INdAM Program on Serre Conjectures and the p-adic local Langlands program

Konstantin Ardakov (Oxford), The first Drinfeld covering and equivariant D-modules on rigid spaces

Let p be a prime and let F be a p-adic local field. The p-adic upper half plane Omega is obtained from the projective line viewed as a rigid analytic variety by removing the F-rational points.Drinfeld introduced a tower of finite etale Galois coverings of Omega by interpreting Omega asthe rigid generic fibre of the moduli space of certain formal one-dimensional commutative groupswith quaternionic multiplication, and introducing level structures to define the coverings. Thistower is now known to realise both the Jacquet-Langlands and local Langlands correspondences forG = GL2(F) in `-adic etale cohomology, where ` is a prime not equal to p. Coherent cohomology ofthe tower is expected to produce representations of G which are admissible in the sense of Schneiderand Teitelbaum. Using the theory of equivariant D-modules on rigid spaces we can prove that thedual of the global sections of a non-trivial line bundle arising from the first covering of Omegais an irreducible admissible representation of G. Patel, Schmidt and Strauch have also given anargument for the admissibility of these representations using a formal model for the first covering;whilst similar in certain respects, our approach is significantly dierent to theirs. This is joint workwith Simon Wadsley.

June 10th. Aula 1 AD 100, Department of Mathematics.