On the geometric Serre weight conjecture for Hilbert modular forms

TL;DR

This paper proves the geometric Serre weight conjecture for certain totally real fields and primes, establishing a link between geometric and algebraic modularity of Galois representations in these cases.

Contribution

It confirms the geometric Serre weight conjecture for specific real quadratic and totally real fields without parity assumptions.

Findings

Proves the conjecture for real quadratic fields with inert primes p ≥ 5.

Establishes the conjecture for totally real fields with p totally split and p ≥ min{5, [F:Q]}.

Bridges geometric and algebraic modularity in the specified cases.

Abstract

Let $p$ be a prime, $F$ be a totally real field in which $p$ is unramified and $\rho: \mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ 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 $\rho$ arises from a mod $p$ Hilbert modular form and algebraic modularity in the sense of Buzzard-Diamond-Jarvis. Diamond and Sasaki conjecture that if $\rho$ is geometrically modular of weight $(k,l)\in \mathbb{Z}^\Sigma_{\geq 2}\times\mathbb{Z}^\Sigma$ and $k$ lies in the minimal cone, then $\rho$ is algebraically modular of the same weight, where $\Sigma$ 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.