Lattices over Number Fields
- Introduction
- Number Field Lattices
- Creation of Number Field Lattices
NumberFieldLattice(K, d): FldNum, RngIntElt → LatNFNumberFieldLattice(S): [ModTupFldElt] → LatNFNumberFieldLattice(D): ModDed → LatNFNumberFieldLattice(L): Lat → LatNFNumberFieldLatticeWithGram(F): Mtrx → LatNFStandardLattice(V): SpcPlr → LatNFStandardLattice(V): SpcPlr → LatLatticeWithBasis(V, B): SpcPlr, Mtrx[RngOrd] → LatNFLatticeWithBasis(V, B): SpcPlr, Mtrx[FldAlg] → LatNFLatticeWithBasis(V, B): SpcPlr, Mtrx[RngInt] → LatLatticeWithBasis(V, B): SpcPlr, Mtrx[FldRat] → LatLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[RngOrd], [ RngOrdFracIdl ] → LatNFLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldAlg], [ RngOrdFracIdl ] → LatNFLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldQuad], [ RngOrdFracIdl ] → LatNFLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[RngInt], [ RngIntFracIdl ] → LatLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[RngInt], [ RngInt ] → LatLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldRat], [ RngIntFracIdl ] → LatLatticeWithBasis(V, B, J): SpcPlr, AlgMatElt[FldRat], [ RngInt ] → LatLatticeWithPseudobasis(V, P): SpcPlr, PMat → LatNFLattice(V, L): SpcPlr, ModDed → LatNFLattice(V, L): SpcPlr, Lat → LatModule(L): LatNF → ModDedChangeRing(L, R): LatNF, Rng → LatNFsub<L | RHS>: LatNF, Any → LatNF, Mapext<L | RHS>: LatNF, Any → LatNF, MapA + B: LatNF, LatNF → LatNFA meet B: LatNF, LatNF → LatNFr * L: RngElt, LatNF → LatNFL * r: Lat NF, LatNF → LatNFL / r: Lat NF, LatNF → LatNFBasisScaling(L, r): LatNF, RngElt → LatNFInnerProductScaling(L, r): LatNF, RngElt → LatNFScaledLattice(L, r): LatNF, FldElt → LatNFJ * L: RngOrdFracIdl, LatNF → LatNFL * J: LatNF, RngOrdFracIdl → LatNFL / J: LatNF, RngOrdFracIdl → LatNFT * L: Mtrx, LatNF → LatNFTJ * L: PMat, LatNF → LatNFL * T: LatNF, Mtrx → LatNFDirectSum(A, B): LatNF, LatNF → LatNFDirectSum(A): SeqEnum[LatNF] → LatNFOrthogonalComplement(L, v): LatNF, LatNFElt → LatNFOrthogonalComplement(L, S): LatNF, LatNF → LatNFDual(L): LatNF → LatNFSimpleLattice(L): LatNF → LatNFZLattice(L): LatNF → LatRestrictionOfScalarsToZ(L): LatNF → LatMaximalSublattices(L, p): LatNF, RngOrdIdl → [LatNF], [RngIntElt]MinimalSuperlattices(L, p): LatNF, RngOrdIdl → [LatNF], [RngIntElt]MaximalIntegralLattice(Q): Mtrx → LatNFMaximalIntegralLattice(L): LatNF → LatNFMaximalIntegralLattice(L): Lat → LatMaximalIntegralLattice(L, p): LatNF, RngOrdIdl → LatNFMaximalIntegralLattice(L, p): Lat, RngInt → LatNFMaximalIntegralLattice(L, p): Lat, RngIntElt → LatNFMaximalIntegralLattice(V): SpcPlr → LatNF- Example: Creating Number Field Lattices
- Attributes of Number Field Lattices
