arXiv subjects

Jungtaek Hong

Publications and source records attributed to Jungtaek Hong.

5 recordsLinked to original sources

arXiv2026

Global geometrization of local smooth integral models in the Hitchin fibration for $\mathrm{GL}_n$: The Bass case

In previous joint work, we proposed a new method to study local orbital integrals for $\mathrm{GL}_n$ (where $n=3$ or in the Bass case), by employing a smoothening method of a certain scheme defined over a henselian ring. In this paper, we geometrize this local smoothening method within the framework of the global Hitchin fibration for $\mathrm{GL}_n$ in the Bass case. Consequently, we provide a closed formula for the $\ell$-adic cohomology of the compactified Jacobian of a spectral curve over a finite field with double singularities whose local rings are integral domains, provided that the normalization of a spectral curve is isomorphic to $\mathbb{P}^1$.

math.NT↗

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 $R$.

math.NT↗

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 $\Delta_E>3075$.

math.NT↗

arXiv2024

Thing2Reality: Transforming 2D Content into Conditioned Multiviews and 3D Gaussian Objects for XR Communication

During remote communication, participants often share both digital and physical content, such as product designs, digital assets, and environments, to enhance mutual understanding. Recent advances in augmented communication have facilitated users to swiftly create and share digital 2D copies of physical objects from video feeds into a shared space. However, conventional 2D representations of digital objects restricts users' ability to spatially reference items in a shared immersive environment. To address this, we propose Thing2Reality, an Extended Reality (XR) communication platform that enhances spontaneous discussions of both digital and physical items during remote sessions. With Thing2Reality, users can quickly materialize ideas or physical objects in immersive environments and share them as conditioned multiview renderings or 3D Gaussians. Thing2Reality enables users to interact with remote objects or discuss concepts in a collaborative manner. Our user study revealed that the ability to interact with and manipulate 3D representations of objects significantly enhances the efficiency of discussions, with the potential to augment discussion of 2D artifacts.

cs.HC↗

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.

math.NT↗