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On Galois representations associated to mod $p$ Hilbert modular forms

Fred Diamond, Shu Sasaki

TL;DR

The paper develops a comprehensive framework linking mod $p$ Hilbert modular forms over totally real fields (with possible ramification at $p$) to two-dimensional Galois representations, extending Serre-type weight conjectures to the geometric setting. It constructs Galois representations from mod $p$ eigenforms using congruences and Jacquet–Langlands transfers, and formulates a geometric weight conjecture governed by a minimal weight cone and crystalline liftability at primes over $p$. It proves partial results towards the conjectures, notably in real quadratic cases with ramified $p$, and establishes a bridge between geometric and algebraic modularity, with a detailed treatment of partial weight one and quado-Barsotti–Tate phenomena. The work thus advances a Langlands-modulo-$p$ perspective for Hilbert modular forms, combining automorphic, geometric, and $p$-adic Hodge-theoretic techniques to understand weights, modularity, and local-global compatibility in the ramified setting.

Abstract

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the existence of an associated two-dimensional representation of the absolute Galois group of $F$. Furthermore, for any such irreducible Galois representation, we formulate a conjecture predicting the set of weights of eigenforms from which it arises. This generalizes Edixhoven's variant of the weight part of Serre's Conjecture (in the case $F = \mathbb{Q}$), and removes the restriction that $p$ be unramified in $F$ from prior work in this direction. We also establish one direction of a conjectural relation with the algebraic analogue of the weight part of Serre's Conjecture in this context. Finally, we prove results towards our conjecture in the case of partial weight one for real quadratic fields $F$ in which $p$ is ramified.

On Galois representations associated to mod $p$ Hilbert modular forms

TL;DR

The paper develops a comprehensive framework linking mod Hilbert modular forms over totally real fields (with possible ramification at ) to two-dimensional Galois representations, extending Serre-type weight conjectures to the geometric setting. It constructs Galois representations from mod eigenforms using congruences and Jacquet–Langlands transfers, and formulates a geometric weight conjecture governed by a minimal weight cone and crystalline liftability at primes over . It proves partial results towards the conjectures, notably in real quadratic cases with ramified , and establishes a bridge between geometric and algebraic modularity, with a detailed treatment of partial weight one and quado-Barsotti–Tate phenomena. The work thus advances a Langlands-modulo- perspective for Hilbert modular forms, combining automorphic, geometric, and -adic Hodge-theoretic techniques to understand weights, modularity, and local-global compatibility in the ramified setting.

Abstract

We consider mod Hilbert modular forms for a totally real field , viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic . For a Hecke eigenform of arbitrary weight, we prove the existence of an associated two-dimensional representation of the absolute Galois group of . Furthermore, for any such irreducible Galois representation, we formulate a conjecture predicting the set of weights of eigenforms from which it arises. This generalizes Edixhoven's variant of the weight part of Serre's Conjecture (in the case ), and removes the restriction that be unramified in from prior work in this direction. We also establish one direction of a conjectural relation with the algebraic analogue of the weight part of Serre's Conjecture in this context. Finally, we prove results towards our conjecture in the case of partial weight one for real quadratic fields in which is ramified.

Paper Structure

This paper contains 37 sections, 46 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Outline of the paper
  4. Notation and conventions
  5. Hilbert modular varieties over $\mathbb{C}$
  6. Hilbert modular forms over $\mathbb{C}$
  7. The Jacquet--Langlands Correspondence
  8. Automorphic forms on totally definite quaternion algebras over $\mathbb{C}$
  9. Automorphic forms on Shimura curves over $\mathbb{C}$
  10. Hilbert modular forms over ${\mathcal{O}}$-algebras
  11. Hilbert modular varieties over ${\mathcal{O}}$
  12. Hilbert modular forms over ${\mathcal{O}}$-algebras
  13. $q$-expansions
  14. Hecke operators
  15. Twisting
  16. ...and 22 more sections

Key Result

Theorem A

Suppose that $f \in M_{\vec{k}}(U,E)$ is an eigenform for $T_v$ (resp. $S_v$) with eigenvalue $a_v$ (resp. $d_v$) for all $v \not\in Q$, where $Q$ is a finite set of primes containing all those dividing the level or $p$. Then there exists a continuous representation such that if $v\not\in Q$, then $\rho_f$ is unramified at $v$ and the characteristic polynomial of $\rho_f({\operatorname{Frob}}_v)$

Theorems & Definitions (105)

  • Theorem A
  • Conjecture B
  • Theorem C
  • Theorem D
  • Remark 1.3.1
  • Theorem 2.3.1
  • Remark 3.6.1
  • Theorem 4.1.1
  • Corollary 4.1.2
  • Theorem 4.2.1
  • ...and 95 more