\(\varphi\)-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 \(D\).