Local Galois Representations
- Overview
- Creating Galois Representations
ZeroRepresentation(K): FldPad → GalRep- Example: Zero Galois Representation
PrincipalCharacter(K): FldPad → GalRep- Example: Principal Character of a Local Field
CyclotomicCharacter(K): FldPad → GalRep- Example: Cyclotomic Character
UnramifiedCharacter(K,c): FldPad, Any → GalRep- Example: Unramified Characters
UnramifiedRepresentation(K,CharPoly): FldPad, RngUPolElt → GalRep- Example: Unramified Galois Representations
UnramifiedRepresentation(K,dim, dimcomputed,CharPoly): FldPad, RngIntElt, RngIntElt, RngUPolElt → GalRep- Example: Representations Not Fully Computed
SP(K,n): FldPad, RngIntElt → GalRep- Example: Special Galois Representations
SP(K,f,n): FldPad, RngUPolElt, RngIntElt → GalRep- Example: Twisted Special Representations
- Representations from Finite Extensions
GaloisRepresentations(F,K): FldPad, FldPad → SeqEnum[GalRep]- Example: Irreducible Characters of a Dihedral Extension
GaloisRepresentations(f): RngUPolEltFldPad → SeqEnum[GalRep]- Example: One-Dimensional Characters of a Cyclotomic Extension
PermutationCharacter(F,K): FldPad, FldPad → GalRep- Example: Permutation Character of a Cubic Extension
A !! chi: GalRep, AlgChtrElt → GalRepA !! chi: GalRep, SeqEnum → GalRep- Example: Replacing the Character of a Representation
- Local Representations of Global Objects
GaloisRepresentation(chi,p): GrpDrchElt, RngIntElt → GalRep- Example: Local Components of a Dirichlet Character
GaloisRepresentation(A,p): ArtRep, RngIntElt → GalRep- Example: Local Components of an Artin Representation
GaloisRepresentation(E): CrvEll → GalRep- Example: Galois Representation of a Local Elliptic Curve
GaloisRepresentation(E,p): CrvEll, RngIntElt → GalRep- Example: Elliptic Curve 20a1 at Several Primes
GaloisRepresentation(E,P): CrvEll, RngOrdIdl → GalRep- Example: Elliptic Curve over a Cyclotomic Field
GaloisRepresentation(C): CrvHyp[FldPad] → GalRep- Example: Hyperelliptic Curve over a p-adic Field
GaloisRepresentation(C,p): CrvHyp[FldRat], RngIntElt → GalRep- Example: Hyperelliptic Curve over the Rationals
GaloisRepresentation(C,P): CrvHyp[FldNum], RngOrdIdl → GalRep- Example: Hyperelliptic Curve over a Cyclotomic Field
GaloisRepresentation(f,p): ModFrmElt, RngIntElt → GalRep- Example: Local Representations of a Modular Form
- Basic Invariants
BaseField(A): GalRep → FldPadDegree(A): GalRep → RngIntEltDimension(A): GalRep → RngIntElt- Example: Degree and Base Field of Representations
Group(A): GalRep → GrpPermGaloisGroup(A): GalRep → GrpPerm- Example: Galois and Inertia Groups of Representations
FrobeniusElement(A): GalRep → GrpPermElt- Example: Frobenius Element of an Unramified Extension
Character(A): GalRep → AlgChtrElt- Example: Character of a Galois Representation
Field(A): GalRep → FldPadDefiningPolynomial(A): GalRep → RngUPolElt- Example: Defining Polynomial of a Representation
Automorphism(A,g): GalRep, GrpPermElt → Map- Example: Field and Automorphisms of a Representation
EulerFactor(A): GalRep → RngUPolElt- Example: Euler Factors of Galois Representations
IsZero(A): GalRep → BoolEltIsOne(A): GalRep → BoolEltFactorization(A): GalRep → List, GalRep- Example: Factorization of a Galois Representation
- Ramification
InertiaGroup(A): GalRep → GrpPerm- Example: Inertia Groups of Irreducible Representations
InertiaGroup(A,n): GalRep, RngIntElt → GrpPerm- Example: Higher Ramification Groups
IsUnramified(A): GalRep → BoolElt- Example: Testing Whether a Representation Is Unramified
IsRamified(A): GalRep → BoolElt- Example: Ramification of a Galois Representation
IsTamelyRamified(A): GalRep → BoolEltIsWildlyRamified(A): GalRep → BoolElt- Example: Wild Ramification of Elliptic Curves
InertiaInvariants(A): GalRep → GalRep- Example: Inertia Invariants of a Representation
ConductorExponent(A): GalRep → RngIntEltConductor(A): GalRep → FldPadElt- Example: Conductor of a Galois Representation
EpsilonFactor(A): GalRep → FldComEltRootNumber(A): GalRep → FldComElt- Example: Epsilon Factors and Root Numbers
- Semisimplicity and Irreducibles
- Arithmetic
A1 + A2: GalRep, GalRep → GalRep- Example: Sums of Galois Representations
A1 - A2: GalRep, GalRep → BoolElt- Example: Differences of Galois Representations
A1 * A2: GalRep, GalRep → GalRep- Example: Tensor Products of Galois Representations
A1 / A2: GalRep, GalRep → GalRepA1 / A2: RngIntElt, GalRep → GalRepA ^ n: GalRep, Any → GalRep- Example: Powers of Galois Representations
A1 eq A2: GalRep, GalRep → BoolElt- Example: Equality of Galois Representations
Determinant(A): GalRep → GalRep- Example: Determinant of a Galois Representation
TateTwist(A,n): GalRep, RngIntElt → GalRep- Example: Tate Twists of Galois Representations
- Changing Precision
- Changing Fields
- Advanced Examples
- Bibliography