I am a number theorist interested in Galois representations and the Langlands program and related areas in representation theory and geometry.
I was previously a postdoc at the University of Toronto and the IAS and a Ph.D. student at
the University of Chicago.
My CV is available
pdf.
Submitted articles
with
Bao V. Le Hung
Submitted. 2024. 27 pages.
[Abstract ±]
Under some technical assumptions of a global nature, we establish the weight part of Serre's conjecture for mod p Galois representations for CM fields that are tamely ramified and sufficiently generic at p.
Submitted. 2024. 5 pages.
[Abstract ±]
Let p>3 and F be a non-archimedean local field with residue field a proper finite extension of F_p. Let E be an algebraically closed countable field extension of the residue field of F. In this short note, we explain how the methods from [Le19,GLS23] can be used to construct irreducible smooth representations of GLn(F), n>1, over E without a central character.
with
Bao V. Le Hung,
Brandon Levin, and
Stefano Morra
Submitted. 2024. 17 pages.
[Abstract ±]
Let G be a reductive group over 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 of G(F_p), which is optimal. We also prove several results on the ``obvious'' Jordan--Holder factors of general Deligne--Lusztig representations. As an application we improve the weight elimination result of "Weight elimination in Serre-type conjectures".
with
Bao V. Le Hung and
Stefano Morra
Submitted. 2024. 85 pages.
[Abstract ±]
Let F/F^+ be a CM extension and H/F^+ a definite unitary group in three variables that splits over F. We describe Hecke isotypic components of mod p algebraic modular forms on H at first principal congruence level at p and "minimal" level away from p in terms of the restrictions of the associated Galois representation to decomposition groups at p when these restrictions are tame and sufficiently generic. This confirms an expectation of local-global compatibility in the mod p Langlands program. To prove our result, we develop a local model theory for multitype deformation rings and new methods to work with patched modules that are not free over their scheme-theoretic support.
Preprints
Preprint. 2017. 7 pages.
[Abstract ±]
Let F/F^+ be a CM extension in which p is unramified and all places above p in F^+ split.
Let w be a place of F dividing p.
We show, under mild hypotheses related to the Taylor-Wiles method, that the m-torsion in the space of mod p automorphic
forms on a compact U(3) defined over F^+ with infinite level at w and appropriate level and coefficients at other places
is indecomposable when the associated Galois representation representation is irreducible and generic at w.
Journal articles
with
Bao V. Le Hung,
Stefano Morra,
Chol Park, and
Zicheng Qian
To appear in Transactions of the AMS. 34 pages.
[Abstract ±]
Let K/Q_p be a finite unramified extension. We explicitly determine potentially crystalline deformation rings
of minimal regular weight and generic tame inertial types when the shape has colength
at most one. This has applications 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 [PQ22].
with
Bao V. Le Hung,
Brandon Levin, and
Stefano Morra
To appear in JEMS. 65 pages.
[Abstract ±]
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 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.
with
Bao V. Le Hung,
Brandon Levin, and
Stefano Morra
To appear in Algebra & Number Theory. 51 pages.
[Abstract ±]
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 [LLHLM18,LLHLM20] 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 using the local models introduced in [LLHLMa].
with
Bao V. Le Hung,
Stefano Morra,
Chol Park, and
Zicheng Qian
To appear in Memoirs of the AMS. 169 pages.
[Abstract ±]
Let F/F^+ be a CM field and let v be a finite unramified place of F above the prime p.
Let r be an n-dimensional continuous representation of G_F which we assume to be modular for a unitary group over F^+ which splits over F and is compact at all real places.
We prove, under Taylor-Wiles hypotheses, that the smooth GL_n(F_v)-action on the corresponding Hecke isotypical part of the mod p cohomology with infinite level above v|_{F^+} determines the restriction of r to G_{F_v} when this restriction is Fontaine--Laffaille and has a suitably generic semisimplification.
with
Eknath Ghate and
Mihir Sheth
Represent. Theory
27 (2023) 1088-1101.
[Abstract ±]
Let p > 3 and F be a non-archimedean local field with residue field a proper finite extension
of F_p. We construct smooth absolutely irreducible non-admissible representations of GL_2(F) defined
over the residue field of F extending the earlier results of the authors for F unramified over Q_p. This
construction uses the theory of diagrams of Breuil and Paskunas. By parabolic induction, we obtain
smooth absolutely irreducible non-admissible representations of GL_n(F) for n > 2.
with
Bao V. Le Hung,
Brandon Levin, and
Stefano Morra
Inventiones Mathematicae
231 (2023), no. 3, 1277-1488.
[Abstract ±]
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 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-Mezard 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.
with
Stefano Morra and
Benjamin Schraen
J. Institut Math. Jussieu
21 (2022), no. 2, 637-658.
[Abstract ±]
Let F be a totally real field in which p is unramified. Let r be a modular
representation of the absolute Galois of F which satisfies the Taylor--Wiles
hypotheses and is tamely ramified and generic at a place v above p. Let m be
the corresponding Hecke eigensystem. We describe the m-torsion in the mod
p cohomology of Shimura curves with full congruence level at v as a GL_2(k_v)-representation.
In particular, it only depends on the restriction of r to the inertial subgroup at v,
and its Jordan--Holder
factors appear with multiplicity one. The main ingredients are a description
of the submodule structure of generic GL(2,q)-projective envelopes and the
multiplicity one results of [EGS15].
with
Andrea Dotto
Compositio Mathematica
157 (2021), no. 8, 1653-1723.
