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On the geometric Serre weight conjecture for Hilbert modular forms

Siqi Yang

TL;DR

The paper advances the geometric Serre weight program for Hilbert modular forms by proving Diamond–Sasaki type conjectures in new cases: when p totally splits in a totally real field F, and when F is real quadratic with p inert, under mild hypotheses. It develops and combines tools—cohomology vanishing for non-parallel weights, weight-shifting via partial Hasse invariants and Theta operators, and Goren–Oort stratification—to relate geometric modularity of mod p forms to algebraic modularity of the same weights. Central to the approach is translating geometric modularity information into liftable automorphic forms through Jacquet–Langlands correspondences and controlled twists, then transferring back to mod p representations. The results unify and extend the BDJ framework with a geometric perspective, providing concrete pathways to verify Serre-type weight predictions in a broad class of totally real fields and enriching the interaction between-strata geometry, cohomology, and automorphic forms. The methods hold potential for broader generalizations beyond the split/inert cases via stratification and cohomological techniques, contributing to a more robust geometric Langlands-type picture at p.

Abstract

Let $p$ be a prime, $F$ be a totally real field in which $p$ is unramified and $ρ: \mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a totally odd, irreducible, continuous representation. The geometric Serre weight conjecture formulated by Diamond and Sasaki can be viewed as a geometric variant of the Buzzard-Diamond-Jarvis conjecture, where they have the notion of geometric modularity in the sense that $ρ$ arises from a mod $p$ Hilbert modular form and algebraic modularity in the sense of Buzzard-Diamond-Jarvis. Diamond and Sasaki conjecture that if $ρ$ is geometrically modular of weight $(k,l)\in \mathbb{Z}^Σ_{\geq 2}\times\mathbb{Z}^Σ$ and $k$ lies in the minimal cone, then $ρ$ is algebraically modular of the same weight, where $Σ$ is the set of embeddings from $F$ into $\overline{\mathbb{Q}}$. We prove the conjecture without parity hypotheses for real quadratic fields $F$ in which $p \geq 5$ is inert, and for totally real fields $F$ in which $p \geq \min\{5, [F:\mathbb{Q}]\}$ totally splits.

On the geometric Serre weight conjecture for Hilbert modular forms

TL;DR

The paper advances the geometric Serre weight program for Hilbert modular forms by proving Diamond–Sasaki type conjectures in new cases: when p totally splits in a totally real field F, and when F is real quadratic with p inert, under mild hypotheses. It develops and combines tools—cohomology vanishing for non-parallel weights, weight-shifting via partial Hasse invariants and Theta operators, and Goren–Oort stratification—to relate geometric modularity of mod p forms to algebraic modularity of the same weights. Central to the approach is translating geometric modularity information into liftable automorphic forms through Jacquet–Langlands correspondences and controlled twists, then transferring back to mod p representations. The results unify and extend the BDJ framework with a geometric perspective, providing concrete pathways to verify Serre-type weight predictions in a broad class of totally real fields and enriching the interaction between-strata geometry, cohomology, and automorphic forms. The methods hold potential for broader generalizations beyond the split/inert cases via stratification and cohomological techniques, contributing to a more robust geometric Langlands-type picture at p.

Abstract

Let be a prime, be a totally real field in which is unramified and be a totally odd, irreducible, continuous representation. The geometric Serre weight conjecture formulated by Diamond and Sasaki can be viewed as a geometric variant of the Buzzard-Diamond-Jarvis conjecture, where they have the notion of geometric modularity in the sense that arises from a mod Hilbert modular form and algebraic modularity in the sense of Buzzard-Diamond-Jarvis. Diamond and Sasaki conjecture that if is geometrically modular of weight and lies in the minimal cone, then is algebraically modular of the same weight, where is the set of embeddings from into . We prove the conjecture without parity hypotheses for real quadratic fields in which is inert, and for totally real fields in which totally splits.
Paper Structure (27 sections, 47 theorems, 109 equations)

This paper contains 27 sections, 47 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Geometric Hilbert modular forms
  3. Hilbert modular varieties of level prime to p
  4. Toroidal and minimal compactifications
  5. Automorphic line bundles
  6. Hilbert modular forms
  7. Hecke operators
  8. Weight shifting operators
  9. Hilbert modular forms over O
  10. Definite and indefinite quaternionic settings
  11. Twisting by characters
  12. The Jacquet-Langlands Correspondence
  13. Goren-Oort stratification
  14. Unitary and quaternionic Shimura varieties
  15. Stratification
  16. ...and 12 more sections

Key Result

Theorem A

Let $p \geq d$, $p > 3$ and $F$ be a totally real field in which $p$ totally splits. Let $\rho: G_F \rightarrow \mathop{\mathrm{GL}}\nolimits_2(\overline{\mathbb{F}}_p)$ be an irreducible, continuous, totally odd representation. If $p = 5$, assume that the projective image of $\rho|_{G_{F(\zeta_p)}}

Theorems & Definitions (106)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 96 more