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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.
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