[Abstract ±]
We prove a local-global compatibility result in the mod p Langlands program for GL_2(Q_{p^f}).
Namely, given a global residual representation r that is sufficiently generic at p, we prove that the diagram
occurring in the corresponding Hecke eigenspace of completed cohomology is determined by the restrictions of r to
decomposition groups at p.
If these restrictions are moreover semisimple at p, we show that the (phi,Gamma)-modules attached to this diagram
by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of r to
decomposition groups at p.
with
Bao V. Le Hung,
Brandon Levin, and
Stefano Morra
Forum of Math, Pi
8 (2020), e5, 135 pages.
[Abstract ±]
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 GL3 of the lattice conjecture of Breuil.
In the process, we also prove the geometric Breuil-Mezard conjecture for (tamely) potentially crystalline
deformation rings with Hodge-Tate weights (0,1,2) as well as the Serre weight conjectures of
[Her09] over an unramified field extending the results of [LLHLM]. We also prove results in modular
representation theory about lattices in Deligne-Luzstig representations for the group GL(3,q).
Math. Research Letters 26 (2019), no. 6, 1747-1758.
[Abstract ±]
Let p>2 be a prime and let q be p^3. We give examples of smooth absolutely irreducible representations of GL_2(Q_q)
over F_q which are not admissible.
Algebra & Number Theory 13 (2019), no. 8, 1807-1827.
[Abstract ±]
Let F be a totally real field in which p is unramified.
We show, under a Taylor-Wiles hypothesis, that the m-torsion in the cohomology of Shimura curves with full congruence level at v, if nonzero, coincides with a representation constructed by Breuil-Paskunas.
In particular, it depends only on the restriction of the corresponding Galois representation to a decomposition group at v, and its Jordan-Holder factors appear with multiplicity one.
This builds on and extends work of the author with Morra and Schraen and independently of Hu-Wang, which proved these results when the Galois representation was additionally assumed to be tamely ramified.
The main new tool is a method for computing Taylor-Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti-Tate deformation rings and their intersection theory.
with
Bao V. Le Hung and
Brandon Levin
Duke Math. Journal
168 (2019), no. 13, 2433-2506.
[Abstract ±]
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.
with
Stefano Morra and
Chol Park
Proceedings of the LMS
117 (2018), no. 4, 790-848.
[Abstract ±]
Assume that r is a three dimensional modular mod p representation which is non-ordinary and nonsplit reducible (niveau 2) at a totally split place w above p.
Under mild hypotheses, we show that the isomorphism class of r locally at w is determined by the GL_3(Q_p)-action on the space of mod p algebraic
automorphic forms by using the refined Hecke action of [HLM]. We also give a nearly
optimal weight elimination result for niveau two Galois representations compatible with
the explicit conjectures of [Her09] and [GHS]. Moreover, we prove the modularity of
certain Serre weights, in particular, when the Fontaine-Laffaille invariant takes value infinity, our methods provide with the modularity of a certain shadow weight.
with
Bao V. Le Hung,
Brandon Levin, and
Stefano Morra
Inventiones Mathematicae
212 (2018), no. 1, 1-107.
[Abstract ±]
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/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mezard conjectures hold for these deformation rings.
Mathematische Annalen 372 (2018), no. 1-2, 55-89.
[Abstract ±]
Under hypotheses required for the Taylor-Wiles method, we prove for forms of U(3) which are compact at
infinity that the lattice structure on upper alcove algebraic vectors or on principal series types given
by the lambda-isotypic part of completed cohomology is a local invariant of the Galois representation attached
to lambda when this Galois representation is residually irreducible locally at places dividing p. We combine
Hecke theory and weight cycling with the Taylor-Wiles method to establish crucial mod p multiplicity one
results for upper alcove algebraic vectors and principal series types.
with
Florian Herzig and
Stefano Morra
Compositio Mathematica
153 (2017), no. 11, 2215-2286.
[Abstract ±]
Suppose that r is an irreducible modular mod p Galois representation such that r is upper-triangular, maximally non-split, and generic locally at a totally split place w above p.
Under mild hypotheses, we show that r locally at w can be recovered from the GL_3(Q_p)-action on the space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights in Serre's conjecture for r, show the existence of an ordinary lifting of r, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations r to which our main theorem applies.
with Shelly Manber and Shrenik Shah
International Journal of Number Theory 06 (2010), no. 3, 625-653.
[Abstract ±]
We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of filtration p + 1. We derive p-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework, we formulate a precise criterion for p-divisibility of class numbers of imaginary quadratic fields in terms of the existence of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in Fp.
Conference proceedings
with
Bao V. Le Hung
To appear in Proceedings of the International Colloquium on Arithmetic Geometry, TIFR. 24 pages.
[Abstract ±]
We survey some recent progress on generalizations of conjectures of Serre concerning the cohomology of arithmetic groups, focusing primarily on the "weight" aspect. This is intimately related to (generalizations of) a conjecture of Breuil and Mezard relating the geometry of potentially semistable deformation rings to modular representation theory. Recently, B. Levin, S. Morra, and the authors established these conjectures in tame generic contexts by constructing projective varieties (local models) in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of Q_p with small regular Hodge-Tate weights.
Mathematisches Forschungsinstitut Oberwolfach Report No. 18, 2022, 938-940.
Teaching
MA 26500 Spring 2021 Linear Algebra and its Applications
MA 26500 Fall 2022 Linear Algebra and its Applications
MA 26500 Spring 2024 Linear Algebra and its Applications
MA 26500 Spring 2025 Linear Algebra and its Applications
