Chennai Mathematical Institute


3.30 pm, Lecture Hall 6
Hecke operators and the coherent cohomology of Shimura varieties

Najmuddin Fakhruddin


For $X$ a smooth projective scheme over $\mathbb{Z}_p$, results of Faltings and Fontaine (generalising a result of Mazur) imply that for any $i$, the Newton polygon associated to the action of Frobenius on the $i$-th crystalline cohomology of the special fibre of $X$ lies above the Hodge polygon associated to the $i$-th de Rham cohomology of the generic fibre of $X$. In this talk, I will discuss an analogue of this result in the setting of automorphic forms, first considered by Clozel. His results were generalised to cohomological automorphic forms (corresponding to Betti cohomology of locally symmetric spaces) on arbitrary split reductive groups by V.~Lafforgue. In work in progress with Vincent Pilloni, we consider this for automorphic forms associated to the coherent cohomology of automorphic vector bundles on Shimura varieties. I will explain the classical case of modular curves in detail and then discuss some results for unitary Shimura varieites.