Basis(L): LatNF → [ModTupFldElt]BasisMatrix(L): LatNF → MtrxFreeBasis(L): LatNF → [ModTupFldElt]LocalBasis(L, p): LatNF, RngOrdIdl → [ ModTupFldElt ]LocalBasis(L, p): Lat, RngInt → [ ModTupFldElt ]PseudoBasis(L): LatNF → [ModTupFldElt]PseudoBasisMatrix(L): LatNF → MtrxPseudoMatrix(L): LatNF → PMatCoefficientIdeals(L): LatNF → SeqEnumInvolution(L): LatNF → FldAutGenerators(L): LatNF → SeqEnumGeneratorMatrix(L): LatNF → MtrxInnerProductMatrix(L): LatNF → MtrxAmbientSpace(L): LatNF → SpcPlrEmbeddingSpace(L): LatNF → ModMakeAmbientInnerProduct(~L, IP): LatNFGramMatrix(L): LatNF → MtrxPseudoGramMatrix(L): LatNF → MtrxGramMatrix(L, S): LatNF, [ ModTupFldElt ] → AlgMatEltGramMatrix(S): [ LatNFElt ] → AlgMatEltRank(L): LatNF → RngIntEltDimension(L): LatNF → RngIntEltDegree(L): LatNF → RngIntEltBaseRing(L): LatNF → FldNumCoordinateRing(L): LatNF → RngOrdOrder(L): LatNF → RngOrdDeterminant(L): LatNF → FldNumEltDiscriminant(L): LatNF → RngOrdFracIdlVolume(L): LatNF → RngOrdFracIdlNorm(L): LatNF → RngOrdFracIdlScale(L): LatNF → RngOrdFracIdlBadPrimes(L): LatNF → SetAuxiliaryForms(L): LatNF → [ AlgMatElt ]ElementaryDivisors(A, B): LatNF, LatNF → [ RngOrdFracIdl ]Discriminant(A, B): LatNF, LatNF → RngOrdFracIdlIndex(A, B): LatNF, LatNF → RngOrdFracIdlJordanDecomposition(L, p): LatNF, RndOrdIdl → List, List, SeqEnumMaximalNormSplitting(L, p): LatNF, RngOrdIdl → SeqEnum, ListGoodBasisOfNormGenerators(L, p): LatNF, RngOrdIdl → SeqEnum, SeqEnumGenusSymbol(L, p): LatNF, RngOrdIdl → SeqEnum, AnyLocalGenus(L, p): LatNF, RngOrdIdl → SymGenLocGenus(L): LatNF → SymGenHasseInvariant(M, p): AlgMatElt, RngOrdIdl → RngIntEltHasseInvariant(L, p): LatNF, RngOrdIdl → RngIntEltWittInvariant(M, p): AlgMatElt, RngOrdIdl → RngIntEltWittInvariant(L, p): LatNF, RngOrdIdl → RngIntEltSpinorNorm(L, p): LatNF, RngOrdIdl → ModTupFld, Map, BoolEltMass(L): LatNF → FldRatEltMass(L): Lat → FldRatEltLocalFactor(L, p): LatNF, RngOrdIdl → FldRatEltNeighbours(L, p): LatNF, RngOrdIdl → [LatNF], [RngIntElt]IteratedNeighbours(L, p): LatNF, RngOrdIdl → [LatNF]GenusRepresentatives(L): LatNF → [LatNF], Assoc- Example: Attributes of Number Field Lattices
NumberOfIsotropicSubspaces(L, P, k): LatNF, RngOrdIdl, RngIntElt → RngIntEltNumberOfIsotropicSubspaces(L, P, k): Lat, RngInt, RngIntElt → RngIntEltNumberOfIsotropicSubspaces(L, P, k): Lat, RngIntElt, RngIntElt → RngIntEltNumberOfNeighbors(L, P, k): LatNF, RngOrdIdl, RngIntElt → RngIntEltNumberOfNeighbors(L, P, k): Lat, RngInt, RngIntElt → RngIntEltNumberOfNeighbors(L, P, k): Lat, RngIntElt, RngIntElt → RngIntEltNeighborProcess(L, P, k): LatNF, RngOrdIdl, RngIntElt → NeighborProcNeighborProcess(L, P, k): Lat, RngInt, RngIntElt → NeighborProcNeighbor(nProc): NeighborProc → LatNFAdvance(nProc): NeighborProcSkipTo(nProc, V, S): NeighborProc, SeqEnum[ModTupFldElt[FldFin]], AlgMatElt[FldFin]
- Predicates on Number Field Lattices
