Martin Ortiz: Mod p de Rham cohomology of Shimura varieties and Serre's weight conjecture

Number Theory Seminar: Martin Ortiz (MPIM Bonn)

Mod p de Rham cohomology of Shimura varieties and Serre's weight conjecture

Serre's weight conjecture concerns mod p congruences of automorphic forms of different weights. In the context of Shimura varieties one can consider different spaces of mod p automorphic forms, e.g. via etale, coherent, or de Rham cohomology. 
I will explain a conjectural relation between etale and de Rham cohomology, and how the latter is related to coherent cohomology via generalized BGG resolutions. As opposed to the classical characteristic 0 picture, de Rham cohomology is not a direct sum of coherent cohomology groups, and only some pieces cut out by some differential operators contribute to it. Finally, I will mention one application of this approach to Serre's weight conjecture for GSp4. The whole talk will be illustrated by the example of the modular curve, and the Picard modular variety (G=GL3).