Tobias Finis, Erez Lapid
This is a sequel to arXiv:1308.3604. We study applications to limit multiplicity generalizing the results of arXiv:1208.2257.
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This papers deals with congruence subgroups of convex cocompact subgroups of PSL2(Z). We examine the behaviour of the resonance spectrum when the congruence parameter q goes to infinity: we show a lower bound for the counting function in discs and an upper bound in vertical strips. These results show drastically different behaviour on both sides of the critical line $\Re(s)=δ/2$.
Weak Weyl's law for congruence subgroups
设$G$是$\Bbb Q$上的连通和单连通半单代数群，$Γ\subset G(\Bbb Q)$是一个算术子群.设$K_\infty\subset G(\Bbb R)$是极大紧子群，$d$是对称空间$G({\mathbb R})/K_\infty$的维数.设$σ$是$K_\infty$的不可约酉表示.证明了对每个$Γ$，存在一个有限指标的正规子群$Γ_1\subset Γ$，使得$Γ_1$ -权$σ$和$T^{d/2}$的尖谱的计数函数的商有一个正下界为$T\to\infty$ .
In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of ${\rm GL}_n(F)$ for a non-archimedean local field $F$ of characteristic zero. Some global and local applications are also discussed.
Higher Selberg zeta functions for congruence subgroups
作为[KW 3]结果的推广，我们研究了同余子群的高阶Selberg zeta函数的函数方程.为了得到该函数的Gamma因子，我们引入了一个高阶Dirichlet $L$ -函数.然后，利用Barnes三重Γ函数和高阶Dirichlet $L$ -函数明确地确定Gamma因子.
The density conjecture for principal congruence subgroups
本文证明了SL_n(Z)无平方因子级主同余子群的Sarnak密度猜想，借助局部Whittaker函数与Kloosterman和的新上界，并讨论了相关算术应用。
On the Cusp Forms of Congruence Subgroups of an almost Simple Lie group
在本文中，我们解决的问题存在的尖点形式几乎单李群使用的方法，第二作者结合本地信息的超尖点表示$p$ -进群已知的第一作者。我们特别注意的情况下，$SL_M(\Bbb R)$，我们证明了各种存在性结果的主同余子群。
It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).
Congruence testing for odd subgroups of the modular group
给出了SL（2，Z）的有限指数子群是否为同余子群的一个计算有效的判别准则，推广了Hsu关于PSL（2，Z）的子群的工作.
The Multiplicity One Defect and Congruence-Modular Exponent Ratios
本文引入重数一缺陷定量衡量Shimura曲线雅可比重数一性质的失效程度，推导其显式上界，证明其约束模阿贝尔簇同余与模指数比，并给出水平126的Gorenstein Hecke代数实例。
The structure of the normalisers of the congruence subgroups of the Hecke group $G_5$
我们确定了$G_5$的同余子群的正规化子。
Arithmetic expressions of Selberg's zeta functions for congruence subgroups
Sarnak的文章证明了SL（2，Z）的Selbergzeta函数可用本原不定二元二次型的基本单位和类数表示.本文的目的是得到SL（2，Z）的同余子群的Selberg zeta函数的对数导数的类似算术表达式.作为应用，我们研究了Brun-Titchmarsh型素测地线定理，类数和的渐近性质.
We give a simple algorithm that enables us to determine whether a subgroup of finite index of the Hecke group is normal.
Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre
本文综述了Bass、Milnor和Serre之后关于同余子群和元代数问题的研究进展。
We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL_2(q) and PGL_2(q) regularly as Galois groups.
Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.
We give here a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.
We answer positively a question of Church, Miller, Nagpal and Reinhold on existence of a linear bound on the presentation degree of the homology of a complex of FI-modules. This implies a linear stable range for the homology of congruence subgroups of general linear groups.
