Magma - varphi-modules and Galois Representations in Magma

varphi-modules and Galois Representations in Magma

Let us now give an overview of the functionalities of the package.

varphi-modules

PhiModule(M) : AlgMatElt -> PhiMod

    F: SeqEnum                          Default: [1,p]

Create the varphi-module whose matrix is given by M in some basis. The optional argument F describes the action of the Frobenius on coefficients: if F = [s, b] then varphi acts by a |-> a^{p^s} on the residue field and maps the variable u to u^b. The default value is [1, p] where p is the characteristic of the base field, corresponding to the absolute Frobenius.

ElementaryPhiModule(S,d,h) : RngSerLaur, RngIntElt, RngIntElt -> PhiMod

    F: SeqEnum                          Default: [1,p]

Create the varphi-module D(d, s) whose matrix is the companion matrix of T^d - u^s.

PhiModuleElement(x,D) : AlgMatElt, PhiMod -> PhiModElt

Create the element of the varphi-module D whose coordinates are given by the vector x.

Dimension(D) : PhiMod -> RngIntElt

The dimension of a varphi-module.

CoefficientRing(D) : PhiMod -> RngSerLaur

The coefficient ring of a varphi-module.

FrobeniusMatrix(D) : PhiMod -> AlgMatElt

Return the matrix of the action of varphi on D in the current basis.

Basic Operations and Properties of varphi-modules

IsEtale(D) : PhiMod -> BoolElt

Return true if the action of varphi on D is injective. This is only possible up to the precision of the coefficient ring of D.

ChangePrecision(~D, prec) : PhiMod, RngIntElt ->

Change the precision of the coefficient ring of D to prec.

DirectSum(D1, D2) : PhiMod, PhiMod -> PhiMod

The direct sum of two varphi-modules. The coefficient rings and Frobenius action on the coefficients must be the same.

BaseChange(~D, P) : ~PhiMod, AlgMatElt ->

Change the basis of D. The base change matrix is P, meaning that if G is the current matrix of varphi, the new matrix will be P^{- 1}G varphi(P).

RandomBaseChange(~D) : PhiMod ->

Randomly change the basis of D.

Phi(D, x) : PhiMod, PhiModElt -> PhiModElt

Compute the image of x ∈D under the action of varphi.

Reduction of varphi-modules and Galois Representations

SemisimpleDecomposition(D) : PhiMod -> AlgMatElt, AlgMatElt, SeqEnum, SeqEnum

Compute a Jordan-Holder sequence for the varphi-module D. The result G, P, sl, pol is as follows: G is the matrix of varphi in a basis where it is block upper triangular, with diagonal blocks corresponding to simple varphi-modules. The matrix P gives the corresponding basis. The list sl is the list of the slopes of D, and the list pol is a list of polynomials. The isomorphism class of a simple block of G is determined by the corresponding slope and polynomial.

Slopes(D) : PhiMod -> SeqEnum

Compute the list of slopes of D (with multiplicities).

SSGaloisRepresentation(D) : PhiMod -> SSGalRep

Compute the semisimplification of the Galois representation corresponding to D.

Semisimple Galois Representations

This part is dedicated to the study of representations of absolute Galois groups of fields of the form k((u)) with k finite, and with coefficients in finite fields. The implementation is for semisimple representations, and these are described by their tame inertia weights and polynomials giving the action of the Frobenius on the unramified part.

Creation Functions

SSGaloisRepresentation(E,K,w,P) : FldFin, RngSerLaur, SeqEnum, SeqEnum -> SSGalRep

Create the semisimple representation of the absolute Galois group of K with coefficients in E, tame inertia weights given by w, and action of the Frobenius described by the elements of the list P.