SPEAKER: Jacques Tilouine, Paris 13

TITLE: Companion forms and BGG dual complex

ABSTRACT: In the first lecture, we give a new proof of an old result by B. Gross providing a criterion for the existence of a modulo $p$ (resp. $p$-adic overconvergent) companion form of a classical modular form, in terms of the local behaviour at $p$ of its modulo $p$ (resp. $p$-adic) Galois representation,

In the second lecture, we extend our method to more general automorphic forms, and we relate again the existence of modulo $p$ (or $p$-adic overconvergent) companion forms of such an automorphic form to the local behaviour at $p$ of its modulo $p$ (resp. $p$-adic) Galois representation.

Our method to find a companion form is to calculate de Rham cohomology by using a subcomplex (called the BGG dual complex) of the de Rham complex which reflects in a transparent way the extra symmetries under the Weyl group of the restriction at the decomposition group at $p$ of the Galois representation.