ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Colmez, Pierre, CNRS, École Normale Supérieure et Institut de Math. de Jussieu, Paris, France (φ, Γ)-modules and Iwasawa theory of p-adic representations

The talk will try to report on the classification of p-adic representions of a local field in terms of (φ, Γ)-modules. This classification was introduced by Fontaine some ten years ago and brings a new perspective on local class field theory and its generalisations. It is still in progress but a number of interesting results (due to various people) have already been obtained especially in the applications to Iwasawa theory. Before describing a little bit more in detail what the content of the talk should be, let us introduce some notation. Fix an algebraic closure \overline{Q}_{p} of Q_{p}. If K is a finite extension of Q_{p} and n ∈ N, set K_{n} = Q_{p}(μ_{p^{n}}) and let K_{∞} be the cyclotomic extension of K union of all the K_{n}. Let G_{K} be the Galois group Gal(\overline{Q}_{p}/K) and H_{K} = Gal(\overline{Q}_{p}/K_{∞}). Finally, set Γ_{K} = G_{K}/H_{K} = Gal(K_{∞}/K). The theory of (φ, Γ_{K})-modules attaches to any p-adic representation V of H_{K} a module D(V) over a ring of power series in one variable, endowed with an action of the Frobenius φ. The main point of the theory is that one can recover V from D(V) which is a priori much simpler (Γ_{K} being in general procyclic, D(V) is given by two operators (φ and a generator of Γ_{K}) satisfying a commutation relation). The theory of p-adic representations of G_{K} is thus equivalent to the study of (φ, Γ)-modules which naturally gives rise to three types of questions.

– Can one read the properties of the representation directly on its (φ, Γ)- module?

– Can one recover from D(V) other classical invariants of V and use D(V) to attach new invariants to V?

– Can one use (φ, Γ)-modules to construct interesting p-adic representations of G_{K}?

Concerning the first question, the idea is that, the nicer the representation is, the nicer its (φ, Γ)-module must be. In this direction one can prove that representations of G_{K} are nicer than general representations of H_{K}: they are overconvergent. If K is unramified over Q_{p}, crystalline representations of G_{K} are as nice as possible: they are of finite height. Concerning the second question, the most interesting result is a description of the Galois cohomology of V in terms of D(V) from which one can recover Tate’s local duality and describe the Iwasawa modules attached to V. One obtains in this way a vast generalisation of Coleman’s isomorphism and, if the representation is assumed to be de Rham, of Coates-Wiles homomorphisms as well as an explicit reciprocity law relating these Coates-Wiles homomorphisms to Bloch-Kato exponential maps (these last results use the overconvergence of representations of G_{K}). It is not unreasonable to expect that these results will share some new light on p-adic L-functions. One can also, using the infinitesimal action of Γ_{K}, attach to V an overconvergent differential module endowed with a Frobenius and use this module to recover the generalised Hodge-Tate decomposition of Sen and characterise de Rham representations. Finally, one can construct (φ, Γ)-modules having good properties and recover the results of Fontaine and Laffaille on crystalline representations from this construction.