Galois representations and Lubin-Tate groups

Abstract

Using Lubin-Tate groups, we develop a variant of Fontaine's theory of (ϕ,)-modules, and we use it to give a description of the Galois stable lattices inside certain crystalline representations. K of K. Fontaine's theory starts with an infinite extension K∞/K which is required to have certain ramification properties. Miraculously, these properties ensure that GK1 = Gal( ¯ K/K∞) can be identified with the absolute Galois group of a local field of equal characteristic p, X(K). It is well known that representations of such a Galois group on finite dimensional Fp-vector spaces can be classified rather concretely in terms of finite dimensional vector spaces over X(K) equipped with anFrobenius. If K∞/K is Galois, then = Gal(K∞/K) acts naturally on X(K), and one obtains a classification of GK- representations on finite dimensional Fp-vector spaces by adding a semi-linear action of to the ´ ϕ-modules over X(K). To obtain a classification of GK-representations on finite Zp-modules, one needs to lift the action of ϕ and on X(K) to commuting operators on a Cohen ring for X(K). This is probably not always possible, but can be done when K∞ is the p-cyclotomic extension of K. Much of the work on Fontaine's theory by Berger, Colmez, Wach and others has focused on this case. In this paper we focus on the case when K∞ is generated by the p-power torsion points of a Lubin-Tate group for a finite extension L/Qp contained in K. As an application we obtain a description of the GK-stable lattices in a certain class of crystalline GK-representations. This is possible using the p-cyclotomic theory only when K is an unramified extension of some Qp(� pn).