Maass Wave Forms Research Articles

On degree 2 Siegel cusp forms and its Fourier coefficients

We present a set of diagonal matrices which index enough Fourier coefficients for a complete characterization of all Siegel cusp forms of degree 2, weight k, level N and character χ, where k is an even integer ≥4, N is an odd, square-free positive integer, and χ has conductor equal to N. As an application, we show that the Koecher-Maass series of any F∈Sk2 twisted by the set of Maass waveforms whose eigenvalues are in the continuum spectrum of the hyperbolic Laplacian determines F. We also generalize a result due to Skogman about the non-vanishing of all theta components of a Jacobi cusp form of even weight and prime index, which may have some independent interest.

Noncongruence subgroups and Maass waveforms

The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke–Selberg and Phillips–Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of “new” discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) certify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.

Maass wave forms, quantum modular forms and Hecke operators

We prove that Cohen's Maass wave form and Li-Ngo-Rhoades' Maass wave form are Hecke eigenforms with respect to certain Hecke operators. As a corollary, we find new identities of the $p$th coefficients of these Maass wave forms in terms of $p$th root of unities.

A study on twisted Koecher–Maass series of Siegel cusp forms via an integral kernel

We find an explicit integral kernel for the twisted Koecher–Maass series of any degree two Siegel cusp form F, where the twist is realized by an arbitrary Maass waveform whose eigenvalue is in the continuum spectrum. We also obtain the analytic properties of such a kernel (functional equations and analytic continuation), as well as a series representation of it in terms of the degree two Siegel Poincare series. From these properties we deduce the analytic properties of the twisted Koecher–Maass series. Moreover, we express the later as a multiple Dirichlet series involving the Dirichlet series associated to the Fourier–Jacobi coefficients of F. Finally, we get the integral kernel of the untwisted Koecher–Maass series (first studied by Kohnen and Sengupta in any degree) as a limit case of our construction.

A class of non-holomorphic modular forms III: real analytic cusp forms for mathrm {SL}_2(mathbb {Z})

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit ‘weak harmonic lifts’ for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n^{1-k}(a_n^{prime } + rho a_n) for nne 0, where rho is the normalised permanent of the period matrix of the corresponding motive, and a_n, a_n^{prime } are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

On Some Special Families of q-hypergeometric Maass Forms

Using special polynomials introduced by Hikami and the second author in their study of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new families of quantum modular forms, and their Fourier coefficients encode distinguished Maass cusp forms. The cuspidality of these Maass waveforms is proven by making use of the Habiro ring representations of the associated quantum modular forms. Thus, we provide an example of using the $q$-hypergeometric structure of associated series to establish modularity properties which are otherwise non-obvious. We conclude the paper with a number of motivating questions and possible connections with Hecke characters, combinatorics, and still mysterious relations between $q$-hypergeometric series and {the passage} from positive to negative coefficients of Maass waveforms.

On a relation between certain $q$-hypergeometric series and Maass waveforms

In this paper, we answer a question of Li, Ngo, and Rhoades concerning a set of $q$-series related to the $q$-hypergeometric series $\sigma$ from Ramanajun's lost notebook. Our results parallel a theorem of Cohen which says that $\sigma$, along with its partner function $\sigma^*$, interpolate the coefficients of a Maass waveform of eigenvalue $1/4$.

Spectral correspondences for Maass waveforms on quaternion groups

We prove that in most cases the Jacquet–Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e. when a bijection is possible. The proof uses the Selberg trace formula.

Central limit theorems for classical cusp forms

An abstract Central Limit Theorem is applied to give weighted Central Limit Theorems for classical holomorphic and Maass wave forms. Examples and conjectural related results are considered.

A reduction point algorithm for cocompact Fuchsian groups and applications

In the present article we propose a reduction point algorithm for any Fuchsian group in the absence of parabolic transformations.We extend to this setting classical algorithms for Fuchsian groups with parabolic transformations, such as the flip flop algorithm known for the modular group $\mathbf{SL}(2, \mathbb{Z})$ and whose roots go back to [9].The research has been partially motivated by the need to design more efficient codes for wireless transmission data and for the study of Maass waveforms under a computational point of view.

