Ramanujan Graphs on Cosets of $PGL_2(\mathbb{F}_q)$

$PGL_4$的标准非一致算术商的弱拉马努金性质

Weak Ramanujan property of the standard non-uniform arithmetic quotient of $PGL_4$

该研究在有限域形式级数域的仿射厦与格的框架下，定义了弱拉马努金商复形的谱条件，证明了$PGL_4$的标准非一致算术商满足弱拉马努金性质。

拉马努金双图

Ramanujan Bigraphs

本文研究了双图的拉马努金性质，提出了双图的构造方法，并探讨了其在伪随机性、扩展器等领域的应用。

Type II$_1$ von Neumann algebra representations of Hecke operators on Maass forms and the Ramanujan-Petersson conjectures

Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantized representation of the Hecke operators. in this representation, the Hecke operators act on the Berezin's quantization, deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequa…

$PGL_3$ 的非均匀算术商上加权邻接算子的谱

Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of $PGL_3$

本文研究了 $PGL(3,\mathbb{F}_q[t])$ 商复形上加权邻接算子的自守谱，证明了其非平凡近似特征值集合包含同时谱与内摆线，并从组合角度重新证明了该商不是 Ramanujan 复形。

扭曲的 $\hbox{PGL}(2)$ 和 $\hbox{PGL}(3)$ $\mathrm{L}$-函数的临界零点与无条件均值定理

Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$-functions

本文利用Levinson方法证明，当Q→∞时，至少1/9的L函数L(s, Π₀×χ)的零点位于临界线上，其中χ是模不超过Q的本原狄利克雷特征，该结果在Π₀自对偶时无条件成立，否则在温和条件下成立。

On computing quaternion quotient graphs for function fields

Let $Λ$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units $Λ^*$ on the Bruhat-Tits tree $\mathcal{T}$ associated to $PGL_2(\mathbb{F}_q((1/T)))$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group $Λ^*$ in terms of generators and relations. Moreover we determine an upper bound for its run…