Let $A$ be the ring of elements in an algebraic function field $K$ over a finite field $F_q$ which are integral outside a fixed place $\infty$. In an earlier paper we have shown that the Drinfeld modular group $G=GL_2(A)$ has automorphisms which map congruence subgroups to non-congruence subgroups. Here we prove the existence of (uncountably many) normal genuine non-congruence subgroups, defined to be those which remain non-congruence under the action of every automorphism of $G$. In addition, for all but finitely many cases we evaluate $ngncs(G)$, the smallest index of a normal genuine non-co…
On the trace of Hecke operators for Maass forms for congruence subgroups II
设E_lambda为非欧拉普拉斯算子的本征函数所张成的希尔伯特空间，其本征函数具有正的离散本征值λ.本文对非平方自由水平的Hecke同余子群Gamma 0（N）计算了作用在空间E λ上的Hecke算子T n的迹。这推广了Conrey-Li [2]的计算，其中只考虑了无平方数水平N的Hecke同余子群Gamma_0（N）.
On scattering constants for a non-congruence subgroup
散射常数是狄利克雷级数与非全纯艾森斯坦级数相联系的特殊值。本文给出了由椭圆曲线的Belyi映射得到的与非同余子群有关的散射常数的闭公式。
Eisenstein cohomology for congruence subgroups of SO(n,2)
定义在Q上的连通约化代数群的自守上同调分解为尖点上同调与Eisenstein上同调的直代数和。本文构造了SO（n，2）的Q秩为2的有理形式G的同余子群的正则Eisenstein上同调类。
In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and recently Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true) and conjectured the congruence for the sixth sequence. In this paper…
On the cohomology of congruence subgroups of SL_4 (\Z)
本文综述了我们与Avner Ash和Mark McConnell的联合工作，即计算研究了SL_4（\Z）的同余子群的上同调。这是基于2009年RIMS会议上的演讲“自守表示，自守L函数和算术。"
We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2 and the discriminant kernels of special orthogonal groups SO 0 (2, n), n = 3, 4, 6. The proof is based on an application of linear algebra adapted to the number theoretical needs.
Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups
本文证明了对于Bianchi群的适当的同余子群序列，包括同余子群的标准穷举序列，以及Sl_2（C）的标准表示的偶对称幂，第一同调的挠部分的大小指数增长.这将Bergeron和Venkatesh的结果推广到非均匀格的情况。
Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maass cusp forms of weight 0 or 1 for the congruence subgroups $Γ_0(q)$, $Γ_1(q)$, and $Γ(q)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.
A density theorem for Borel-Type Congruence subgroups and arithmetic applications
利用预Kuznetsov型公式证明Borel型同余子群的密度结果，并应用于最优提升和计数，扩展了Kamber和Lavner在GL_3上的研究。
Computing Classical Modular Forms for Arbitrary Congruence Subgroups
本文提出了一种高效算法，用于计算任意同余子群的模形式q展开式，并讨论了相关理论和实际应用。
On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels I
本文研究了亏格为零且水平至多为12的同余子群的正规化子所对应的Hecke型Faber多项式的Eisenstein级数的零点和模函数的位置.在第一部分，我们将考虑规范化子的模函数的一般理论。
We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed $ϕ$-congruence subgroups, are obtained by reducing homomorphisms $ϕ$ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi-unipotent case we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower …
The Modular number, Congruence number, and Multiplicity One
本文研究与新形式对应 Abel 簇的模参数数和同余数，证明在相关 Hecke 代数极大理想满足重数为一时，二者的 p-adic 赋值相同。
Congruence and Noncongruence Subgroups of Gamma(2) via Graphs on Surfaces
在Gamma（2）的有限指数子群Gamma和曲面上的二部图之间存在一个已建立的双射，或者等价地，某些三重排列。利用这种关系，我们研究了相应的图的同余和非同余子群。我们给出了一些基本的判别准则，这些准则可以用来判别许多非同余子群。给定一个曲面上的图，我们有一个方法来产生相应的群Gamma的生成元，根据Gamma（2）的生成元。给定Gamma（2n）的生成元，我们展示了如何确定2n级图是否对应于同余子群。
On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels II
本文研究了亏格为零且水平至多为12的同余子群的正规化子所对应的Hecke型Faber多项式的Eisenstein级数的零点和模函数的位置.在第二部分中，我们将通过数值计算观察上述函数零点的位置。
We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),
Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups
在Zagier证明了奇异模$j(z)$的迹是弱全纯模形式的傅里叶系数之后，主要研究了满模群$PSL_2(\mathbb{Z})$上模函数奇异值的迹的各种性质，如它们的精确公式、极限分布、对偶性和同余性.本文的目的是将满模群上弱全纯模函数奇异值迹的算术性质推广到同余子群$Γ_0(N)$上.
