Galois Representations and Automorphic Forms

The theory of (\varphi, \Gamma)-modules, which was introduced by Fontaine in the early 90's, classifies local Galois representations into modules over certain power series rings carrying certain extra structures (\varphi and \Gamma). In a recent joint work with Kedlaya, we generalize Fontaine's theory to geometric families of local Galois representations. Namely, we exhibit an equivalence between etale local systems over nonarchimedean analytic spaces and certain modules over commutative period rings carrying similar extra structures.

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine--Mazur conjecture, and to a conjecture of Kisin.

The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms. Such methods, however, fail (at least naively) to capture information about torsion classes in integral cohomology. We discuss a formalism -- completed cohomology -- which provides a way of studying the cohomology in a p-power congruence tower.The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms.

The usual Katz-Mazur model for the modular curve X(p^n) has horribly singular reduction. For large n there isn't any model of X(p^n) which has good reduction, but after extending the base one can at least find a semistable model, which means that the special fiber only has normal crossings as singularities. We will reveal a new picture of the special fiber of a semistable model of the entire tower of modular curves. We will also indicate why this problem is important from the point of view of the local Langlands correspondence for GL(2) .

We give a new local characterization of the Local Langlands Correspondence, using deformation spaces of p-divisible groups, and show its existence by a comparison with the cohomology of some Shimura varieties. This reproves results of Harris-Taylor on the compatibility of local and global correspondences, but completely avoids the use of Igusa varieties and instead relies on the classical method of counting points a la Langlands and Kottwitz.

The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms. Such methods, however, fail (at least naively) to capture information about torsion classes in integral cohomology. We discuss a formalism -- completed cohomology -- which provides a way of studying the cohomology in a p-power congruence tower.The cohomology of arithmetic groups (with real coefficients) is usually understood in terms of automorphic forms. Such methods, however, fail (at least naively) to capture information about torsion classes in integral cohomology.

We will discuss a generalisation of Serre's conjecture on the possible weights of modular mod p Galois representations for a broad class of reductive groups. In good cases (essentially when the Galois representation is tamely ramified at p) the predicted weight set can be made explicit and compared to previous conjectures. This is joint work with Toby Gee and David Savitt.

Let p be an odd prime number and let F be a totally real field. Let F_cyc be the cyclotomic extension of F generated by the roots of unity of order a power of p . From the maximal abelian extension of F_cyc which is unramified (resp. unramified outside auxiliary primes), we get exact sequences of Iwasawa modules. We will discuss how splitting of these exact sequences are linked to Leopoldt conjecture for F and p . (JW with C. Khare)