arXiv subjects

Sug Woo Shin

Publications and source records attributed to Sug Woo Shin.

At least 19 recordsLinked to original sources

arXiv2026

Torsion-vanishing for Siegel modular varieties via supercuspidal congruences

We prove that the generic part of the cohomology of Siegel modular varieties with torsion coefficients is concentrated above the middle degree, under a suitable notion of genericity that is optimal in the unramified case. Our method relies on the semi-perversity of the relative cohomology of the Igusa stack and on a trace formula computation that is made possible by a congruence technique introduced by Scholze and Fintzen--Shin. Compared with the work of Yang--Zhu, which handles Shimura varieties of abelian type via categorical local Langlands, our method is adapted to Siegel modular varieties but handles coefficients of arbitrarily small characteristic.

math.NT↗

arXiv2026

Local Intertwining Relations and Co-tempered $A$-packets of Classical Groups

The local intertwining relation is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. We prove several instances of the local intertwining relation for quasi-split classical groups and the twisted general linear group, as they are required in the inductive proof of the endoscopic classification for quasi-split classical groups due to Arthur and Mok. In addition, we construct the co-tempered local $A$-packets by Aubert duality and verify their key properties by purely local means, which provide the seed cases needed as an input to the inductive proof. Together with further technical results that we establish, this makes the endoscopic classification conditional only on the validity of the twisted weighted fundamental lemma.

math.NT↗

arXiv2026

To be or not to be local

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbf{Q}_p$. For a smooth representation $π$ of $\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different strategies (inspired by the case $K=\mathbf{Q}_p$) to attack the locality question: does $π$ depend only on the underlying $2$-dimensional representation $\overlineρ$ of ${\rm Gal}(\overline K/K)$? In particular when $[K:\mathbf{Q}_p]=2$, crucially using perfectoid geometry, we associate to $\overlineρ$ an infinite-dimensional mod $p$ smooth representation of $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ which we hope is the restriction to $\begin{pmatrix}K^\times&K\\0&1\end{pmatrix}$ of the (irreducible) supersingular subquotient of $π$.

math.NT↗

arXiv2025

The stable trace formula for Igusa varieties, II

Assuming the trace formula for Igusa varieties in characteristic p, which is known by Mack-Crane in the case of Hodge type with good reduction at p, we stabilize the formula via Kaletha's theory of rigid inner twists when the reductive group in the underlying Shimura datum is quasi-split at p. This generalizes our earlier work under more restrictive hypotheses.

math.NT↗

arXiv2023

Recent progress on Langlands reciprocity for $\mathrm{GL}_n$: Shimura varieties and beyond

The goal of these lecture notes is to survey progress on the global Langlands reciprocity conjecture for $\mathrm{GL}_n$ over number fields from the last decade and a half. We highlight results and conjectures on Shimura varieties and more general locally symmetric spaces, with a view towards the Calegari-Geraghty method to prove modularity lifting theorems beyond the classical setting of Taylor-Wiles.

math.NT↗

arXiv2022

Galois representations for even general special orthogonal groups

We prove the existence of $\mathrm{GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of $\mathrm{GSO}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb{H}}$, arising from forms of $\mathrm{GSO}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin $L$-functions, and improve on the construction of $\mathrm{SO}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.

math.NT↗

arXiv2022

Galois representations for general symplectic groups

We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G_2 in the cohomology of Siegel modular varieties of genus three. Under some additional local hypotheses we also show automorphic multiplicity one as well as meromorphic continuation of the spin L-functions.

math.NT↗

arXiv2021

$H^0$ of Igusa varieties via automorphic forms

Our main theorem describes the degree 0 cohomology of Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well known analogue for complex Shimura varieties. As an application, we obtain a completely new approach to two geometric questions. (See Sect. 1.5 for a comparison with independent results by van Hoften and Xiao via a different approach.) Firstly, we verify the discrete part of the Hecke orbit conjecture, which amounts to irreducibility of central leaves, generalizing preceding works by Chai, Oort, Yu, et al. Secondly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extending previous results by Igusa, Ribet, Faltings--Chai, Hida, and others. Our proof is based on a Langlands--Kottwitz-type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in representation theory of $p$-adic groups due to Howe--Moore and Casselman.

