### The local Langlands conjecture for GL(n) over a p-adic field, n < p, by Michael Harris

Let F be a p-adic field and n a positive integer.
The local Langlands conjecture asserts the existence of a bijection
between irreducible admissible representations of GL(n,F)
and n-dimensional admissible representations of the Weil-Deligne
group of F. This bijection is required to satisfy certain
natural compatibilities, of which the most important is compatibility
with local functional equations (preservation of L and epsilon
factors of pairs). It is enough to construct a bijection
with these properties between supercuspidal representations
of GL(n,F) and n-dimensional irreducible representations of
the Weil group of F. In a previous paper, the author constructed
a canonical bijection on the etale cohomology of the rigid-analytic
coverings of the p-adic upper half space constructed by Drinfeld.
(That the map in the previous paper is a bijection was
actually proved by Henniart.) However, the compatibility of
epsilon factors of pairs was not shown. The present article uses
a technique of non-Galois automorphic induction to show that the
bijection previously constructed is compatible with epsilon factors of
pairs of representations of GL(n,F) and GL(m,F) when n and
m are prime to p (the tame case). This implies the local
Langlands conjecture in degree < p. It is also shown how the
local Langlands conjecture in general would follow from a generalization
of Carayol's theorem on the bad reduction of Shimura curves to
certain Shimura varieties of higher dimension.

Michael Harris <harris@math.jussieu.fr>