Let $N>1$ be an integer, and let $Γ= Γ_0 (N) \subset \SL_4 (\Z)$ be the subgroup of matrices with bottom row congruent to $(0,0,0,*)\mod N$. We compute $H^5 (Γ; \C) $ for a range of $N$, and compute the action of some Hecke operators on many of these groups. We relate the classes we find to classes coming from the boundary of the Borel-Serre compactification, to Eisenstein series, and to classical holomorphic modular forms of weights 2 and 4.
The automorphic cohomology of a connected reductive algebraic group defined over Q decomposes as a direct algebraic sum of cuspidal and Eisenstein cohomology. In the present paper we construct regular Eisenstein cohomology classes for congruence subgroups of a rational form G of Q-rank 2 of SO(n,2).
We give a computationally effective criterion for determining whether a finite-index subgroup of SL(2, Z) is a congruence subgroup, extending earlier work of Hsu for subgroups of PSL(2, Z).
In a previous paper [3] we computed cohomology groups H^5 (Gamma_0 (N), \C), where Gamma_0 (N) is a certain congruence subgroup of SL (4, \Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology.
We survey our joint work with Avner Ash and Mark McConnell that computationally investigates the cohomology of conguence subgroups of SL_4 (\Z). This is based on a talk given at the 2009 RIMS conference "Automorphic representations, automorphic L-functions and arithmetic."
In this survey article we give an overview of the developments on the congruence subgroup and the metaplectic problems after the work of Bass, Milnor and Serre.
We describe torsion classes in the first cohomology group of $\text{SL}_2(\mathbb{Z})$. In particular, we obtain generalized Dickson's invariants for p-power polynomial rings. Secondly, we describe torsion classes in the zero-th homology group of $\text{SL}_2(\mathbb{Z})$ as a module over the torsion invariants. As application, we obtain various congruences between cuspidal forms of level one and Eisenstein series.
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.
We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then prove that associated to each torsion cohomology class, there exists a infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent.
Let E_lambda be the Hilbert space spanned by the eigenfunctions of the non-Euclidean Laplacian associated with a positive discrete eigenvalue lambda. In this paper, the trace of Hecke operators T_n acting on the space E_lambda is computed for Hecke congruence subgroups Gamma_0(N) of non-square free level. This extends the computation of Conrey-Li [2], where only Hecke congruence subgroups Gamma_0(N) of square free level N were considered.
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…
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.
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.
We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part I, we will consider the general theory of modular functions for the normalizers.
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 …
We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part II, we will observe the location of the zeros of the above functions by numerical calculation.
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),
After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize these arithmetic properties of traces of singular values of a weakly holomorphic modular function on the full modular group to those on a congruence subgroup $Γ_0(N)$.
This is basically a summary of [Mu]. The focus of the paper is the explicit computation of Hecke operators for period functions. In particular we compute the matrix representations of the 2nd Hecke operator on period functions for the full modular group $SL(2,Z)$ and the congruence subgroup $Γ_0(2)$. [Mu] Tobias Mühlenbruch. Hecke operators on period functions for $Γ_0(N)$. Journal of Number Theory. http://dx.doi.org/10.1016/j.jnt.2005.09.003
We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.
Let F be the imaginary quadratic field of discriminant -3 and OF its ring of integers. Let Gamma be the arithmetic group GL_3 (OF), and for any ideal n subset OF let Gamma_0 (n) be the congruence subgroup of level n consisting of matrices with bottom row (0,0,*) bmod n. In this paper we compute the cohomology spaces H^{nu - 1} (Gamma_0 (n); C) as a Hecke module for various levels n, where nu is the virtual cohomological dimension of Gamma. This represents the first attempt at such computations for GL_3 over an imaginary quadratic field, and complements work of Grunewald--Helling--Mennicke and …
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…
Modules for sesquiads and congruence schemes are introduced. It is shown that the corresponding categories are belian and that base change functors establish an ascent datum which allows for a cohomology theory to be established.
Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups and modular forms will be discussed. We will mainly focus on three interesting aspects of modular forms for noncongruence subgroups: the unbounded denominator property, modularity of the Galois represen…
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.
Let E_lambda be a Hilbert space, whose elements are functions spanned by the eigenfunctions of the Laplace-Beltrami operator associated with an eigenvalue lambda>0. The norm of elements in this space is given by the Petersson inner product. In this paper, the trace of Hecke operators T_n acting on the space E_lambda is computed for congruence subgroups of Gamma_0(N) of square free level, which may be considered as the analogue of the Eichler-Selberg trace formula [11] for non-holomorphyic cusp forms of weight zero.
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.
We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cusp…
The principal aim of this paper is to construct torsion cohomology classes in the initial terms of a spectral sequence computing the cohomology of a Kottwitz-Harris-Taylor Shimura variety. Beside we produce some global congruences between automorphic representations.
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.
In this paper we exhibit a noncongruence subgroup $\G$ whose space of weight 3 cusp forms $S_3(\G)$ admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to $S_3(\G)$ by A. Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.
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…
In a previous work we apply lattice point theorems on hyperbolic spaces obtaining asymptotic formulas for the number of integral representations of negative integers by quadratic and hermitian forms of signature (n,1) lying in Euclidean balls of increasing radius. This formula involves an error term that depends on the first nonzero eigenvalue of the Laplace-Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations, with the error term mentioned above, for several choices of quadratic and her…
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 $\Re(s)=δ/2$.
Scattering constants are special values of Dirichlet series associated to non-holomorphic Eisenstein series. In this paper we give closed formulas for the scattering constants related to a non-congruence subgroup obtained via a Belyi map of an elliptic curve.
We investigate the homology of a congruence subgroup Gamma of SL_3(Z) with coefficients in the Steinberg modules St(Q^3) and St(E^3), where E is a real quadratic field and the coefficients are Q. By Borel-Serre duality, H_0(Gamma, St(Q^3)) is isomorphic to H^3(Gamma,Q). Taking the image of the connecting homomorphism H_1(Gamma, St(E^3)/St(Q^3)) \to H_0(Gamma, St(Q^3)), followed by the Borel-Serre isomorphism, we obtain a naturally defined Hecke-stable subspace H(Gamma,E) of H^3(Gamma,Q). We conjecture that H(Gamma,E) is independent of E and consists of the cuspidal cohomology H_cusp^3(Gamma,Q)…
We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when varying the weight or the level in the trace formula.
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.
Prime Geodesic Theorem for Arithmetic Compact Surfaces: Principal Congruence Case
Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of the Chinese Academy of Sciences（中国科学院大学）；Hangzhou International Innovation Institute, Beihang University（北京航空航天大学杭州创新研究院）；Qiuzhen College, Tsinghua University（清华大学求真学院）；School of Mathematical Sciences, University of Scinece and Technology of China（中国科学技术大学数学科学学院）
In this paper we will use experimental and computational methods to find modular forms for non-congruence subgroups, and the modular forms for congruence subgroups that they are associated with via the Atkin--Swinnerton-Dyer correspondence. We also prove a generalization of a criterion due to Ligozat for an eta-quotient to be a modular function.
We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.
As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.
Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.
There is an established bijection between finite-index subgroups Gamma of Gamma(2) and bipartite graphs on surfaces, or, equivalently, certain triples of permutations. We utilize this relationship to study both congruence and noncongruence subgroups in terms of the corresponding graphs. We show some elementary criteria which can be used to identify many noncongruence subgroups. Given a graph on a surface, we have a method to produce generators for the corresponding group Gamma in terms of the generators of Gamma(2). Given generators for Gamma(2n), we show how to determine whether or not a grap…
Cuspidal subgroups associated with non-rational Eisenstein maximal ideals
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）；Indian Institute of Technology Hyderabad（印度理工学院海德拉巴分校）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；Institut für Mathematik, Universität Heidelberg（海德堡大学数学研究所）
