Classicality of Hilbert modular forms

TL;DR

This paper proves modularity of certain Hecke algebra characters linked to Galois representations in the context of Hilbert modular varieties, advancing the Langlands program over totally real fields.

Contribution

It generalizes existing methods to establish modularity results and proves new cases of the Langlands conjecture for GL(2) over totally real fields.

Findings

Proved modularity of Hecke algebra characters under specific conditions.

Established a locally analytic Jacquet-Langlands transfer.

Calculated geometric partial Fontaine operators and studied related cohomology.

Abstract

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely irreducible, and de Rham of regular parallel weights. As an application, we prove some new cases of the Langlands-Clozel-Fontaine-Mazur conjecture of $\mathrm{GL}_2$ over totally real fields. For the proof, we generalize the method in [Pan26], calculate geometric partial Fontaine operators, and study the cohomology of the associated Koszul-type partial de Rham complexes. The key step is the establishment of a locally analytic Jacquet-Langlands transfer, whose proof consists of several novel ingredients including a comparison of Igusa stacks for different quaternionic Shimura data constructed by [DvHKZ26], and the Grothendieck-Messing theory for locally analytic infinite level Shimura varieties established in [Jiang26a].