arXiv subjects

Bao Le Hung

Publications and source records attributed to Bao Le Hung.

2 recordsLinked to original sources

arXiv2025

Mirror symmetry and the Breuil-Mézard Conjecture

The Breuil-Mézard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ that should govern congruences between mod $p$ automorphic forms. For generic parameters, we propose a construction of Breuil-Mézard cycles in arbitrary rank, and verify that they satisfy the Breuil-Mézard Conjecture for all sufficiently generic tame types and small Hodge-Tate weights. Our method is purely local and group-theoretic, and completely distinct from previous approaches to the Breuil-Mézard Conjecture. In particular, we leverage new connections between the Breuil-Mézard Conjecture and phenomena occurring in homological mirror symmetry and geometric representation theory.

math.NT↗

arXiv2023

Colength one deformation rings

Let $K/\mathbf{Q}_p$ be a finite unramified extension, $\overlineρ:\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ a continuous representation, and $τ$ a tame inertial type of dimension $n$. We explicitly determine, under mild regularity conditions on $τ$, the potentially crystalline deformation ring $R^{η,τ}_{\overlineρ}$ in parallel Hodge--Tate weights $η=(n-1,\cdots,1,0)$ and inertial type $τ$ when the \emph{shape} of $\overlineρ$ with respect to $τ$ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre's conjecture. Along the way we make unconditional the local-global compatibility results of \cite{PQ} and further study the geometry of moduli spaces of Fontaine--Laffaille representations in terms of colength one weights.

math.NT↗