Computation of Quantum Bound States on a Singly Punctured Two-Torus

We study a quantum mechanical system on a singly punctured two-torus with bound states described by the Maass waveforms which are eigenfunctions of the hyperbolic Laplace—Beltrami operator. Since the discrete eigenvalues of the Maass cusp form are not known analytically, they are solved numerically using an adapted algorithm of Hejhal and Then to compute Maass cusp forms on the punctured two-torus. We report on the computational results of the lower lying eigenvalues for the punctured two-torus and find that they are doubly-degenerate. We also visualize the eigenstates of selected eigenvalues using GridMathematica.

Generalized Maass wave forms

We initiate the study of generalized Maass wave forms, those Maass wave forms for which the multiplier system is not necessarily unitary. We then prove some basic theorems inherited from the classical theory of modular forms with a generalization of some examples from the classical theory of Maass forms.

Symmetries of the transfer operator for Γ0(N) and a character deformation of the Selberg zeta function for Γ0(4)

The transfer operator for Γ0(N) and trivial character χ0 possesses a finite group of symmetries generated by permutation matrices P with P2= id. Every such symmetry leads to a factorization of the Selberg zeta function in terms of Fredholm determinants of a reduced transfer operator. These symmetries are related to the group of automorphisms in GL(2,ℤ) of the Maass wave forms of Γ0(N). For the group Γ0(4) and Selberg’s character χα there exists just one nontrivial symmetry operator P. The eigenfunctions of the corresponding reduced transfer operator with eigenvalue λ=±1 are related to Maass forms that are even or odd, respectively, under a corresponding automorphism. It then follows from a result of Sarnak and Phillips that the zeros of the Selberg function determined by the eigenvalue λ=−1 of the reduced transfer operator stay on the critical line under deformation of the character. From numerical results we expect that, on the other hand, all the zeros corresponding to the eigenvalue λ=+1 are off this line for a nontrivial character χα.

MOCK MAASS THETA FUNCTIONS

The purpose of this paper is to use a general class of indefinite theta functions to explain and generalize an example of a Maass waveform that was constructed by Cohen from two functions σ and σ*, studied by Andrews, Dyson and Hickerson. For this, we construct certain functions attached to an indefinite binary quadratic form and show that they are ‘nearly’ Maass waveforms. We call these functions mock Maass theta functions. In certain special cases we obtain actual Maass waveforms. These are the same as in the work of Maass.

On Siegel-Eisenstein series attached to certain cohomological representations

We introduce a Siegel-Eisenstein series of degree 2 which generates a cohomological representation of Saito-Kurokawa type at the real place. We study its Fourier expansion in detail, which is based on an investigation of the confluent hypergeometric functions with spherical harmonic polynomials. We will also consider certain Mellin transforms of the Eisenstein series, which are twisted by cuspidal Maass wave forms, and show their holomorphic continuations to the whole plane.

Lewis–Zagier correspondence for higher-order forms

The Lewis-Zagier correspondence attaches period functions to Maass wave forms. We extend this correspondence to wave forms of higher order, which are higher-order invariants of the Fuchsian group in question. The key ingredient, an identification of higher-order invariants with ordinary invariants of unipotent twists, makes it possible to apply standard methods of automorphic forms to higher-order forms.

Remarks on modular symbols for Maass wave forms

In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions (pseudo--measures in the sense of [MaMar2]). After explaining the basic issues and analogies in the extended Introduction, I construct modular symbols in the sec. 1 and the related Levy--Mellin transforms in the sec. 2. The whole paper is an extended footnote to the Lewis--Zagier fundamental study [LZ2].

Computation of the Rankin–Selberg L-functions of Maass wave forms

This note outlines a method for numerically computing the Rankin–Selberg convolutions of Maass wave forms L-functions and reports on the computation of zeros of some of them. Bibliography: 11 titles.

THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function

In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, L-series, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Despite the fact that there are various nice results about the nature of ?(x), these investigations are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. At the end of the paper it is shown that certain object associated with ?(x) establish a bridge between realms of imaginary and real quadratic irrationals.