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.
Let K be a p-adic field and let GK 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 Qp-representation of GK, i.e. a Qp-vector space endowed with a continuous, linear action of GK. Now let T ⊂V be any Zp-lattice stable under the action of GK. There always exists such a lattice. Moreover, the quotient T/pT has a natural structure of Fp-representation of GK. 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.
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≥0 and b≥2 be integers. We define a "Frobenius" map σ on K by the following formula: σ( ∑i ∈Z ai ui) = ∑i ∈Z aipsubi. A varphi-module over K is the data of a finite-dimensional K-vector space D, endowed with an endomorphism varphi : D -> D that is semilinear with respect to σ. This means that for all λ ∈K, x ∈D, we have the identity varphi(λ x) = σ(λ)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.
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 Kur of K.
Let d ≥1, h ∈Z, λ ∈bar k. We define the varphi-module D(d, s, λ) as the varphi-module of dimension d whose matrix in some basis is the companion matrix of the polynomial Td - uh. We also write D(d, h) = D(d, h, 1). Note that in general there are several ways to extend the action of σ on Kur, but we may only distinguish the cases where σ 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 (bd' - 1)/(bd - 1) is a divisor of h. The main classification results are the following:
If σ != id, the simple objects of the category of {étale varphi-modules over Kur are the D(d, h) for (d, h) reduced, and if σ = id, the simple objects of the category of {étale varphi-modules over Kur are the D(d, h, λ) for (d, h) reduced.
By definition, the slope of a simple varphi-module isomorphic to D(d, h, λ) is the rational number (h)/(bd - 1), up to the equivalence relation "x ~y <=> exists m,n ∈N such that bm x - bny ∈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 Kur tensor K D (this notion does not depend on how σ is extended to Kur). 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.
Let us explain the link between Galois representations and varphi-modules over K. In this section, we assume that σ is the classical Frobenius x |-> xp. Let Ksep be a separable closure of K and let GK = Gal(Ksep/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 Fp-representations of GK.
Under this equivalence of categories, the varphi-module D(d, h) corresponds to the "fundamental character of level d" to the power h, ωdh, seen as a Fp-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_∞ is the extension of F generated by a compatible sequence of pn-th roots of the uniformizer for all n, then GF_∞ is isomorphic to GK. Moreover, the tame inertia weights of a Fp-representation of GF are the same as the tame inertia weights of its restriction to GF_∞, seen as a representation of GK. Hence, working with varphi-modules will enable us to study representations of p-adic Galois groups.