p-adic Cohomology and classicality of overconvergent Hilbert modular forms

TL;DR

This paper proves that overconvergent Hilbert modular forms with small slopes are classical, using a cohomological approach and explicit stratification descriptions, advancing understanding of p-adic properties of modular forms.

Contribution

It extends classicality results to Hilbert modular forms over totally real fields with unramified p, employing explicit Goren-Oort stratification analysis.

Findings

Small slope overconvergent forms are classical.

Explicit description of Goren-Oort stratification.

Rigid cohomology of the ordinary locus matches classical forms.

Abstract

Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke modules.