Hecke Operators for Period Functions for Congruence Subgroups
这基本上是一个[穆]的总结。本文的重点是Hecke算子的周期函数的显式计算。特别地，我们计算了全模群$SL(2,Z)$和同余子群$Γ_0(2)$的第二Hecke算子在周期函数上的矩阵表示. [Mu]托拜厄斯·米伦布鲁赫$Γ_0(N)$的周期函数上的Hecke算子。数论杂志。http://dx.doi.org/10.1016/j.jnt.2005.09.003
On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups
证明了关于同余子群的2次Siegel尖点形式的傅里叶系数的Petersson型公式，并作为推论给出了个别傅里叶系数的上界估计.本文中的方法基本上是一个概括的北冈以前的工作，研究了充分的模块化的情况下，但一些修改是必要的，以获得估计是尖锐的水平方面。
We prove a representation stability result for the codimension-one cohomology of the level three congruence subgroup of $\mathbf{SL}_n(\mathbb{Z})$. This is a special case of a question of Church-Farb-Putman which we make more precise. Our methods involve proving several finiteness properties of the Steinberg module for the group $\mathbf{SL}_n(K)$ for $K$ a field. This also lets us give a new proof of Ash-Putman-Sam's homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church-Putman's homological vanishing theorem for the Steinberg module for the gr…
For a positive integer $k$, we extend the surjectivity results from special linear groups (Type $A_k$) and symplectic linear groups (Type $C_k$) onto product of generalized projective spaces by associating the rows or columns, to certain congruence ideal subgroups of special linear groups and symplectic linear groups.
Cohomology of Congruence Subgroups of SL(4,\Z) II
在文献[3]中我们计算了上同调群H ^5（Gamma_0（N），\C），其中Gamma_0（N）是SL（4，\Z）的一个同余子群，对于一定的水平范围N.在本文中，我们通过扩展能级的范围来更新这一早期工作，并描述自[3]出版以来发现的尖点上同调类和额外的边界现象。本文的尖点上同调类是GL（4）的第一个尖点型，它是用Betti上同调具体构造的.
Counting elements of the congruence subgroup
本文研究同余子群 Γ_0(Q) 中朴素高度不超过 X 的矩阵计数问题，给出了在 Q 与 X 的广泛范围内一致成立的渐近公式。
Eigenvalues of congruence covers of geometrically finite hyperbolic manifolds
设G=SO（n，1），Gamma是G的一个包含在G的一个等差子群中的几何有限的Zebraki稠密子群.用Gamma（q）表示水平q的Gamma的主同余子群，并固定一个严格小于（n-1）^2/4的正数λ 0，证明了当q沿素数趋于无穷大时，小于小于λ da_0的同余覆盖Gamma（q）\ H^n的Laplacian特征值的个数至多为[Gamma：Gamma（q）]^c阶，其中c=c（λ_0）>0.
