p-adic Cohomology and classicality of overconvergent Hilbert modular 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.

70 pages, to appear in Asterisque

References in corpus (3)

• Triangulation of refined families
• Cohomology of arithmetic families of (phi,Gamma)-modules
• Classicality of overconvergent Hilbert eigenforms: Case of quadratic residue degree

Cited by in corpus (15)

• Rankin--Eisenstein classes for modular forms
• Euler systems for Hilbert modular surfaces
• The universal $p$-adic Gross-Zagier formula
• Syntomic regulators of Asai--Flach classes
• Analytic continuation on Shimura varieties with $\mu$-ordinary locus
• On some partially de Rham Galois representations
• Modular sheaves of de Rham classes on Hilbert formal modular schemes for unramified primes
• $\mathcal{L}$-invariants, partially de Rham families and local-global compatibility
• Partial Hasse invariants on splitting models of Hilbert modular varieties
• $\mathcal{L}$-invariants and local-global compatibility for the group ${\mathrm{GL}}_2/F$
• Congruences between Hilbert modular forms of weight $2$, and special values of their $L$-functions
• Partial Classicality of Hilbert Modular Forms
• Higher Hida theory for Hilbert modular varieties in the totally split case
• Ramification of Hilbert eigenvarieties at classical points
• A trace formula approach to control theorems for overconvergent automorphic forms