ModAbVar, Tup → ModTupFldElt- Example: Modular Symbols in Integral Homology
- Example: Modular Symbols for Higher Weight
- Embeddings
- Base Change
- Additional Examples
- Homology
- Homomorphisms
- Creation
- Restriction, Evaluation, and Other Manipulations
Restriction(phi, B): MapModAbVar, ModAbVar → MapModAbVarRestrictEndomorphism(phi, B): MapModAbVar, ModAbVar → MapModAbVarRestrictEndomorphism(phi, i): MapModAbVar, MapModAbVar → MapModAbVarRestrictionToImage(phi, i): MapModAbVar, MapModAbVar → MapModAbVarEvaluate(f, phi): RngUPolElt, MapModAbVar → MapModAbVarDivideOutIntegers(phi): MapModAbVar → MapModAbVar, RngIntEltSurjectivePart(phi): MapModAbVar → MapModAbVarUniversalPropertyOfCokernel(pi, f): MapModAbVar, MapModAbVar → MapModAbVar- Example: Cutting Out a Subvariety
- Example: Universal Property of the Cokernel
- Example: Dividing a Homomorphism by an Integer
- Example: Restricting Homomorphisms to Factors
- Kernels
- Images
- Cokernels
- Matrix Structure
- Arithmetic
Inverse(phi): MapModAbVar → MapModAbVar, RngIntEltphi * psi: MapModAbVar, MapModAbVar → MapModAbVara * phi: FldRatElt, MapModAbVar → MapModAbVara * phi: RngIntElt, MapModAbVar → MapModAbVarphi * psi: MapModAbVar, AlgMatElt → AlgMatEltphi * psi: MapModAbVar, ModMatFldElt → ModMatFldEltpsi * phi: AlgMatElt, MapModAbVar → AlgMatEltpsi * phi: ModMatFldElt, MapModAbVar → ModMatFldEltphi ^ n: MapModAbVar, RngIntElt → MapModAbVarphi + psi: MapModAbVar, MapModAbVar → MapModAbVarn + phi: FldRatElt, MapModAbVar → MapModAbVarn + phi: RngIntElt, MapModAbVar → MapModAbVarphi + n: MapModAbVar, RngIntElt → MapModAbVarphi + psi: MapModAbVar, AlgMatElt → AlgMatEltphi + psi: MapModAbVar, ModMatFldElt → ModMatFldEltpsi + phi: AlgMatElt, MapModAbVar → AlgMatEltpsi + phi: ModMatFldElt, MapModAbVar → ModMatFldEltphi - psi: MapModAbVar, MapModAbVar → MapModAbVarn - phi: FldRatElt, MapModAbVar → MapModAbVarn - phi: RngIntElt, MapModAbVar → MapModAbVarphi - n: MapModAbVar, FldRatElt → MapModAbVarphi - n: MapModAbVar, RngIntElt → MapModAbVarphi - psi: MapModAbVar, AlgMatElt → AlgMatEltphi - psi: MapModAbVar, ModMatFldElt → ModMatFldEltpsi - phi: AlgMatElt, MapModAbVar → AlgMatEltpsi - phi: ModMatFldElt, MapModAbVar → ModMatFldElt- Example: Arithmetic with Hecke Operators
- Polynomials
- Invariants
- Predicates
- Endomorphism Algebras and Hom Spaces
- Creation
Hom(A, B): ModAbVar, ModAbVar → HomModAbVarHom(A, B, oQ): ModAbVar, ModAbVar, BoolElt → HomModAbVarEnd(A): ModAbVar → HomModAbVarEnd(A, oQ): ModAbVar, BoolElt → HomModAbVarBaseExtend(H, R): HomModAbVar, Rng → HomModAbVarHeckeAlgebra(A): ModAbVar → HomModAbVar- Example: Creating Homomorphism Groups
- 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
DirectSum(A, B): ModAbVar, ModAbVar → ModAbVar, List, ListDirectProduct(A, B): ModAbVar, ModAbVar → ModAbVar, List, ListA * B: ModAbVar, ModAbVar → ModAbVar, List, ListDirectSum(X): [ModAbVar] → ModAbVar, List, ListDirectProduct(X): [ModAbVar] → ModAbVar, List, ListA ^ n: ModAbVar, RngIntElt → ModAbVar- Example: Products of Modular Abelian Varieties
- Sum in an Ambient Variety
- Intersections
- Quotients
- Decomposing and Factoring Abelian Varieties
- Building Blocks
- Orthogonal Complements
- New and Old Subvarieties and Natural Maps
- Elements of Modular Abelian Varieties
- Subgroups of Modular Abelian Varieties
- Creation
- Elements
- Arithmetic
Quotient(A, G): ModAbVar, ModAbVarSubGrp → ModAbVar, MapModAbVarQuotient(G): ModAbVarSubGrp → ModAbVar, MapModAbVar, MapModAbVarA / G: ModAbVar, ModAbVarSubGrp → ModAbVar, MapModAbVar, MapModAbVarA meet G: ModAbVar, ModAbVarSubGrp → ModAbVarSubGrpG meet A: ModAbVarSubGrp, ModAbVar → ModAbVarSubGrpG1 + G2: ModAbVarSubGrp, ModAbVarSubGrp → ModAbVarSubGrpG1 meet G2: ModAbVarSubGrp, ModAbVarSubGrp → ModAbVarSubGrp- Example: Arithmetic of Finite Subgroups
- Underlying Abelian Group and Lattice
- Invariants
- Predicates and Comparisons
- Rational Torsion Subgroups
- Hecke and Atkin-Lehner Operators
- \(L\)-series
- Creation
- Invariants
- Characteristic Polynomials of Frobenius Elements
- Values at Integers in the Critical Strip
L(s): RngIntElt, ModAbVarLSer → RngEltEvaluate(L, s): ModAbVarLSer, RngIntElt → FldReEltEvaluate(L, s, prec): ModAbVarLSer, RngIntElt, RngIntElt → FldReEltLRatio(A, s): ModAbVar, RngIntElt → FldRatEltLRatio(L, s): ModAbVarLSer, RngIntElt → FldRatEltIsZeroAt(L, s): ModAbVarLSer, RngIntElt → BoolElt- Example: Evaluating L-Series at Integers
- Leading Coefficient
- Complex Period Lattice
- Tamagawa Numbers and Component Groups of Neron Models
- Elliptic Curves
- Bibliography