arXiv subjects

Yuchan Lee

Publications and source records attributed to Yuchan Lee.

9 recordsLinked to original sources

arXiv2026

Ideal class monoids of cubic orders

Let $R$ be an order in a number field, let $\overline{\mathrm{Cl}}(R)$ be its ideal class monoid, and let $\mathrm{Cl}(R)$ act on it by multiplication. The local-global product formula identifies the orbit set $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}}(R)$ with a product of local orbit sets; in this sense, it is the genus set of fractional $R$-ideals. For a Gorenstein order $R$ in a cubic extension of number fields, we give a closed Euler product formula for the cardinality of this genus set. The local factors come from an explicit classification of local cubic overorders: for arbitrary local cubic orders, we parametrize all overorders, determine their inclusion relations, and identify the Gorenstein ones. As an application to Bhargava's parametrization of $2\times3\times3$ cubes, our formula gives the exact number of $\mathrm{Cl}(R)$-equivalence classes of integral $\mathrm{GL}_2(\mathbb Z)\times\mathrm{SL}_3(\mathbb Z)\times\mathrm{SL}_3(\mathbb Z)$-orbits whose associated cubic ring is the prescribed Gorenstein order $R$.

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arXiv2026

Beyond Endoscopy for $\mathrm{GL}_3(\mathbb{Q})$: Functional equation for the $L$-function of a cubic order

The Beyond Endoscopy strategy, proposed by Langlands, aims to establish the principle of functoriality by analyzing the trace formula. Recently, Deng and Espinosa advanced this program for $\mathrm{GL}_3(\mathbb{Q})$ by isolating the contribution of the trivial representation from the elliptic regular part. Their work relies on a conjectural factorization formula for the $L$-function associated with a cubic order, which yields the functional equation for the completed $L$-function. In this paper, we provide an unconditional proof of this functional equation for every Gorenstein order in a cubic number field. As a consequence, their isolation of the trivial representation for $\mathrm{GL}_3(\mathbb{Q})$ becomes fully unconditional.

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arXiv2025

Diophantine analysis and Arthur's trace formula

Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_\gamma\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $\gamma\in G(k)$ whose centralizer $G_\gamma$ is a maximal torus in $G$ which is anisotropic over $k$. We formulate the asymptotic for the number of integral points on $X$ bounded by a fixed norm $T>0$ as $T\rightarrow \infty$ in terms of $\kappa$-orbital integrals, which play a role in the stabilization of Arthur's trace formula. This formula coincides with the contribution of the stable conjugacy class of $\gamma$ to the geometric side of the trace formula. As an application, we obtain an asymptotic formula for the number of $n \times n$ matrices over the ring of integers $\mathcal{O}_k$ whose characteristic polynomial equals a fixed irreducible polynomial $\chi(x)$ of degree $n$. This result generalizes a case studied by Eskin-Mozes-Shah (1996).

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arXiv2025

An upper bound for the size of the ideal class monoid

The ideal class monoid for an order $R$ in a finite field extension $E/F$ of a number field, denoted by $\overline{\mathrm{Cl}}(R)$, is a fundamental object to study in number theory which has useful applications in algebraic geometry and topology. In this paper, we describe an upper bound for $\#\overline{\mathrm{Cl}}(R)$, in terms of the class number of $E$ and (local) orbital integrals for $\mathfrak{gl}_n$. We also describe an upper bound for the class number of $E$ in terms of the Minkowski bound. When $[E:F]\leq 3$ or when $R$ is a Bass order, we refine our upper bound, using a known formula for local orbital integrals in the authors' previous work. In particular, if $R=\mathbb{Z}[x]/(x^3-mx^2+(m-1)x-1)$ with $m\in \mathbb{Z}$ which arises in a study of Cappell-Shaneson homotopy 4-spheres in topology, then we further refine our upper bound in terms of the discriminants of $R$ and $E$, which is $\frac{2}{3^5} \Delta_R^{\frac{1}{2}}\cdot \Delta_E^{\frac{3}{2}}$, when $\Delta_E>3075$.

