在SO(n, 1)的Schottky和连分数半群中的同余计数

$\mathrm{SL}(2,\mathbb{Z})$ 的 Schottky 子群的显式谱隙

Explicit spectral gap for Schottky subgroups of $\mathrm{SL} (2,\mathbb{Z})$

本文研究整数特殊线性群 Schottky 子群对应双曲曲面同余覆盖的谱隙，在极限集足够厚时给出 Laplace–Beltrami 算子第二特征值的一致显式下界。

解析 torsion 和上同调 torsion 对于有理数秩为 1 的算术群的渐进行为

Asymptotic behaviour of analytic torsion and cohomological torsion for $\mathbb{Q}$-rank $1$ arithmetic groups

本文扩展了先前工作中关于 SL(n) 同余子群的解析 torsion 的精细渐进行为，应用于获得 SL(2, O_F) 同余子群和 SO(n,1) 同余子群的上同调 torsion 增长的新渐进行为和界。

On spt-crank type functions

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function $\operatorname{spt}_ω(n)$, which is closely related to Ramanujan's third order mock theta function $ω(q)$. Garvan and Jennings-Shaffer introduce a crank function which explains congruences for $\operatorname{spt}_ω(n)$. In this note, we study asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for $\operatorname{spt}_ω(n)$.

关于由两个抛物元素生成的$\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$的同余子群

On congruence subgroups of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ generated by two parabolic elements

该研究聚焦两个抛物矩阵生成的SL₂(ℤ[1/p])子群结构，提出相关同余子群猜想，并在r≤4、r=5部分情形及强Artin本原根猜想下的特定r值情形中证明猜想，明确其代数结构与有限指数性质。

PSL₂(F_{p²})的Galois实现 via 非单位理Hilbert不可约性

Galois Realisations of $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ via non-unirational Hilbert Irreducibility

本文通过非单位理Hilbert不可约性方法，证明了PSL₂(F_{p²})在特定同余条件下满足正则反Galois问题。

Twisted correlations of the divisor function via discrete averages of $\operatorname{SL}_2(\mathbb{R})$ Poincaré series

We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups. By keeping track of interactions between multiple orbits we get advantages over the widely used sums of Kloosterman sums techniques. We showcase this with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical lin…

关于数字段上 $SL_2$ 合同子群加权尖点计数的衰减

Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields

研究数字段上 $SL_2$ 合同子群的加权尖点计数，证明总加权尖点计数与群指数的比值被合同水平范数的负幂所限制，推广了 Cox--Parry 关于有理数域的结果，并支持尖点项在拓扑、算术和表示论公式中为次主导项的猜想。

eta-quotient新形式的例外同余

Exceptional Congruences for Eta-quotient newforms

本文研究了eta-quotient新形式系数的I型和II型同余，利用模形式理论证明了这些同余及对素数幂的扩展。

关于大水平新形式在PGL(n)上的上范数问题

The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$

本文研究了大水平新形式在PGL(n)上的上范数问题，通过新的降维理论和Atkin-Lehner算子推广，证明了次下界并基于GRH条件扩展结果。

$\operatorname{SL}_2(\mathbb{R})$ 自守核的加权平均，第一部分：非振荡函数

Weighted averages of $\operatorname{SL}_2(\mathbb{R})$ automorphic kernel, Part I: non-oscillatory functions

本文证明了一个计算由 $\operatorname{SL}_2(\mathbb{Z})$ 同余子群参数化的和的加权平均的定理，通过将谱方法直接应用于自守核以更好地保留对称性，并应用于二次多项式获得了关于最大素因子和根等分布的新结果。

Selberg $3/16$ 定理对 $\operatorname{SO}(n, 1)$ 的几何有限薄子群的推广

Generalization of Selberg's $3/16$ theorem for geometrically finite thin subgroups of $\operatorname{SO}(n, 1)$

针对 $\operatorname{SO}(n,1)$ 中带尖点的几何有限薄子群，在临界指数 $\delta_\Gamma > 1/2$ 时建立了均匀谱隙，填补了 $n \geq 3$ 时 $\delta_\Gamma \in (1/2, n-2]$ 区域的空白。

Elliptic curves over a finite field and the trace formula

We prove formulas for power moments for point counts of elliptic curves over a finite field $k$ such that the groups of $k$-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$. As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$ which include as special cases the groups $Γ_1(N)$ and $Γ(N)$. Our formulas generalize results of Birch and Ihara (the case of the trivial s…

$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-轨道在无限体积的 $p$-进齐性空间中的闭包

$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-orbit closures on a $p$-adic homogeneous space of infinite volume

本文研究了无限体积 $p$-进齐性空间上的轨道闭包问题，通过引入高分支性条件，推广了 Ratner 定理并对 $\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ 的 $Γ$-轨道进行了分类。

Deformations of Reducible Galois Representations to Hida-Families

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\barρ:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ lifts to a Hida line for which the weights range over a congruence class modulo-$p^2$. The advantage of the purely Galois theoretic approach is that it allows us to construct $p$-adic families of Galois representations lifting the actual representation $\barρ$, and not just the semisimplification.

两块奇数分拆函数的二元结构及 $pod(n)$ 的 theta 加权同余式

Dyadic Structure of the Two-Block Odd Partition Function and Theta-Weighted Congruences for $pod(n)$

本文通过二元结构分解和theta函数求值，解释两块奇数分拆函数a(n)的递推关系，并利用生成函数因式分解建立模任意奇素数的同余式，获得模3、5、13的无限族加权同余。

三次序环类群中2-挠元平均个数

The mean number of $2$-torsion elements in the class groups of cubic orders

本文确定了三次序类群中2-挠元的平均个数，并将其推广到任意奇数维情形，给出了可约SL_N(Z)-轨道的渐近计数公式。

置换逆bent函数猜想的证明

A proof of a permutation-inverse bent-function conjecture

本文证明当α不是F_q中的立方时，f_α是bent函数，解决了Li等人提出的猜想，并通过计算Walsh值和使用二元指数和及Hasse同余式得到非立方情形下Walsh系数为±q。