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LOCAL GALOIS REPRESENTATIONS
Acknowledgements
Overview
Notation and Printing
Conventions
Implementation Notes
Creating Galois Representations
Representations from Finite Extensions
Local Representations of Global Objects
Basic Invariants
Ramification
Semisimplicity and Irreducibles
Arithmetic
Changing Precision
Changing Fields
Advanced Examples
Example: Local and Global Epsilon Factors for Dirichlet Characters
Example: Reconstructing a Galois Representation from its Euler Factors
Bibliography
Overview
Notation and Printing
Example GalRep_galrep-printing (H58E1)
Conventions
Implementation Notes
Creating Galois Representations
ZeroRepresentation(K) : FldPad -> GalRep
Example GalRep_galrep-zerorepresentation (H58E2)
PrincipalCharacter(K) : FldPad -> GalRep
Example GalRep_galrep-principalcharacter (H58E3)
CyclotomicCharacter(K) : FldPad -> GalRep
Example GalRep_galrep-cyclotomiccharacter (H58E4)
UnramifiedCharacter(K,c) : FldPad, Any -> GalRep
Example GalRep_galrep-unramifiedcharacter (H58E5)
UnramifiedRepresentation(K,CharPoly) : FldPad,RngUPolElt -> GalRep
Example GalRep_galrep-unramifiedrepresentation (H58E6)
UnramifiedRepresentation(K,dim,dimcomputed,CharPoly) : FldPad,RngIntElt,RngIntElt,RngUPolElt -> GalRep
Example GalRep_galrep-notfullycomputed (H58E7)
SP(K,n) : FldPad,RngIntElt -> GalRep
Example GalRep_galrep-sp (H58E8)
SP(K,f,n) : FldPad,RngUPolElt,RngIntElt -> GalRep
Example GalRep_galrep-sp-1 (H58E9)
Representations from Finite Extensions
GaloisRepresentations(F,K) : FldPad,FldPad -> SeqEnum[GalRep]
Example GalRep_galrep-galoisrepresentations-1 (H58E10)
GaloisRepresentations(f) : RngUPolEltFldPad -> SeqEnum[GalRep]
Example GalRep_galrep-galoisrepresentations (H58E11)
PermutationCharacter(F,K) : FldPad,FldPad -> GalRep
Example GalRep_galrep-permutationcharacter (H58E12)
A !! chi : GalRep, AlgChtrElt -> GalRep
Example GalRep_galrep-smash (H58E13)
Local Representations of Global Objects
GaloisRepresentation(chi,p) : GrpDrchElt,RngIntElt -> GalRep
Example GalRep_galrep-dirichlet (H58E14)
GaloisRepresentation(A,p) : ArtRep,RngIntElt -> GalRep
Example GalRep_galrep-artin (H58E15)
GaloisRepresentation(E) : CrvEll -> GalRep
Example GalRep_galrep-E (H58E16)
GaloisRepresentation(E,p) : CrvEll,RngIntElt -> GalRep
Example GalRep_galrep-E-p (H58E17)
GaloisRepresentation(E,P) : CrvEll,RngOrdIdl -> GalRep
Example GalRep_galrep-E-P (H58E18)
GaloisRepresentation(C) : CrvHyp[FldPad] -> GalRep
Example GalRep_galrep-crvhyp-Qp (H58E19)
GaloisRepresentation(C,p) : CrvHyp[FldRat], RngIntElt -> GalRep
Example GalRep_galrep-crvhyp-Q (H58E20)
GaloisRepresentation(C,P) : CrvHyp[FldNum],RngOrdIdl -> GalRep
Example GalRep_galrep-crvhyp-nf (H58E21)
GaloisRepresentation(f,p) : ModFrmElt,RngIntElt -> GalRep
Example GalRep_galrep-f-p (H58E22)
Basic Invariants
BaseField(A) : GalRep -> FldPad
Degree(A) : GalRep -> RngIntElt
Example GalRep_galrep-degree (H58E23)
Group(A) : GalRep -> GrpPerm
Example GalRep_galrep-group (H58E24)
