2nd seminar : On the bad reduction of certain Shimura varieties
Suppose that K is a finite extension of Q_p. The local Langlands conjecture asserts the existence of a natural bijection between the irreducible smooth representations of GL_n(K) and the n-dimensional representations of the Weil-Deligne group of K (this latter group is a modification of the absolute Galois group of K). The content of this conjecture is all in the word ``natural'' - and its meaning was made precise by Henniart who also proved that there is a most one such natural bijection. In the case n=1 this conjecture is a reformulation of local class field theory for K.
In these talks I'll sketch a proof of this conjecture, which generalises the ``formal groups'' approach to local class field theory. Following Carayol, Deligne and Drinfeld we consider the etale cohomology of a certain rigid space obtained by considering the deformations of one dimensional formal O_K modules over the algebraic closure of F_p. This cohomology (essentially) has an action of the Weil group of K, of GL_n(K) and of the group of units of a division algebra with centre K. As Carayol conjectured, these actions can be used to define a correspondence of the desired sort which we prove also satisfies Henniart's ``natural'' conditions.
The key ingredient of the proof is to relate the etale cohomology groups considered by Carayol to the etale cohomology of certain Shimura varieties, which are defined over number fields and so susceptible to global arguments.
In the first seminar I'll discuss the local Langlands conjecture, and discuss its proof in general terms. In particular I will state a theorem on the bad reduction of certain Shimura varieties from which one can deduce the local Langlands conjecture. In the second seminar I will discuss the proof of this theorem.