Introduction

This package provides tools to work with \(\varphi\)-modules over \(k((u))\) where \(k\) is a finite field, and representations of the absolute Galois group of \(k((u))\) with coefficients in a finite field. The main functionality of the package computes the semisimplification of a given \(\varphi\)-module, and the semisimplification of the Galois representation that is naturally attached to it. In particular, the slopes of the \(\varphi\)-module, corresponding to the tame inertia weights of the Galois representation, can be computed using this package.

Motivation

Let \(K\) be a \(p\)-adic field and let \(G_K\) be the absolute Galois group of \(K\). Representations of this group naturally arise from geometry, namely from the \(p\)-adic étale cohomology of a scheme over \(K\).

The study of these representations is a central topic in arithmetic, and a motivation for creating this package is the following: let \(V\) be a \({\mathbb{Q}}_p\)-representation of \(G_K\), i.e. a \({\mathbb{Q}}_p\)-vector space endowed with a continuous, linear action of \(G_K\). Now let \(T \subset V\) be any \({\mathbb{Z}}_p\)-lattice stable under the action of \(G_K\). There always exists such a lattice. Moreover, the quotient \(T/pT\) has a natural structure of \({\mathbb{F}}_p\)-representation of \(G_K\). This representation depends on the choice of \(T\), but its semisimplification \((T/pT)^{ss}\) does not, according to the Brauer-Nesbitt theorem. Recall the semisimplification of a representation is the direct sum of the composition factors appearing in any Jordan-Holder sequence of this representation. Therefore, it is an interesting question to determine properties of \((T/pT)^{ss}\) in terms of \(V\). Although the Fontaine-Laffaille theory completely addresses this question for some \(V\), the general case remains an open question. Some computations concerning this problem can be performed in Magma using this package.

Definitions

Let \(k\) be a finite field of characteristic \(p\), and let \(K = k((u))\) be the field of Laurent series with coefficients in \(k\). Let \(s\geq 0\) and \(b\geq 2\) be integers. We define a “Frobenius” map \(\sigma\) on \(K\) by the following formula:

\[\sigma\left( \sum_{i \in {\mathbb{Z}}} a_i u^i\right) = \sum_{i \in {\mathbb{Z}}} a_i^{p^s}u^{bi}.\]

A \(\varphi\)-module over \(K\) is the data of a finite-dimensional \(K\)-vector space \(D\), endowed with an endomorphism \(\varphi\) : \(D \rightarrow D\) that is semilinear with respect to \(\sigma\). This means that for all \(\lambda \in K\), \(x \in D\), we have the identity \(\varphi(\lambda x) = \sigma(\lambda)\varphi(x)\).

A \(\varphi\)-module is said to be étale if the map \(\varphi\) is injective. A \(\varphi\)-module can be described by the matrix representing the action of \(\varphi\) on some basis of \(D\), and it is étale if and only if this matrix is invertible.

Classification of \(\varphi\)-modules

Some \(\varphi\)-modules play a crucial role in the theory because they are the simple objects in the category of étale \(\varphi\)-modules over the maximal unramified extension \(K^{ur}\) of \(K\).

Let \(d \geq 1\), \(h \in {\mathbb{Z}}\), \(\lambda \in \bar k\). We define the \(\varphi\)-module \(D(d,s,\lambda)\) as the \(\varphi\)-module of dimension \(d\) whose matrix in some basis is the companion matrix of the polynomial \(T^d - u^h\). We also write \(D(d,h) = D(d,h,1)\). Note that in general there are several ways to extend the action of \(\sigma\) on \(K^{ur}\), but we may only distinguish the cases where \(\sigma\) acts as identity on \(k\), and the case where is does not. We say that a couple \((d,h)\) is reduced if there is no divisor \(d'\) of \(d\) (except \(d\)) such that \({b^{d'} -1}\over{b^d -1}\) is a divisor of \(h\). The main classification results are the following:

If \(\sigma \neq id\), the simple objects of the category of étale \(\varphi\)-modules over \(K^{ur}\) are the \(D(d,h)\) for \((d,h)\) reduced, and if \(\sigma = id\), the simple objects of the category of étale \(\varphi\)-modules over \(K^{ur}\) are the \(D(d,h,\lambda)\) for \((d,h)\) reduced.

By definition, the slope of a simple \(\varphi\)-module isomorphic to \(D(d,h,\lambda)\) is the rational number \({h}\over{b^d - 1}\), up to the equivalence relation “\(x \sim y \Leftrightarrow \exists m,n \in {\mathbb{N}}\) such that \(b^m x - b^ny \in {\mathbb{Z}}\)”. With this equivalence relation, the definition does not depend on the choice of \((d,h)\).

If \(D\) is a \(\varphi\)-module over \(K\), the slopes of \(D\) are the collection of the slopes of the composition factors of \(K^{ur} \otimes_K D\) (this notion does not depend on how \(\sigma\) is extended to \(K^{ur}\)). Note that even though the algorithms that we present can give decompositions over \(K\), for most practical uses the knowledge of the slopes should be sufficient.

Connection with Galois Representations

Let us explain the link between Galois representations and \(\varphi\)-modules over \(K\). In this section, we assume that \(\sigma\) is the classical Frobenius \(x\mapsto x^p\). Let \(K^{sep}\) be a separable closure of \(K\) and let \(G_K = Gal(K^{sep}/K)\) be the absolute Galois group of \(K\).

A theorem of Katz states that there is an equivalence of categories between the étale \(\varphi\)-modules over \(K\) and the \({\mathbb{F}}_p\)-representations of \(G_K\).

Under this equivalence of categories, the \(\varphi\)-module \(D(d,h)\) corresponds to the “fundamental character of level \(d\)” to the power \(h\), \(\omega_d^h\), seen as a \({\mathbb{F}}_p\)-representation. The figures of \(h\) in base \(p\) are called the tame inertia weights of the representation, because they describe the action of the tame inertia group on the representation. These weights can be recovered from the slope of the \(\varphi\)-module. It is worth noting that if \(F\) is a \(p\)-adic field whose residue field is \(k\), and \(F_\infty\) is the extension of \(F\) generated by a compatible sequence of \(p^n\)-th roots of the uniformizer for all \(n\), then \(G_{F_\infty}\) is isomorphic to \(G_K\). Moreover, the tame inertia weights of a \({\mathbb{F}}_p\)-representation of \(G_F\) are the same as the tame inertia weights of its restriction to \(G_{F_\infty}\), seen as a representation of \(G_K\). Hence, working with \(\varphi\)-modules will enable us to study representations of \(p\)-adic Galois groups.