FrobeniusElement(A) : GalRep -> GrpPermElt
Example GalRep_galrep-frobeniuselement (H58E25)
Character(A) : GalRep -> AlgChtrElt
Example GalRep_galrep-character (H58E26)
Field(A) : GalRep -> FldPad
DefiningPolynomial(A) : GalRep -> RngUPolElt
Example GalRep_galrep-definingpolynomial (H58E27)
Automorphism(A,g) : GalRep,GrpPermElt -> Map
Example GalRep_galrep-automorphism (H58E28)
EulerFactor(A) : GalRep -> RngUPolElt
Example GalRep_galrep-eulerfactor (H58E29)
IsZero(A) : GalRep -> BoolElt
IsOne(A) : GalRep -> BoolElt
Factorization(A) : GalRep -> List,GalRep
Example GalRep_galrep-factorization (H58E30)
Ramification
InertiaGroup(A) : GalRep -> GrpPerm
Example GalRep_galrep-inertia (H58E31)
InertiaGroup(A,n) : GalRep,RngIntElt -> GrpPerm
Example GalRep_galrep-higherinertia (H58E32)
IsUnramified(A) : GalRep -> BoolElt
Example GalRep_galrep-isunramified (H58E33)
IsRamified(A) : GalRep -> BoolElt
Example GalRep_galrep-isramified (H58E34)
IsTamelyRamified(A) : GalRep -> BoolElt
IsWildlyRamified(A) : GalRep -> BoolElt
Example GalRep_galrep-iswildlyramified (H58E35)
InertiaInvariants(A) : GalRep -> GalRep
Example GalRep_galrep-inertiainvariants (H58E36)
ConductorExponent(A) : GalRep -> RngIntElt
Conductor(A) : GalRep -> FldPadElt
Example GalRep_galrep-conductor (H58E37)
EpsilonFactor(A) : GalRep -> FldComElt
RootNumber(A) : GalRep -> FldComElt
Example GalRep_epsilon-rootnumber (H58E38)
Semisimplicity and Irreducibles
IsIrreducible(A) : GalRep -> BoolElt
Example GalRep_galrep-isirreducible (H58E39)
IsIndecomposable(A) : GalRep -> BoolElt
IsSemisimple(A) : GalRep -> BoolElt
Semisimplification(A) : GalRep -> GalRep
Decomposition(A) : GalRep -> SeqEnum[GalRep]
Example GalRep_galrep-decomposition (H58E40)
Arithmetic
A1 + A2 : GalRep,GalRep -> GalRep
Example GalRep_galrep-sum (H58E41)
A1 - A2 : GalRep,GalRep -> BoolElt
Example GalRep_galrep-difference (H58E42)
A1 * A2 : GalRep,GalRep -> GalRep
Example GalRep_galrep-product (H58E43)
A1 / A2 : GalRep,GalRep -> GalRep
A ^ n : GalRep,Any -> GalRep
Example GalRep_galrep-power (H58E44)
A1 eq A2 : GalRep,GalRep -> BoolElt
Example GalRep_galrep-eq (H58E45)
Determinant(A) : GalRep -> GalRep
Example GalRep_galrep-determinant (H58E46)
TateTwist(A,n) : GalRep,RngIntElt -> GalRep
Example GalRep_galrep-tatetwist (H58E47)
Changing Precision
Precision(A) : GalRep -> RngIntElt
ChangePrecision(~A,Prec) : GalRep,RngIntElt ->
ChangePrecision(A,Prec) : GalRep,RngIntElt -> GalRep
Example GalRep_galrep-precision (H58E48)
Changing Fields
Minimize(A) : GalRep -> GalRep
Example GalRep_galrep-minimize (H58E49)
Restriction(A,L) : GalRep,FldPad -> GalRep
Example GalRep_galrep-basechange (H58E50)
Induction(A,K0) : GalRep,FldPad -> GalRep
Example GalRep_galrep-induction (H58E51)
Advanced Examples
Example: Local and Global Epsilon Factors for Dirichlet Characters
Example GalRep_galrep-funeq (H58E52)
Example: Reconstructing a Galois Representation from its Euler Factors
Example GalRep_galrep-reconstruction (H58E53)
Bibliography
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