Geometric methods in the mod p local Langlands correspondence

Funded by the European Research Council (ERC) Advanced Grant "Arithmetic of Automorphic MOTives (AAMOT)"

6 June 2016 - 10 June 2016

Aims

For a preliminary program please link to the Documents section.

The purpose of the workshop is to study geometric structures connected to the two sides of the mod p local Langlands correspondence, and to see whether or not concrete properties can be identified on both sides that can be placed in parallel, in order to formulate precise conjectures in the spirit of the work of Arinkin and Gaitsgory on the geometric Langlands program.

Let G be a reductive algebraic group over a local field K of residue characteristic p, and let k be an algebraically closed field of characteristic p. On the automorphic side of the correspondence, the geometric object is an appropriate category of modules over one of the differential graded Hecke algebras of G introduced by Schneider. On the Galois side of the correspondence, the geometric object is an appropriate category of quasi-coherent sheaves on the moduli space, introduced by Emerton and Gee, of homomorphisms from the Galois group of K into the L-group of G over k. In both cases, categories as well as spaces should be considered in a derived sense.

Both of these objects have a rich geometric structure but it appears to be difficult to make useful calculations. One purpose of the workshop will be to try to identify invariants on each side that can be compared both to similar invariants on the other side and to structures that have been studied in the global p-adic Langlands program. It will also be of interest to compare these structures to the conjectures formulated by Fargues involving the Fargues-Fontaine curve.