Let $Γ<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $Γ(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of $L^2(Γ(q) \backslash \mathrm{SL}_2(\mathbb{R}))$ with a {\it uniform} exponential error term for all square-free $q$ with no small prime divisors. As an application we establish a uniform resonance-free half plane for the resolvent of the Laplacian on $Γ(q) \backslash \mathbb{H}^2$ over $q$ as above. Our approach is to extend Dolgopyat's dynamical proof of exponential m…
A uniform spectral gap for congruence covers of a hyperbolic manifold
令$G$是$\SO(n,1)$或$\SU(n,1)$，并且令$Γ\subset G$表示算术格。双曲流形$Γ\backslash \calH$带有一个自然的覆盖族，来自$Γ$的同余子群。在许多应用中，对于这个族，具有一致的谱隙的界限是有用的。当$Γ$本身是一个同余格时，从已知的结果来看，Ramanujan拓扑有很好的界。本文建立了非同余格的同余子群的一致有效界。
In a previous paper, we obtained a general trace formula for double coset operators acting on modular forms for congruence subgroups, expressed as a sum over conjugacy classes. Here we specialize it to the congruence subgroups $Γ_0(N)$ and $Γ_1(N)$, obtaining explicit formulas in terms of class numbers for the trace of a composition of Hecke and Atkin-Lehner operators. The formulas are among the simplest in the literature, and hold without any restriction on the index of the operators. We give two applications of the trace formula for $Γ_1(N)$: we determine explicit trace forms for $Γ_0(4)$ wi…
In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL_2(C) grows exponentially in m^2. We give upper and lower bounds for the growth rate. Our result extends a result of Mueller and Marshall, who proved the corresponding statement for closed arithmetic 3-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for twisted combinatorial Reidemeister torsion…
Finite index subgroups of the modular group and their modular forms
经典上，模群的同余子群在群论和模形式中起着重要的作用，它们可以用同余关系来描述。实际上，模群的大多数有限指数子群是非同余的。这些组以及它们的模形式是这篇调查文章的主要参与者。将讨论全等子群和非全等子群以及模形式之间的区别。我们将主要集中在非同余子群的模形式的三个有趣的方面：无界分母性质，由非同余尖形产生的伽罗瓦表示的模性，Atkin和Swinnerton-Dyer同余。
There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets are quasimodular forms. We extend these families so that the corresponding q-brackets are quasimodular for a congruence subgroup. Moreover, we find subspaces of these families for which the q-bracket is a modular form. These results follow from the properties of Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.
The congruence subgroup property for $S$-arithmetic subgroups of simple algebraic groups when $S$ has positive Dirichlet density
本文研究了当S具有正狄利克雷密度时，简单代数群的S-算术子群的同余子群性质，证明了同余核为平凡群，为塞雷的同余子群猜想提供了证据。
On the trace of Hecke Operators for Maass forms for congruence subgroups
设E_lambda是一个希尔伯特空间，其元素是Laplace-Beltrami算子的本征函数所张成的函数，本征值λ>0.这个空间中元素的范数由Petersson内积给出。本文计算了空间E_lambda上的Hecke算子T_n对无平方水平的Gamma_0（N）的同余子群的迹，它可以看作是权为零的非全纯尖点形式的Eichler-Selberg迹公式[11]的类似。
Lifts of projective congruence groups
