p-adic cohomology and classicality of overconvergent Hilbert modular forms

Abstract:  Coleman showed in 1996 by a cohomological method that an overconvergent modular form of small slope is classical. In this talk, I will explain how to generalize Coleman's cohomological approach to overconvergent Hilbert cusp forms. Let F be a totally real field of degree $g>1$ in which p is unramified. Consider the Goren-Oort stratification of the special fiber of the Hilbert modular variety with hyperspecial level at p. We show first that each such stratum  is a certain (P^1)^r-bundle over (the special fiber) of another quaternionic Shimura  variety. Using this geometric result, we show that the rigid cohomology of the ordinary locus of the Hilbert modular variety has the same Hecke spectrum as the space of classical cusp forms of Iwahori level at p. Then we deduce that a p-adic overconvergent Hilbert cuspidal eigenform of small slope is classical.  We point out that the slope condition for classicality we obtained is better than that obtained by Pilloni-Stroh, and was previously conjectured by Breuil.This is a joint work with Liang Xiao.