math.NT↗

arXiv2021

The stable trace formula for Shimura varieties of abelian type

We express the Frobenius-Hecke traces on the compactly supported cohomology of a Shimura variety of abelian type in terms of elliptic parts of stable Arthur-Selberg trace formulas for the endoscopic groups. This confirms predictions of Langlands and Kottwitz at primes where the level is hyperspecial.

math.NT↗

arXiv2021

Congruences of algebraic automorphic forms and supercuspidal representations

Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(\mathbb A_F)$ and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of $G$. We illustrate how such congruences can be applied in the construction of Galois representations. Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton--Paškūnas on density of supercuspidal points from definite unitary groups to general $G$ as above.

math.NT↗

arXiv2019

Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element.

math.RT↗

arXiv2018

Endoscopy and cohomology of U(n,1)

By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to $U(n,1)$. In the case of lattices arising from Hermitian forms, we conjecture that the growth exponents we obtain are sharp in all degrees.

math.NT↗

arXiv2017

Patching and the p-adic Langlands program for GL(2, Q_p)

We present a new construction of the p-adic local Langlands correspondence for GL(2, Q_p) via the patching method of Taylor--Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author's local-global compatibility theorem, and relaxes a hypothesis on the local mod p representation in that theorem.

math.NT↗

arXiv2016

Patching and the p-adic local Langlands correspondence

We use the patching method of Taylor--Wiles and Kisin to construct a candidate for the p-adic local Langlands correspondence for GL_n(F), F a finite extension of Q_p. We use our construction to prove many new cases of the Breuil--Schneider conjecture.

math.NT↗

arXiv2016

Asymptotics and local constancy of characters of p-adic groups

In this paper we study quantitative aspects of trace characters $Θ_π$ of reductive $p$-adic groups when the representation $π$ varies. Our approach is based on the local constancy of characters and we survey some other related results. We formulate a conjecture on the behavior of $Θ_π$ relative to the formal degree of $π$, which we are able to prove in the case where $π$ is a tame supercuspidal. The proof builds on J.-K.~Yu's construction and the structure of Moy-Prasad subgroups.

math.RT↗

arXiv2014

Endoscopic Classification of Representations: Inner Forms of Unitary Groups

We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split unitary groups. The classification of automorphic representations is given in terms of automorphic representations of general linear groups. The classification of irreducible admissible representations is given in terms of Langlands parameters.

math.NT↗

arXiv2014

Sato-Tate theorem for families and low-lying zeros of automorphic $L$-functions

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at infinity. Let $^{L}G=\hat G \rtimes \mathrm{Gal}(\bar F/F)$ be the associated $L$-group and $r:{}^L G\to \mathrm{GL}(d,\mathbb{C})$ a continuous homomorphism which is irreducible and does not factor through $\mathrm{Gal}(\bar F/F)$. The families under consideration consist of discrete automorphic representations of $G(\mathbb{A}_F)$ of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak [Progr. Math. 70 (1987), 321--331.] and Serre [J. Amer. Math. Soc. 10 (1997), no. 1, 75--102.]. As an application we study the distribution of the low-lying zeros of the associated family of $L$-functions $L(s,π,r)$, assuming from the principle of functoriality that these $L$-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the Unitary, Symplectic and Orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius--Schur indicator to determine this Symmetry type. If $r$ is not isomorphic to its dual $r^\vee$ then the Symmetry type is Unitary. Otherwise there is a bilinear form on $\mathbb{C}^d$ which realizes the isomorphism between $r$ and $r^\vee$. If the bilinear form is symmetric (resp. alternating) then $r$ is real (resp. quaternionic) and the Symmetry type is Symplectic (resp. Orthogonal).

math.NT↗

arXiv2014

On Fields of rationality for automorphic representations

This paper proves two results on the field of rationality $\Q(π)$ for an automorphic representation $π$, which is the subfield of $\C$ fixed under the subgroup of $\Aut(\C)$ stabilizing the isomorphism class of the finite part of $π$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations $π$ such that $π$ is unramified away from a fixed finite set of places, $π_\infty$ has a fixed infinitesimal character, and $[\Q(π):\Q]$ is bounded. The second main result is that for classical groups, $[\Q(π):\Q]$ grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions.

math.NT↗