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arXiv · 1308.0779

p-adic Cohomology and classicality of overconvergent Hilbert modular forms

Yichao Tian·Liang Xiao
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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.

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Yichao Tian, Liang Xiao. 2013-08-04. p-adic Cohomology and classicality of overconvergent Hilbert modular forms. https://arxiv.org/abs/1308.0779

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