arXiv subjects

David C. Luo

Publications and source records attributed to David C. Luo.

7 recordsLinked to original sources

arXiv2026

The Local Langlands Correspondence for Middle Supercuspidal Representations of $p$-adic $\text{GL}(2n)$

Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper we give an explicit description of the local Langlands correspondence for middle supercuspidal representations of $\text{GL}(2n,\text{F})$, under the tameness condition $p\nmid 2n$, in terms of the maximal simple types that define them. We achieve this by explicitly computing and comparing the gamma factors on the automorphic and Galois sides of the local Langlands correspondence. The computation on the automorphic side requires neither the tameness condition nor the characteristic zero condition.

math.RT↗

arXiv2026

Distinguished Simple Supercuspidal Representations of $p$-adic $\text{GL}(n)$

Let $\text{E}/\text{F}$ be a quadratic extension of non-archimedean local fields with odd residual characteristic. In this paper, we give equivalent conditions for a simple supercuspidal representation $\pi$ of $\text{GL}(n, \text{E})$ to be distinguished by $\text{GL}(n, \text{F})$ in terms of its defining maximal simple type and twisted gamma factors. Furthermore, we prove that the collection of twisted gamma factors evaluated at $\frac{1}{2}$ between $\pi$ and all unitary, tamely ramified quasi-characters of $\text{E}^{\times}$ that are trivial on $\text{F}^{\times}$ is sufficient to determine whether $\pi$ is distinguished by $\text{GL}(n, \text{F})$.

math.RT↗

arXiv2025

On the Local Langlands Functorial Transfer from $\text{SO}(5)$ to $\text{GL}(4)$

Let $\text{F}$ be a non-archimedean local field of characteristic zero. We study the local Langlands functorial transfer from $\text{SO}(5, \text{F})$ to $\text{GL}(4, \text{F})$ for several families of supercuspidal representations of $\text{GL}(4, \text{F})$. In doing so, we give equivalent conditions for such representations to be functorial transfers from $\text{SO}(5, \text{F})$, expressed in terms of the Bushnell--Kutzko construction of supercuspidal representations, by studying poles of local exterior square $L$-functions and the existence of non-zero local Shalika models.

math.RT↗

arXiv2025

On the Local Converse Theorem for Depth $\frac{1}{N}$ Supercuspidal Representations of $\text{GL}(2N, F)$

In this paper, we use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $\text{GL}(2N, F)$ which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$.

math.RT↗

arXiv2021

On Zeckendorf Related Partitions Using the Lucas Sequence

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive terms in the Lucas sequence.

math.NT↗

arXiv2020

Gordian Adjacency for Positive Braid Knots

A knot $K_1$ is said to be Gordian adjacent to a knot $K_2$ if $K_1$ is an intermediate knot on an unknotting sequence of $K_2$. We extend previous results on Gordian adjacency by showing sufficient conditions for Gordian adjacency between classes of positive braid knots through manipulations of braid words. In addition, we explore unknotting sequences of positive braid knots and give a proof that there are only finitely many positive braid knots for a given unknotting number.

math.GT↗

arXiv2019

Generalizing the Abundancy of an Integer

The abundancy index of a positive integer is the ratio between the sum of its divisors and itself. We generalize previous results on abundancy indices by defining a two-variable abundancy index function as $I(x,n)\colon\mathbb{Z^+}\times\mathbb{Z^+}\to\mathbb{Q}$ where $I(x,n)=\frac{σ_x(n)}{n^x}$. Specifically, we extend limiting properties of the abundancy index and construct sufficient conditions for rationals greater than one that fail to be in the image of the function $I(x,n)$.

math.NT↗