IsSimple(L): LatNF → BoolEltIsFree(L): LatNF → BoolEltIsZero(L): LatNF → BoolEltIsFull(L): LatNF → BoolEltIsHermitian(L): LatNF → BoolEltIsQuadratic(L): LatNF → BoolEltIsTotallyPositiveDefinite(L): LatNF → BoolEltIsDefinite(L): LatNF → BoolElt, RngOrdEltSignature(F): Mtrx[FldAlg] → ModTupRngElt[RngInt], ModTupRngElt[RngInt], RngIntEltA eq B: LatNF, LatNF → BoolEltA ne B: LatNF, LatNF → BoolEltIsIdentical(A, B): LatNF, LatNF → BoolEltIsSublattice(S, L): LatNF, LatNF → BoolElt, MtrxS subset L: LatNF, LatNF → BoolEltIsMaximal(L): LatNF → BoolElt, LatNFIsMaximal(L, p): LatNF → BoolElt, LatNFIsIntegral(L): LatNF → BoolEltIsIntegral(L): Lat → BoolEltIsIntegral(L, p): LatNF, RngOrdIdl → BoolEltIsIntegral(L, p): Lat, RngInt → BoolEltIsIntegral(L, p): Lat, RngIntElt → BoolEltIsEven(L): LatNF → BoolEltIsMaximalIntegral(L): LatNF → BoolElt, LatNFIsMaximalIntegral(L): Lat → BoolElt, LatIsMaximalIntegral(L, p): LatNF, RngOrdIdl → BoolElt, LatNFIsMaximalIntegral(L, p): Lat, RngInt → BoolElt, LatNFIsMaximalIntegral(L, p): Lat, RngIntElt → BoolElt, LatIsModular(L): LatNF → BoolElt, RngOrdFracIdlIsModular(L, p): LatNF, RngOrdIdl → BoolElt, RngIntEltIsIsotropic(L, p): LatNF, RngOrdIdl → BoolEltIsIsotropic(L, p): LatNF, PlcNumElt → BoolEltIsLocallyIsometric(L1, L2, p): LatNF, LatNF, RngOrdIdl → BoolEltIsSameGenus(L1, L2): LatNF, LatNF → BoolEltIsRationallyEquivalent(L1, L2, p): LatNF, LatNF, RngOrdIdl → BoolEltIsRationallyEquivalent(L1, L2, p): LatNF, LatNF, PlcNumElt → BoolEltIsRationallyEquivalent(L1, L2): LatNF, LatNF → BoolElt
- Totally Positive Definite Lattices
AutomorphismGroup(L): LatNF → GrpMatProperAutomorphismGroup(L): LatNF → GrpMatIsIsometric(A, B): LatNF, LatNF → BoolElt, MtrxIsProperlyIsometric(A, B): LatNF, LatNF → BoolElt, MtrxIsSimilar(A, B): LatNF, LatNF → BoolElt, Mtrx, FldNumEltSphere(L, e): LatNF, RngElt → SetqIsRepresented(L, e): LatNF, RngElt → BoolElt, LatNFElt
- Number Field Lattice Elements
- Creation
- Parent and Element Relations
- Arithmetic
v + w: LatNFElt, LatNFElt → LatNFEltv - w: LatNFElt, LatNFElt → LatNFElt- v: LatNFElt → LatNFEltv eq w: LatNFElt, LatNFElt → BoolEltv ne w: LatNFElt, LatNFElt → BoolEltIsZero(v): LatNFElt → BoolElts * v: RngElt, LatNFElt → LatNFEltv * s: LatNFElt, RngElt → LatNFEltv / s: LatNFElt, RngElt → LatNFEltT * v: Mtrx, LatNFElt → LatNFEltv * T: LatNFElt, Mtrx → LatNFEltv ^ M: LatNFElt, Mtrx → LatNFEltv ^ G: LatNFElt, GrpMat → Setq[LatNFElt]Orbit(G, v): GrpMat, LatNFElt → Setq[LatNFElt]Stabilizer(G, v): GrpMat, LatNFElt → GrpMatNorm(v): LatNFElt → FldNumEltInnerProduct(v, w): LatNFElt, LatNFElt → FldNumElt- Example: Arithmetic of Number Field Lattice Elements
- Access Functions
- Examples
- Lorentzian Lattices
- Special Intrinsics
IsLorentzian(L): LatNF → BoolElt, ModTupFldElt, RngIntEltIsTimelike(v): LatNFElt → BoolEltIsSpacelike(v): LatNFElt → BoolEltAutomorphismGroup(L, v): LatNF, LatNFElt → GrpMat, GrpMatIsIsometric(L, v, w): LatNF, LatNFElt, LatNFElt → BoolElt, Mtrx- Example: Isometries of a Lorentzian Lattice
- Bibliography