Algebraic Modular Forms
- Introduction
- Creation of Spaces of Algebraic Modular Forms
AlgebraicModularForms(G): GrpRed → ModFrmAlgAlgebraicModularForms(G, W): GrpRed, GrpRep → ModFrmAlgAlgebraicModularForms(G, W, L): GrpRed, GrpRep, AlgMatElt → ModFrmAlgAlgebraicModularForms(G, W, L): GrpRed, GrpRep, LatNF → ModFrmAlgAlgebraicModularForms(G, W, L): GrpRed, GrpRep, Lat → ModFrmAlgAlgebraicModularForms(data): List → ModFrmAlgOrthogonalModularForms(Q, W): AlgMatElt, GrpRep → ModFrmAlgOrthogonalModularForms(Q, W): AlgMatElt, [RngIntElt] → ModFrmAlgOrthogonalModularForms(Q): AlgMatElt → ModFrmAlgOrthogonalModularForms(F, Q, W, p): Fld, AlgMatElt, [RngIntElt], RngIntElt → ModFrmAlgOrthogonalModularForms(L): Lat → ModFrmAlg- Example: Spaces of Orthogonal Modular Forms
UnitaryModularForms(Q, W): AlgMatElt, GrpRep → ModFrmAlgUnitaryModularForms(Q, W): AlgMatElt, [RngIntElt] → ModFrmAlgUnitaryModularForms(Q): AlgMatElt → ModFrmAlgUnitaryModularForms(F, Q, W, p): Fld, AlgMatElt, [RngIntElt], RngIntElt → ModFrmAlgUnitaryModularForms(F, n, W, p): Fld, RngIntElt, [RngIntElt], RngIntElt → ModFrmAlgUnitaryModularForms(F, n, W): Fld, RngIntElt, GrpRep → ModFrmAlgUnitaryModularForms(F, n): Fld, RngIntElt → ModFrmAlg
- Basic Properties
BaseRing(M): ModFrmAlg → RngCoefficientRing(M): ModFrmAlg → RngFieldOfDefinition(M): ModFrmAlg → FldOrdInnerForm(M): ModFrmAlg → AlgMatEltWeight(M): ModFrmAlg → GrpRepLevel(M): ModFrmAlg → LatNFIsOrthogonal(M): ModFrmAlg → BoolEltIsSpecialOrthogonal(M): ModFrmAlg → BoolEltDimension(M): ModFrmAlg → RngIntEltDimensionsOfDirectSummands(M): ModFrmAlg → [RngIntElt]VectorSpace(M): ModFrmAlg → ModTupFldRepresentation(M): ModFrmAlg → ModTupFld- Example: Level, Weight, and Dimension
Genus(M): ModFrmAlg → SymGenGenusReps(M): ModFrmAlg → [ Lat ]GenusReps(M): ModFrmAlg → [ LatNF ]SetGenus(M, reps): ModFrmAlg, [ Lat ]AutomorphismGroups(M): ModFrmAlg → [GrpMat]SetAutomorphismGroups(M, groups): ModFrmAlg, [GrpMat]
- Elements
- Operators
HeckeOperator(M, P, k): ModFrmHil, RngOrdIdl, RngIntElt → AlgMatEltHeckeOperator(M, P, k): ModFrmHil, RngInt, RngIntElt → AlgMatEltHeckeOperator(M, P, k): ModFrmHil, RngIntElt, RngIntElt → AlgMatEltHeckeOperator(M, P): ModFrmHil, RngOrdIdl → AlgMatEltHeckeOperator(M, P): ModFrmHil, RngInt → AlgMatEltHeckeOperator(M, P): ModFrmHil, RngIntElt → AlgMatElt- Example: Hecke Operators on Unitary Forms
HeckeOperator(M, P, k): ModFrmHil, RngOrdIdl, RngIntElt → AlgMatEltHeckeOperator(M, P, k): ModFrmHil, RngInt, RngIntElt → AlgMatEltHeckeOperator(M, P, k): ModFrmHil, RngIntElt, RngIntElt → AlgMatEltHeckeOperator(M, P): ModFrmHil, RngOrdIdl → AlgMatEltHeckeOperator(M, P): ModFrmHil, RngInt → AlgMatEltHeckeOperator(M, P): ModFrmHil, RngIntElt → AlgMatEltPerestroikaOperator(M, P): ModFrmHil, RngOrdIdl → AlgMatEltPerestroikaOperator(M, P): ModFrmHil, RngInt → AlgMatEltPerestroikaOperator(M, P): ModFrmHil, RngIntElt → AlgMatEltPlusOperator(M, P): ModFrmHil, RngOrdIdl → AlgMatEltPlusOperator(M, P): ModFrmHil, RngInt → AlgMatEltPlusOperator(M, P): ModFrmHil, RngIntElt → AlgMatEltHeckeOperators(M, k): ModFrmAlg, RngIntElt → [ AlgMatElt ], [ RngOrdIdl ]HeckeOperators(M, k): ModFrmAlg, RngIntElt → [ AlgMatElt ], [ RngOrdIdl ]HeckeOperators(M): ModFrmAlg, RngIntElt → [ AlgMatElt ], [ RngOrdIdl ]HeckeOperators(M): ModFrmAlg, RngIntElt → [ AlgMatElt ], [ RngOrdIdl ]SetHeckeOperator(M, T, P, k): ModFrmAlg, AlgMatElt, RngOrdIdl, RngIntEltSetHeckeOperator(M, T, P, k): ModFrmAlg, AlgMatElt, RngInt, RngIntEltSetHeckeOperator(M, T, P, k): ModFrmAlg, AlgMatElt, RngIntElt, RngIntEltSetHeckeOperator(M, T, P): ModFrmAlg, AlgMatElt, RngOrdIdlSetHeckeOperator(M, T, P): ModFrmAlg, AlgMatElt, RngIntSetHeckeOperator(M, T, P): ModFrmAlg, AlgMatElt, RngIntEltHeckeImages(M, i, n, k): ModFrmAlg, RngIntElt, RngIntElt, RngIntElt → AssocHeckeImages(M, i, ps, k): ModFrmAlg, RngIntElt, SeqEnum, RngIntElt → Assoc
- Creation of Subspaces
- Eigenspace Decomposition and Eigenforms
Decomposition(M, B): ModFrmAlg, RngIntElt → [ ModTupFld ], BoolEltDecomposition(M): ModFrmAlg → [ ModTupFld ]HeckeEigenforms(M): ModFrmAlg → [ ModFrmAlgElt ]EisensteinSeries(M): ModFrmAlg → [ ModFrmAlgElt ]Eigenvectors(M, D): ModFrmAlg, [ ModTupFld ] → List, BoolEltHeckeEigenform(M, data): ModFrmAlg, Tup → ModFrmAlgEltHeckeEigenvalue(f, P): ModFrmAlgElt, RngOrdIdl → FldEltHeckeEigenvalue(f, P): ModFrmAlgElt, RngInt → FldEltHeckeEigenvalue(f, P): ModFrmAlgElt, RngIntElt → FldEltHeckeEigenvalues(f, P): ModFrmAlgElt, RngOrdIdl → [ FldElt ]HeckeEigenvalues(f, P): ModFrmAlgElt, RngInt → [ FldElt ]HeckeEigenvalues(f, P): ModFrmAlgElt, RngIntElt → [ FldElt ]HeckeEigensystems(M, k): ModFrmAlg, RngIntElt → List, [ RngOrdIdl ]HeckeEigensystems(M, k): ModFrmAlg, RngIntElt → List, [ RngInt ]DisplayHeckeEigensystem(f): ModFrmAlgEltHeckeEigensystem(f, k): ModFrmAlgElt, RngIntElt → [ FldElt ], [ RngOrdIdl ]HeckeEigensystem(f, k): ModFrmAlgElt, RngIntElt → [ FldElt ], [ RngInt ]HeckeEigensystem(f): ModFrmAlgElt → [ FldElt ], [ RngOrdIdl ]HeckeEigensystem(f): ModFrmAlgElt → [ FldElt ], [ RngInt ]LPolynomial(f, P, d): ModFrmAlgElt, RngOrdIdl, RngIntElt → RngSerPowEltLPolynomial(f, P, d): ModFrmAlgElt, RngInt, RngIntElt → RngSerPowEltLPolynomial(f, P, d): ModFrmAlgElt, RngIntElt, RngIntElt → RngSerPowEltLPolynomial(f, P): ModFrmAlgElt, RngOrdIdl → RngUPolEltLPolynomial(f, P): ModFrmAlgElt, RngInt → RngUPolEltLPolynomial(f, P): ModFrmAlgElt, RngIntElt → RngUPolEltLPolynomials(f): ModFrmAlgElt → [ RngUPolElt ]SatakePolynomialUnramified(M): ModFrmAlg → RngUPolEltSatakePolynomial(f, p): ModFrmAlgElt, RngIntElt → RngSerPowEltSatakePolynomial(f, p): ModFrmAlgElt, RngIntElt → RngUPolEltSatakePolynomialBallot(r, a): RngIntElt, RngIntElt → RngUPolElt- Example: Eigenforms for Half-Discriminant 11
- Relation to Other Modular Forms
ModularForm(f): ModFrmAlgElt → ModFrmEltThetaSeries(f): ModFrmAlgElt → RngSerPowEltTheta1(f): ModFrmAlgElt → RngSerPowEltThetaSiegel(f, g): ModFrmAlgElt, RngIntElt → AssocTheta2(f): ModFrmAlgElt → AssocShimuraLift(f, k, N): RngSerPowElt, RngIntElt, RngIntElt → RngSerPowElt- Example: Shimura Lift of a Ternary Form
- Further Examples
- Bibliography