Local Galois Representations

• Overview
  • Notation and Printing
    • Example: Galrep Printing
  • Conventions
  • Implementation Notes
• Creating Galois Representations
  • ZeroRepresentation(K): FldPad → GalRep
  • Example: Galrep Zerorepresentation
  • PrincipalCharacter(K): FldPad → GalRep
  • Example: Galrep Principalcharacter
  • CyclotomicCharacter(K): FldPad → GalRep
  • Example: Galrep Cyclotomiccharacter
  • UnramifiedCharacter(K,c): FldPad, Any → GalRep
  • Example: Galrep Unramifiedcharacter
  • UnramifiedRepresentation(K,CharPoly): FldPad, RngUPolElt → GalRep
  • Example: Galrep Unramifiedrepresentation
  • UnramifiedRepresentation({K,dim,}{dimcomputed,CharPoly}): FldPad, RngIntElt, RngIntElt, RngUPolElt → GalRep
  • Example: Galrep Notfullycomputed
  • SP(K,n): FldPad, RngIntElt → GalRep
  • Example: Galrep Sp
  • SP(K,f,n): FldPad, RngUPolElt, RngIntElt → GalRep
  • Example: Galrep Sp 1
  • Representations from Finite Extensions
    • GaloisRepresentations(F,K): FldPad, FldPad → SeqEnum[GalRep]
    • Example: Galrep Galoisrepresentations 1
    • GaloisRepresentations(f): RngUPolEltFldPad → SeqEnum[GalRep]
    • Example: Galrep Galoisrepresentations
    • PermutationCharacter(F,K): FldPad, FldPad → GalRep
    • Example: Galrep Permutationcharacter
    • A !! chi: GalRep, AlgChtrElt → GalRep
    • A !! chi: GalRep, SeqEnum → GalRep
    • Example: Galrep Smash
  • Local Representations of Global Objects
    • GaloisRepresentation(chi,p): GrpDrchElt, RngIntElt → GalRep
    • Example: Galrep Dirichlet
    • GaloisRepresentation(A,p): ArtRep, RngIntElt → GalRep
    • Example: Galrep Artin
    • GaloisRepresentation(E): CrvEll → GalRep
    • Example: Galrep E
    • GaloisRepresentation(E,p): CrvEll, RngIntElt → GalRep
    • Example: Galrep E P
    • GaloisRepresentation(E,P): CrvEll, RngOrdIdl → GalRep
    • Example: Galrep E P
    • GaloisRepresentation(C): CrvHyp[FldPad] → GalRep
    • Example: Galrep Crvhyp Qp
    • GaloisRepresentation(C,p): CrvHyp[FldRat], RngIntElt → GalRep
    • Example: Galrep Crvhyp Q
    • GaloisRepresentation(C,P): CrvHyp[FldNum], RngOrdIdl → GalRep
    • Example: Galrep Crvhyp Nf
    • GaloisRepresentation(f,p): ModFrmElt, RngIntElt → GalRep
    • Example: Galrep F P
• Basic Invariants
  • BaseField(A): GalRep → FldPad
  • Degree(A): GalRep → RngIntElt
  • Dimension(A): GalRep → RngIntElt
  • Example: Galrep Degree
  • Group(A): GalRep → GrpPerm
  • GaloisGroup(A): GalRep → GrpPerm
  • Example: Galrep Group
  • FrobeniusElement(A): GalRep → GrpPermElt
  • Example: Galrep Frobeniuselement
  • Character(A): GalRep → AlgChtrElt
  • Example: Galrep Character
  • Field(A): GalRep → FldPad
  • DefiningPolynomial(A): GalRep → RngUPolElt
  • Example: Galrep Definingpolynomial
  • Automorphism(A,g): GalRep, GrpPermElt → Map
  • Example: Galrep Automorphism
  • EulerFactor(A): GalRep → RngUPolElt
  • Example: Galrep Eulerfactor
  • IsZero(A): GalRep → BoolElt
  • IsOne(A): GalRep → BoolElt
  • Factorization(A): GalRep → List, GalRep
  • Example: Galrep Factorization
  • Ramification
    • InertiaGroup(A): GalRep → GrpPerm
    • Example: Galrep Inertia
    • InertiaGroup(A,n): GalRep, RngIntElt → GrpPerm
    • Example: Galrep Higherinertia
    • IsUnramified(A): GalRep → BoolElt
    • Example: Galrep Isunramified
    • IsRamified(A): GalRep → BoolElt
    • Example: Galrep Isramified
    • IsTamelyRamified(A): GalRep → BoolElt
    • IsWildlyRamified(A): GalRep → BoolElt
    • Example: Galrep Iswildlyramified
    • InertiaInvariants(A): GalRep → GalRep
    • Example: Galrep Inertiainvariants
    • ConductorExponent(A): GalRep → RngIntElt
    • Conductor(A): GalRep → FldPadElt
    • Example: Galrep Conductor
    • EpsilonFactor(A): GalRep → FldComElt
    • RootNumber(A): GalRep → FldComElt
    • Example: Epsilon Rootnumber
  • Semisimplicity and Irreducibles
    • IsIrreducible(A): GalRep → BoolElt
    • Example: Galrep Isirreducible
    • IsIndecomposable(A): GalRep → BoolElt
    • IsSemisimple(A): GalRep → BoolElt
    • Semisimplification(A): GalRep → GalRep
    • Decomposition(A): GalRep → SeqEnum[GalRep]
    • Example: Galrep Decomposition
• Arithmetic
  • A1 + A2: GalRep, GalRep → GalRep
  • Example: Galrep Sum
  • A1 - A2: GalRep, GalRep → BoolElt
  • Example: Galrep Difference
  • A1 * A2: GalRep, GalRep → GalRep
  • Example: Galrep Product
  • A1 / A2: GalRep, GalRep → GalRep
  • A1 / A2: RngIntElt, GalRep → GalRep
  • A ^ n: GalRep, Any → GalRep
  • Example: Galrep Power
  • A1 eq A2: GalRep, GalRep → BoolElt
  • Example: Galrep Eq
  • Determinant(A): GalRep → GalRep
  • Example: Galrep Determinant
  • TateTwist(A,n): GalRep, RngIntElt → GalRep
  • Example: Galrep Tatetwist
• Changing Precision
  • Precision(A): GalRep → RngIntElt
  • ChangePrecision(~A,Prec): GalRep, RngIntElt → {}
  • ChangePrecision(A,Prec): GalRep, RngIntElt → GalRep
  • Example: Galrep Precision
• Changing Fields
  • Minimize(A): GalRep → GalRep
  • Example: Galrep Minimize
  • Restriction(A,L): GalRep, FldPad → GalRep
  • BaseChange(A,L): GalRep, FldPad → GalRep
  • Example: Galrep Basechange
  • Induction(A,K0): GalRep, FldPad → GalRep
  • Example: Galrep Induction
• Advanced Examples
  • Example: Local and Global Epsilon Factors for Dirichlet Characters
    • Example: Galrep Funeq
  • Example: Reconstructing a Galois Representation from its Euler Factors
    • Example: Galrep Reconstruction