We prove certain relations between Satake parameters of cuspidal representations of $\GL_2(\mathbb{A}_{\mathbb{Q}})$ at finite and archimedean places. Consequently, we show that the Ramanujan-Petersson conjecture at a fixed prime $p\nmid N$ for \textit{non-exceptional} Maass forms of level $N$ implies the conjecture at $p$ for \textit{all} Maass forms of level $N$ and the Selberg's $1/4$-eigenvalue conjecture simultaneously. As an application, we improve Kim and Sarnak's $7/64$-bound towards the Satake parameters at all $p\nmid N$ for exceptional Maass forms.
We compute the twisted Koecher-Maass series of the first and second kind of the Ikeda type lift for the exceptional group of type $E_{7,3}$. As an application, we obtain their rationality result.
Recently Bringmann, Raum and Richter generalised the definition of Jacobi forms and Skoruppa's skew-holomorphic Jacobi forms by intertwining with harmonic Maass forms. We prove the isomorphism of the Kohnen's plus space analogue of harmonic Maass forms of weight $k-1/2$ for $Γ_0(4m)$ and the space of these skew-holomorphic harmonic Maass-Jacobi forms of weight $k$ and index $m$ for $k$ odd and $m=1$ or a prime.
We construct theta liftings from half-integral weight weak Maass forms to even integral weight weak Maass forms by using regularized theta integral. Moreover it gives an extension of Niwa's theta liftings on harmonic weak Maass forms. And we obtain the similar results to those by Niwa.
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.
The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.
We construct two families of harmonic Maass Hecke eigenforms. This construction answers a question of Mazur about the existence of an "eigencurve-type" object in the world of harmonic Maass forms. Using these families, we construct $p$-adic harmonic Maass forms in the sense of Serre.
A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level $N$ is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass 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.
We extend a formula of Duke, Imamōglu, and Tóth (which itself is a generalization of the Katok-Sarnak formula) to prove the Kohnen-Zagier formula for Maass forms for $Γ_0(4)$.
We prove identities between cycle integrals of non-holomorphic modular forms arising from applications of various differential operators to weak Maass forms.
We prove an orthogonality relation for the Fourier-Whittaker coefficients of a thin family of $GL(3)$ Maass forms containing all self-dual forms. This is obtained by analysing the Kuznetsov trace formula on $GL(3)$ for a certain family of test functions. The method also yields Weyl's law for the same family of Maass forms.
In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.
We show the existence of "Zagier duality" between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass forms as developed in recent works by Bruinier, Funke, Ono, and Rhoades. Concerning the isomorphism between the spaces of scalar and vector valued harmonic weak Maass forms of integral weight, "Zagier duality" between scalar valued ones is derived.
We present examples of Maass forms on Hecke congruence groups, giving low eigenvalues on $Γ_0(p)$ for small prime $p$, and the first 1000 eigenvalues for $Γ_0(11)$. We also present calculations of the $L$-functions associated to the Maass forms and make comparisons to the predictions from random matrix theory.
The space of polyharmonic Maass forms was introduced by Lagarias-Rhoades, recently. They constructed its basis from the Taylor coefficients of the real analytic Eisenstein series. In this paper, we introduce polyharmonic weak Maass forms, that is, we relax the moderate growth condition at cusp, and we construct a basis as a generalization of Lagarias-Rhoades' works. As a corollary, we can obtain a preimage of an arbitrary polyharmonic weak Maass form under the xi-operator.
We prove a power saving over the local bound for the sup norm of Hecke-Maass forms on any quasi-split semisimple real group that is not isogenous to a product of odd special unitary groups.
We prove an asymptotic formula for the twisted first moment of Maass form symmetric square L-functions on the critical line and at the critical point. The error term is estimated uniformly with respect to all parameters.
In this paper, we introduce the notion of Maass-Jacobi forms and investigate some properties of these new automorphic forms. We also characterize these automorphic forms in several ways.
In this paper, we investigate traces of cycle integrals of certain meromorphic modular forms. By relating them to regularised theta lifts we provide explicit formulae for them in terms of coefficients of harmonic Maass forms.
The classical Maass lift is a map from holomorphic Jacobi forms to holomorphic scalar-valued Siegel modular forms. Automorphic representation theory predicts a non-holomorphic and vector-valued analogue for Hecke eigenforms. This paper is the first part of a series of papers. In this series of papers, we provide an explicit construction of the non-holomorphic Maass lift that is linear and also applies to non-eigenforms. In this first part, we develop new techniques to study Fourier series expansions of Siegel modular forms, which allow us to construct a Maass lift from harmonic Maass-Jacobi fo…
The Maass-form twisted Shintani $\mathscr{L}$-functions are introduced, and some of their analytic properties are studied. These functions contain data regarding the distribution of shapes of cubic rings.
We address the problem of identifying a Hecke-Maass cusp form $f$ of full level from the central values of the Rankin-Selberg $L$-functions $L(1/2,f\otimes h)$ where $h$ runs through the set of Hecke-Maass eigenforms of full level. We prove a quantitative result in this direction.
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.
Using that the overpartition rank function is the holomorphic part of a harmonic Maass form, we deduce formulas for the rank differences modulo 7. To do so we make improvements on the current state of the overpartition rank function in terms of harmonic Maass forms by giving simple formulas for the transformations under $\mbox{SL}_2(\mathbb{Z})$ as well as formulas for orders at cusps.
