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Gyujin Oh

Publications and source records attributed to Gyujin Oh.

8 recordsLinked to original sources
arXiv2025

Aubert duals of strongly positive representations for metaplectic groups

We determine the Aubert duals of strongly positive representations of the metaplectic group \(\widetilde{Sp}(n)\) over a non-Archimedean local field $F$ of characteristic different from two. Using the classification of Mati\'c and an explicit analysis of Jacquet modules, we describe these duals in terms of precise inducing data. Our results extend known descriptions for classical groups to the metaplectic groups case and clarify the role of Aubert duality for non-linear covering groups, providing a foundation for future applications to the study of unitary representations for those cases. Furthermore, We are able to show that the same method applies to odd general spin groups $GSpin(2n+1)$, yielding an explicit description of Aubert duals in that setting as well.

math.NT↗
arXiv2025

Generalized Whittaker models beyond $\mathfrak{sl}_{2}$-triples

We extend the notion of generalized Whittaker models by allowing them to be built upon smooth irreducible representations of unipotent subgroups of a $p$-adic reductive group that are not necessarily characters, nor induced from Weil representations. This notion generalizes the usual notion of generalized Whittaker models, which include the Bessel and Fourier--Jacobi models. Using Kirillov's orbit method for unipotent groups, we define and prove basic properties of the corresponding Jacquet functors. We also provide new instances of generalized Whittaker models of multiplicity one.

math.RT↗
arXiv2025

Moduli stacks of crystals and isocrystals

Given a liftable smooth proper variety over $\mathbb{F}_p$, we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over $\mathbb{Z}_p$ and the latter is an adic stack -- Artin stack in rigid geometry -- over $\mathbb{Q}_p$. Both stacks come equipped with the Verschiebung endomorphism $V$ corresponding to the Frobenius pullback of (iso)crystals. We study the geometry of the $V$-fixed points over the open substack of irreducible isocrystals, which we use to geometrically count the rank one $F$-isocrystals. Along the way, we carefully develop the theory of adic stacks.

math.NT↗
arXiv2025

Arithmetic quantum local systems over the moduli of curves

We construct an arithmetic analogue of the quantum local systems on the moduli of curves, and study its basic structure. Such an arithmetic local system gives rise to a uniform way of assigning a Galois cohomology class of the first geometric \'etale cohomology of a smooth proper curve over a number field.

math.NT↗
arXiv2024

Theta characteristics and noncongruence modular forms

The Hodge bundle $ω$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta characteristic $ν$ (or any odd power of it) different from $ω$ is a noncongruence modular form. On the other hand, we show how $ν\neω$ gives rise to a ``twisted'' analogue of the diagonal period map to a Siegel threefold, whose difference attributes to the stackiness of the moduli of abelian surfaces $\mathcal{A}_{2}$. Some questions on the Brill--Noether theory of the modular curves are answered.

math.NT↗
arXiv2023

A proof of Néron--Ogg--Shafarevich criterion via its archimedean analogue

In this short note, we deduce the classical Néron--Ogg--Shafarevich criterion on good reduction of abelian varieties from its archimedean analogue: a holomorphic family of abelian varieties over a punctured disc extends to the whole unit disc if and only if the topological monodromy representation is trivial.

math.AG↗
arXiv2022

Coherent cohomology of Shimura varieties, motivic cohomology, and archimedean $L$-packets

We formulate an analogue of the archimedean motivic action conjecture of Prasanna--Venkatesh for irregular cohomological automorphic forms on Shimura varieties, which appear on multiple degrees of coherent cohomology of Shimura varieties. Such multiple appearances are due to many infinity types in a single $L$-packet with equal minimal $K$-types. Accordingly, we formulate the conjecture comparing periods of forms of different automorphic representations. We provide evidences for the conjecture by showing its compatibility with existing conjectures on periods of automorphic forms. The conjectures suggest the existence of certain operations which move between different infinity types in an $L$-packet.

math.NT↗