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arXiv2024

An asymptotic formula for the number of integral matrices with a fixed characteristic polynomial via orbital integrals

For an irreducible polynomial $\chi(x)\in \mathcal{O}_k[x]$ of degree $n$, where $k$ is a number field and $\mathcal{O}_k$ its ring of integers, let $N(X, T)$ denote the number of $n \times n$ integral matrices whose characteristic polynomial is $\chi(x)$, bounded by a positive real number $T$ with respect to a certain norm. In this paper, we provide an asymptotic formula for $N(X,T)$ as $T\to \infty$ in terms of the orbital integrals of $\mathfrak{gl}_n$. This result extends the work of A. Eskin, S. Mozes, and N. Shah \cite{EMS} (1996) to a broader setting, thereby further developing the generalization initiated by the second author in arXiv:2509.22314. Our approach is based on the interpretation of local Brauer evaluations for $X$ via local class field theory, and on the Langlands-Shelstad fundamental lemma for $\mathfrak{sl}_n$. In particular, we observe that local Brauer evaluations for $X$ determine local endoscopic data for $\mathrm{SL}_n$, suggesting a deeper conceptual connection between these two notions.

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arXiv2024

Stable orbital integrals for classical Lie algebras and smooth integral models

A main goal of this paper is to introduce a new description of the stable orbital integral for a regular semisimple element and for the unit element of the Hecke algebra in the case of $\mathfrak{gl}_{n,F}$, $\mathfrak{u}_{n,F}$, and $\mathfrak{sp}_{2n,F}$, by assigning a certain stratification and then smoothening each stratum, where $F$ is a non-Archimedean local field of any characteristic. As applications, we will provide a closed formula for the stable orbital integral for $\mathfrak{gl}_{2,F}$, $\mathfrak{gl}_{3,F}$, and $\mathfrak{u}_{2,F}$. We will also provide a lower bound for the stable orbital integral for $\mathfrak{gl}_{n,F}$, $\mathfrak{u}_{n,F}$, and $\mathfrak{sp}_{2n,F}$ with all $n$. Finally we will propose conjectures that our lower bounds are optimal in a sense of the second leading term for $\mathfrak{gl}_{n,F}$ and the first leading term for $\mathfrak{u}_{n,F}$ and $\mathfrak{sp}_{2n,F}$. There is a restriction about the factorization of the characteristic polynomial arising from the parabolic descent when we work with $\mathfrak{u}_{n,F}$ and $\mathfrak{sp}_{2n,F}$, whereas this assumption does not appear in $\mathfrak{gl}_{n,F}$ case.

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arXiv2024

Orbital integrals and ideal class monoids for a Bass order

A Bass order is an order in a number field for which every fractional ideal can be generated by two elements. Examples include quadratic orders, orders containing the maximal order of a subfield $F$ with $[E:F]=2$, and orders whose discriminant is fourth-power-free. We prove a closed conductor formula for the number of fractional ideals of a Bass order $R$ modulo multiplication by invertible ideals. For a Bass order, this number is also the number of overorders of $R$, and we give an explicit conductor parametrization of all overorders. The proof combines the classification of local Bass overorders with a local--global argument. Orbital integrals give the corresponding weighted mass formulas, including the split local case. We also prove, by a smoothening procedure, a geometric orbital integral theorem for the relevant integral model; this theorem is presented separately in Section 4.

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arXiv2024

An explicit formula for the orbital integrals on the spherical Hecke algebra of $\mathrm{GL}_3$

We provide the explicit formula for orbital integrals associated with elliptic regular semisimple elements in $\mathrm{GL}_n(F) \cap \mathrm{M}_n(\mathfrak{o})$ and associated with arbitrary elements of the spherical Hecke algebra of $\mathrm{GL}_n(F)$ when $n=2, 3$, using results of [CKL]. Here $F$ is a non-Archimedean local field of any characteristic with $\mathfrak{o}$ its ring of integers.

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arXiv2023

On a Kostant section for the unitary group

For the unitary group defined over the ring of integers in a non Archimedean local field, we give a correction for a Kostant section provided in G.Laumon and B.C. Ngô's paper; Le lemme fondamental pour les groupes unitaires.

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