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Brandon Levin

Publications and source records attributed to Brandon Levin.

At least 19 recordsLinked to original sources
arXiv2026

Inclusions between p-bounded crystalline loci in dimension two

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations). GPT-5.5 Pro was used extensively in the course of this work.

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arXiv2026

Resolutions of spaces of crystalline representations and modularity

We introduce a new partial resolution of crystalline spaces of Galois representations when the gaps in Hodge--Tate weights are smaller than $p$, with no bound on ramification. Furthermore, when $n =3$ in the case of minimal regular weight, we are able to show that the resolution is normal (assuming the ramification index is divisible by 3). Employing base change techniques and further analysis of the resolution, we are able to show that all the components of the crystalline deformation rings are potentially diagonalizable. As a consequence, we deduce automorphy lifting, the weight part of Serre's conjecture, and the Breuil-M\'ezard conjecture in dimension three for minimal regular weight.

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arXiv2024

Generic decompositions of Deligne--Lusztig representations

Let $G_0$ be a reductive group over $\mathbb{F}_p$ with simply connected derived subgroup, (geometrically) connected center and Coxeter number $h+1$. We extend Jantzen's generic decomposition pattern from $(2h-1)$-generic to $h$-generic Deligne--Lusztig representations, which is optimal. We also prove several results on the ``obvious'' Jordan--H\"older factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of arXiv:1610.04819 [math.NT]

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arXiv2023

Irregular loci in the Emerton-Gee stack for GL_2

Let K/Q_p be unramified. Inside the Emerton-Gee stack X_2, one can consider the locus of two-dimensional mod p representations of the absolute Galois group of K having a crystalline lift with specified Hodge-Tate weights. We study the case where the Hodge-Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms. We prove that if the gap between each pair of weights is bounded by p (the irregular analogue of a Serre weight), then this locus is irreducible. We also establish various inclusion relations between these loci.

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arXiv2022

Extremal weights and a tameness criterion for mod $p$ Galois representations

We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of our conjecture. We then introduce a new class of weights associated to $n$-dimensional local mod $p$ representations which we call \emph{extremal weights}. Using a ``Levi reduction" property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre's conjecture for unit groups of some division algebras in generic situations.

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arXiv2022

Serre weights for three-dimensional wildly ramified Galois representations

We formulate and prove the weight part of Serre's conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption in \cite{arXiv:1512.06380}, \cite{arXiv:1608.06570} that the representation be tamely ramified at $p$. We also prove a version of Breuil's lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton--Gee stacks \cite{arXiv:2012.12719} using the local models introduced in \cite{arXiv:2007.05398}.

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arXiv2020

G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range

Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf{F}$. Fix a representation $\overline{\rho}$ of the absolute Galois group of an unramified extension of $\mathbf{Q}_p$, valued in $G(\mathbf{F})$. We study the crystalline deformation ring for $\overline{\rho}$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine-Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.

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arXiv2020

Local models for Galois deformation rings and applications

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil-M\'ezard conjecture in arbitrary dimension for tamely potentially crystalline deformation rings with small Hodge-Tate weights (with appropriate genericity conditions), the weight part of Serre's conjecture for $U(n)$ as formulated by Herzig (for global Galois representations which satisfy the Taylor-Wiles hypotheses and are sufficiently generic at $p$), and an unconditional formulation of the weight part of Serre's conjecture for wildly ramified representations.

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arXiv2020

Reductions of $2$-dimensional semi-stable representations with large $\mathcal L$-invariant

We determine reductions of 2-dimensional, irreducible, semi-stable, and non-crystalline representations of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ with Hodge--Tate weights $0 < k-1$ and with $\mathcal L$-invariant whose $p$-adic norm is sufficiently large, depending on $k$. Our main result provides the first systematic examples of the reductions for $k \geq p$.

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arXiv2019

Reductions of some two-dimensional crystalline representations via Kisin modules

We determine rational Kisin modules associated with two-dimensional, irreducible, crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ of Hodge-Tate weights $0, k-1$. If the slope is larger than $\lfloor \frac{k-1}{p} \rfloor$, we further identify an integral Kisin module, which we use to calculate the semisimple reduction of the Galois representation. In that range, we find that the reduction is constant, thereby improving on a theorem of Berger, Li, and Zhu.

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arXiv2016

Weight elimination in Serre-type conjectures

We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular weights for a mod $p$ Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the "obvious" weights.

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arXiv2016

Serre weights and Breuil's lattice conjecture in dimension three

We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$-arithmetic manifold is purely local, i.e., only depends on the Galois representation at places above $p$. This is a generalization to $\mathrm{GL}_3$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-M\'ezard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights $(0,1,2)$ as well as the Serre weight conjectures over an unramified field extending our previous results. We also prove results in modular representation theory about lattices in Deligne-Luzstig representations for the group $\mathrm{GL}_3(\mathbb{F}_q)$.

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arXiv2016

A Harder-Narasimhan theory for Kisin modules

We develop a Harder-Narasimhan theory for Kisin modules generalizing a similar theory for finite flat group schemes due to Fargues. We prove the tensor product theorem, i.e., that the tensor product of semi-stable objects is again semi-stable. We then apply the tensor product theorem to the study of Kisin varieties for arbitrary connected reductive groups.

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arXiv2015

Potentially crystalline deformation rings and Serre weight conjectures

We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-M\'ezard conjectures hold for these deformation rings.

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arXiv2015

Kisin modules with descent data and parahoric local models

We construct a moduli space $Y^{\mu, \tau}$ of Kisin modules with tame descent datum $\tau$ and with fixed $p$-adic Hodge type $\leq \mu$, for some finite extension $K/\mathbb{Q}_p$. We show that this space is smoothly equivalent to the local model for $\mathrm{Res}_{K/\mathbb{Q}_p} \mathrm{GL}_n$, cocharacter $\{ \mu \}$, and parahoric level structure. We use this to construct the analogue of Kottwitz-Rapoport strata on the special fiber $Y^{\mu, \tau}$ indexed by the $\mu$-admissible set. We also relate $Y^{\mu, \tau}$ to potentially crystalline Galois deformation rings.

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arXiv2015

Potentially crystalline deformation rings in the ordinary case

We study potentially crystalline deformation rings for a residual, ordinary Galois representation $\overline{\rho}: G_{\mathbb{Q}_p}\rightarrow \mathrm{GL}_3(\mathbb{F}_p)$. We consider deformations with Hodge-Tate weights $(0,1,2)$ and inertial type chosen to contain exactly one Fontaine-Laffaille modular weight for $\overline{\rho}$. We show that, in this setting, the potentially crystalline deformation space is formally smooth over $\mathbb{Z}_p$ and any potentially crystalline lift is ordinary. The proof requires an understanding of the condition imposed by the monodromy operator on Breuil modules with descent datum, in particular, that this locus mod p is formally smooth.

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arXiv2014

Local models for Weil-restricted groups

We extend the group theoretic construction of local models of Pappas and Zhu to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when $p \geq 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

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