# Thursday 26 Sep 2019: NT Seminar: The first Drinfeld covering and equivariant D-modules on rigid spaces

### Prof. Konstantin Ardakov - University of Oxford

**H103** 14:30-16:30

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 as the rigid generic fibre of the moduli space of certain formal one-dimensional commutative groups with quaternionic multiplication, and introducing level structures to define the coverings. This tower is now known to realise both the Jacquet-Langlands and local Langlands correspondences for G = GL_2(F) in l-adic etale cohomology, where l is a prime not equal to p. Coherent cohomology of the tower is expected to produce representations of G which are admissible in the sense of Schneider and Teitelbaum. I will try to explain how to use the theory of equivariant D-modules on rigid spaces to prove that the dual of the global sections of a non-trivial line bundle arising from the first covering of Omega is an irreducible admissible representation of G. This is joint work with Simon Wadsley.