\(\varphi\)-modules and Galois Representations in Magma

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

\(\varphi\)-modules

Category

In Magma, \(\varphi\)-modules have type PhiMod. Elements of \(\varphi\)-modules have type PhiModElt.

Creation Functions

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 \(a \mapsto a^{p^s}\) 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 \(x\).

Attributes of \(\varphi\)-modules

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 \(x \in D\) 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.

Category

In Magma, semisimple Galois representations have type SSGalRep.

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 \(P\).

Basic Operations

CoefficientRing(V): SSGalRep -> FldFin

The coefficient ring.

FixedField(V): SSGalRep -> RngSerLaur

The fixed field of the absolute Galois group of which \(V\) is a representation.

Weights(V): SSGalRep -> SeqEnum

The tame inertia weights of \(V\).

Representation Associated to a \(\varphi\)-Module

SSGaloisRepresentation(D): PhiMod -> SSGalRep

If \(D\) is a \(\varphi\)-module over a field \(K\) of Laurent series this returns the semisimplification of the representation associated to \(D\).