我们证明了SL_2（Z）的投射等价于同余子群的非同余子群是普遍存在的。更确切地说，如果所讨论的同余子群是水平N>2的主同余子群Gamma（N），则它们总是存在，并且它们在许多情况下也存在于Gamma_0（N）。提出这个问题的动机与模形式有关：射影等价群对所有偶数权具有相同的尖点形式空间，而奇数权的尖点形式空间一般是不同的。我们通过对所附椭圆模曲面的几何考虑，对重量为3的这种现象进行了一些初步观察。我们还开发了算法，构造所有的子群投影等价于一个给定的同余子群，并决定其中的同余。这方面的一个关键工具是沃尔法特的广义层次概念。
A note on the spaces of Eisenstein series on general congruence subgroups
本文提出利用主同余子群的Hecke艾森斯坦级数理论研究一般同余子群上谱艾森斯坦级数的新方法，得到相关空间的基与系数性质，并简化证明了Eichler-Shimura同构定理。
The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain
本文将经典模群SL_2（Z）的一些代数性质推广到Drinfeld模理论中的等价群，特别是推广到模曲线理论中的一些重要性质.本文研究了作为D的理想的任意戴德金整环的SL 2（D）的尖点振幅和（同余）子群的水平。特别地，我们推广了Larcher的一个显著结果.我们引入更精细的准振幅和准水平的概念，这是不需要是理想和编码更多的信息的子群。我们的结果还提供了SL 2（D）的子群是同余子群的几个新的必要条件。
N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces
我们建立了一个对应的广义的四维规范理论和同余子群的模块组，铰链上的三价图形，出现在这两个。列举了各种规范理论和图形，并对它们的数量进行了比较。对应关系是特别引人注目的属零扭自由同余子群的例子出现在月光。我们详细分析的情况下，指数24，模块化椭圆K3表面出现：在这里，椭圆j-不变量可以被重铸为dessins d 'enfant，这决定了塞伯格-威滕曲线。
Let $G$ be a reductive algebraic group over $\mathbb{Q}$ and $Γ\subset G(\mathbb{Q})$ an arithmetic subgroup. Let $K_\infty\subset G(\mathbb{R})$ be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of $G(\mathbb{R})$ in $L^2(Γ\backslash G(\mathbb{R}))$ of a fixed $K_\infty$-type $σ$. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type $σ$ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Ar…
Generalized Analytic Automorphic Forms for some Arithmetic Congruence subgroups of the Vahlen group on the n-Dimensional Hyperbolic Space
本文研究了一类新的解析型向量值和Clifford代数值的一元和二元向量值自守形式。对于经典模群及其算术同余子群的超复推广，讨论了在$R^n$的上半空间上被Dirac算子零化的Eisenstein型和Poincaré型级数，以及更一般地被迭代Dirac算子零化的Eisenstein型和Poincaré型级数.特别是，我们介绍（多）单演模形式超复杂的推广的经典θ组。
Lie algebras and cohomology of congruence subgroups for SL_n(R)
设$R$是交换环，$k$是交换群，$p$是素环，$SL(n,R)$是特殊线性群.证明了$SL(n,R)$通过$p$ -同余子群的滤子所对应的李代数同构于张量积$\mathfrak{sl}_n(R\otimes_{\Z}\Z/p)\otimes_{\F_p}t\F_p[t]$，即$\F_p$上的由零常数项多项式和系数无迹矩阵$n\times n$组成的李代数.我们使用李代数结构以及Lyndon-Hochschild-Serre谱序列来计算涉及$p$ -同余子群的同调子的某些中心扩张的$d^2$同调微分。我们还利用基本的群结构得到了几个同调结果。例如，我们计算$n\geq3$的水平$p$ -同余子群的第一同调群。我们证明了$n=2$和$R=\Z[t]$的水平$p^r$ -同余子群的上同调群不是可生成的.最后，我们证明了对于高斯整数$n=2$和$R=\Z[i]$，作为$\F_p$ -向量空间，水平$p^r$ -同余子群的第二上同调群的维数至少为2.
In this article, the authors give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. They also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus were dealt in an earlier work of Choie, the first author and Kohnen.
Cohomology of congruence subgroups of SL(4,Z)
令$N>1$为整数，且令$Γ= Γ_0 (N) \subset \SL_4 (\Z)$为底行与$(0,0,0,*)\mod N$全等的矩阵的子群。我们计算$H^5 (Γ; \C) $的范围内的$N$，并计算的行动，一些赫克运营商对许多这些群体。我们将我们发现的类与来自Borel-Serre紧化边界的类、与Eisenstein级数的类以及与权重为2和4的经典全纯模形式的类联系起来。
Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations
本文研究SL（4，Z）的同余子群的某些同调论中挠率的计算.其中包括通常的群上同调，泰特-法雷尔上同调，以及sharbly复形的同调。所有这些理论都产生Hecke模。我们猜想这些理论中的Hecke本征类具有附加的伽罗瓦表示。解释了我们在挠素数2，3，5处的计算结果。我们提供的证据，我们的猜想在15例奇扭转，我们发现在水平高达31。
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