Brandt Modules

• Introduction
• Brandt Module Creation
  • BrandtModule(D): RngIntElt → ModBrdt
  • BrandtModule(D, m): RngIntElt, RngIntElt → ModBrdt
  • BrandtModule(A): AlgQuatOrd → ModBrdt
  • BrandtModule(A, R): AlgQuatOrd, Rng → ModBrdt
  • BaseExtend(M, R): ModBrdt, Rng → ModBrdt
  • BrandtModule(M, N): AlgQuatOrd, RngElt → ModBrdt
  • Example: Brandt Module over a Finite Field
  • Creation of Elements
    • M ! x: ModBrdt, . → ModBrdtElt
    • M . i: ModBrdt, RngIntElt → ModBrdtElt
  • Operations on Elements
    • a * x: RngElt, ModBrdtElt → ModBrdtElt
    • x * a: ModBrdtElt, RngElt → ModBrdtElt
    • x * T: ModBrdtElt, AlgMatElt → ModBrdtElt
    • x + y: ModBrdtElt, ModBrdtElt → ModBrdtElt
    • x - y: ModBrdtElt, ModBrdtElt → ModBrdtElt
    • x eq y: ModBrdtElt, ModBrdtElt → BoolElt
    • Eltseq(x): ModBrdtElt → SeqEnum
    • InnerProduct(x, y): ModBrdtElt, ModBrdtElt → RngElt
    • Norm(x): ModBrdtElt → RngElt
  • Categories and Parent
    • Category(M): ModBrdt → Cat
    • Type(M): ModBrdt → Cat
    • Category(x): ModBrdtElt → Cat
    • Type(x): ModBrdtElt → Cat
    • Parent(x): ModBrdtElt → ModBrdt
    • x in M: ModBrdtElt, ModBrdt → BoolElt
  • Elementary Invariants
    • Level(M): ModBrdt → RngIntElt
    • Discriminant(M): ModBrdt → RngIntElt
    • Conductor(M): ModBrdt → RngIntElt
    • BaseRing(M): ModBrdt → Rng
    • Basis(M): ModBrdt → SeqEnum
  • Associated Structures
    • AmbientModule(M): ModBrdt → ModBrdt
    • IsAmbient(M): ModBrdt → BoolElt
    • Dimension(M): ModBrdt → RngIntElt
    • Rank(M): ModBrdt → RngIntElt
    • Degree(M): ModBrdt → RngIntElt
    • GramMatrix(M): ModBrdt → AlgMatElt
    • InnerProductMatrix(M): ModBrdt → AlgMatElt
    • Example: Recovering the Ambient Module
    • Ideals(M): ModBrdt → []
  • Verbose Output
    • Example: Verbose Brandt Module Creation
• Subspaces and Decomposition
  • EisensteinSubspace(M): ModBrdt → ModBrdt
  • CuspidalSubspace(M): ModBrdt → ModBrdt
  • OrthogonalComplement(M): ModBrdt → ModBrdt
  • M meet N: ModBrdt, ModBrdt → ModBrdt
  • Decomposition(M, B): ModBrdt, RngIntElt → [ModBrdt]
  • SortDecomposition(D): [ModBrdt] → SeqEnum
  • Example: Decomposing a Brandt Module
  • Boolean Tests on Subspaces
    • IsEisenstein(M): ModBrdt → BoolElt
    • IsCuspidal(M): ModBrdt → BoolElt
    • IsIndecomposable(M, B): ModBrdt, RngIntElt → BoolElt
    • M1 subset M2: ModBrdt, ModBrdt → BoolElt
    • M1 lt M2: ModBrdt, ModBrdt → BoolElt
    • M1 gt M2: ModBrdt, ModBrdt → BoolElt
    • Example: Eisenstein and Cuspidal Subspaces
• Hecke Operators
  • HeckeOperator(M, n): ModBrdt, RngIntElt → AlgMatElt
  • AtkinLehnerOperator(M, p): ModBrdt, RngIntElt → AlgMatElt
• \(q\)-Expansions
  • ThetaSeries(x, y, prec): ModBrdtElt, ModBrdtElt, RngIntElt → RngSerElt
  • qExpansionBasis(M, prec): ModBrdt, RngIntElt → SeqEnum
• Dimensions of Spaces
  • BrandtModuleDimension(D, N): RngIntElt, RngIntElt → RngIntElt
  • Example: Dimensions of Brandt Modules
• Brandt Modules Over \(F_q[t]\)
  • BrandtModuleDimension(D, N): RngElt, RngElt → RngIntElt
  • BrandtModuleDimensionOfNewSubspace(D, N): RngElt, RngElt → RngIntElt
  • BrandtModule(M, N): AlgQuatOrd, RngElt → ModBrdt
  • QuaternionOrder(M): ModBrdt → AlgQuatOrd
  • Level(M): ModBrdt → RngElt
  • Discriminant(M): ModBrdt → RngElt
  • Conductor(M): ModBrdt → RngElt
  • Ideals(M): ModBrdt → []
  • InnerProductMatrix(M): ModBrdt → AlgMatElt
  • HeckeOperator(M, n): ModBrdtNew, RngElt → Mtrx
  • HeckeEigenvectors(M): ModBrdt → [ ModBrdt ]
  • HeckeEigenvalue(f, p): ModBrdtElt, RngElt → RngElt
• Bibliography