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.
We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.
We determine the ring structure of Siegel modular forms of degree g modulo a prime p, extending Nagaoka's result in the case of degree g=2. We characterize U(p) congruences of Jacobi forms and Siegel modular forms, and surprisingly find different behaviors of Siegel modular forms of even and odd degrees.
Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.
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.
Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from a paper of Bass-Milnor-Serre.
We attempt to generalize a congruence property of elliptic modular forms proved by Sturm to that of Haupttypus of Siegel modular forms of degree 2 with level. Namely, we give an explicit bound of Fourier coefficients required to determine the congruence of modular forms. We give the analog of Sturm's theorem for Jacobi forms, which is required in the proof. In the case that Nebentypus of Siegel modular forms with prime level, in order to prove the congruence between two modular forms, we show that it suffices to check the congruence of the finitely many Fourier coefficients. Finally, we give t…
In this article, we establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
In this paper we generalize a well-known isomorphism between the space of cusp forms of weight $k$ for a Fuchsian subgroup of the first kind $Γ\subset\mathrm{SL}_{2}(\mathbb{R})$ and the space of certain Maaß forms of weight $k$ for $Γ$ to an isomorphism between the space of Siegel cusp forms of weight $k$ for a subgroup $Γ\subset\mathrm{Sp}_{n}(\mathbb{R})$, which is commensurable with the Siegel modular group $\mathrm{Sp}_{n}(\mathbb{Z})$, and a suitable space of Siegel-Maaß forms of weight $k$ for $Γ$.
Sturm obtained the bounds for the number of the first Fourier coefficients of elliptic modular form $f$ to determine vanishing of $f$ modulo a prime $p$. In this paper, we study analogues of Sturm's bound for Siegel modular forms of genus 2. We show the resulting bound is sharp. As an application, we study congruences involving Atkin's $U(p)$-operator for the Fourier coefficients of Siegel mdoular forms of genus 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 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 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.
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 give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case.
In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.
We carry out some computations of vector valued Siegel modular forms of degree two, weight (k,2) and level one. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. We highlight three experimental results: (1) we identify a rational eigenform in a three dimensional space of cusp forms, (2) we observe that non-cuspidal eigenforms of level one are not always rational and (3) we verify a number of cases of conjectures about congruences between classical modular forms…
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.
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).
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…
Lectures on Siegel modular forms given in Nordfjordeid. Besides a survey on Siegel modular forms it presents the joint work with Carel Faber on vector-valued Siegel modular forms of genus 2 and it gives evidence for a conjecture of Harder on congruences between g=1 and g=2 forms.
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."
We determine test vectors and explicit formulas for all Bessel models for those Iwahori-spherical representations of GSp(4) over a p-adic field that have non-zero vectors fixed under the Siegel congruence subgroup.
We establish a connection between the coefficients of Artin-Mazur zeta-functions and Kummer congruences. This allows to settle positively the question of the existence of a map T such that the number of fixed points of the n-th iterate of T is equal to the absolute value of the 2n-th Euler number. Also we solve a problem of Gabcke related to the coefficients of Riemann-Siegel formula.
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 classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities of these monodromy groups, and give an application to Hilbert's irreducibility theorem.
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
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.
A congruence relation satisfied by Igusa's cusp form of weight 35 is presented. As a tool to confirm the congruence relation, a Sturm-type theorem for the case of odd-weight Siegel modular forms of degree 2 is included.
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…
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.
Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $ρ$ with respect to some congruence subgroup of $\mathrm{Sp}_{2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{Sp}_{2n}(\mathbb R)$ obtained by lifting $F$ has the moderate growth (or "slowly increasing") property. This is a consequence of the following bound that we prove: $\|ρ(Y^{1/2})F(Z) \| \ll \prod_{i=1}^n (μ_i(Y)^{λ_1/2} + μ_i(Y)^{-λ_1/2})$ where $ λ_1 \ge \ldots \ge λ_n$ is the highest weight of $ρ$ and $μ_i(Y)$ are the eigenvalues of the matrix $Y$.
We give a summary of results for dimensions of spaces of cuspidal Siegel modular forms of degree 2. These results together with a list of dimensions of the irreducible representations of the finite groups GSp(4,Fp) are then used to produce bounds for dimensions of spaces of newforms with respect to principal congruence subgroups of odd square-free level.
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.
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.