In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.
The primary goal of this paper is to construct the basis of the space of weight 3/2 mock modular forms which is an extension of the Borcherd-Zagier basis of weight 3/2 weakly holomorphic modular forms. The shadows of the members of this basis form the Borcherds- Zagier basis of the space of weight 1/2 weakly holomorphic modular forms. For the purpose, we use a weak Maass-Poincaré Series. The secondary goal is to provide a full computation of the Fourier coefficients for the weak Maass-Poincaré Series in most general form as a weak Maass-Poincaré Series has played a key role in the recent advan…
We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by Zagier on traces of singular moduli. We utilize a twisted version of the theta lift considered by Bruinier and Funke.
Let $f$ be a Hecke--Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $λ$. It is shown that $\pnorm{f}_\infty \ll_{λ,ε} N^{-1/6}+ε} \pnorm{f}_2$ for any $ε>0$.
Conditionally on the Generalized Lindelöf Hypothesis, we obtain an asymptotic for the fourth moment of Hecke Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the Random Wave Conjecture.
Some generalizations of the Maass relation for Siegel modular forms of higher degrees have been obtained by several authors. In the present article we first give a new generalization of the Maass relation for Siegel-Eisenstein series of arbitrary degrees. Furthermore, we show that the Duke-Imamoglu-Ibukiyama-Ikeda lifts satisfy this generalized Maass relation with some modifications. As an application of the generalized Maass relation in the present article we give a new proof of the Miyawaki-Ikeda lifts of two elliptic modular forms. Namely, we compute the standard L-function of the Miyawaki-…
We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$, and as an application obtain a pointwise estimate and a second moment estimate for the sums in question.
We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as algebraic constrictions allow. We show the analogous statement for the spaces of holomorphic forms. For functions on the universal covering group of PSL(2,R) we introduce the concept of generalized weight. For the resulting spaces of higher order Maass forms with even generalized weight we show that the size is maximal.
In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight 0 on the full modular group introduced by Lewis and Zagier.
We show that certain space of vector valued harmonic weak Maass forms of half integral weight is isomorphic to a space of scalar valued ones whose Fourier coefficients are supported on suitable progressions. This kind of result for holomorphic modular forms was proved by Eichler and Zagier.
We investigate a new family of locally harmonic Maass forms which correspond to periods of modular forms. They transform like negative weight modular forms and are harmonic apart from jump singularities along infinite geodesics. Our main result is an explicit splitting of the new locally harmonic Maass forms into a harmonic part and a locally polynomial part that captures the jump singularities. As an application, we obtain finite rational formulas for suitable linear combinations of periods of meromorphic modular forms associated to positive definite binary quadratic forms.
Suppose $π_1$ and $π_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $χ$ we have $$ L(\tfrac{1}{2},π_1\otimesχ)=L(\tfrac{1}{2},π_2\otimesχ). $$ Then we show that $π_1=π_2$.
We derive an algorithm to rigorously compute and verify Maass cusp forms of squarefree level and trivial character. The main tool we use is an explicit version of the Selberg trace formula with Hecke operators due to Strömbergsson. We use this algorithm to compute several thousand Maass forms for a range of levels and use this data to obtain numerical evidence towards various conjectures.
We present and compare several algorithms for evaluating Jacquet's Whittaker functions for SL(3, Z). The most suitable algorithm is then applied to the problem of evaluating a Maass form for SL(3, Z) with known eigenvalues and Fourier coefficients.
In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between inner products of meromorphic modular forms and higher Green's functions evaluated at CM-points.
We present a method for certifying the existence of an arbitrary Maass cusp form for any level and character. This is accomplished by producing a bound on the difference between the $Δ$-eigenvalue of an authentic Maass cusp form and a purported approximation of a $Δ$-eigenvalue, arrived at by any means. We apply this method to a proposed non-CM level 5 form with quadratic character, to present the first certified $Δ$-eigenvalue of such a form. This work generalises the method for certifying level 1 forms presented by Booker, Strömbergsson and Venkatesh, and is motivated by the production of pu…
We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construc…
We prove an equidistribution result for the Satake parameters of Maass cusp forms on $GL_3$ with respect to the $p$-adic Plancherel measure by using an application of the Kuznetsov trace formula. The techniques developed in this paper deal with the removal of arithmetic weight $L(1,F,Ad)^{-1}$ in the Kuznetsov trace formula on $GL_3$.
In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decompositions of Ramanujan-type congruences for overpartitions as well as the modularity of rank differences in certain arithmetic progressions.
We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.
Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, $H$-groups by employing the theory of supplementary functions introduced and developed by M. I. Knopp and S. Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain p…
In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function $p(n)$, they prove that \[p(n)=\frac{1}{24n-1} \sum P(α_Q),\] where $P$ is a weak Maass form and $α_Q$ ranges over a finite set of discriminant $-24n+1$ CM points. Moreover, they show that $6 (24n-1) P(α_Q)$ is always an algebraic integer, and they conjecture that $(24n-1) P(α_Q)$ is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.
We formulate a conjectured orthogonality relation between the Fourier coefficients of Maass forms on PGL(N) for N>=2. Based on the work of Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N>=4, we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the Satake parameter of Maass forms at a finite prime. For N=3, the rate of convergence for the equidistribution theorem is obtained.
Let $f$ be a normalized Hecke-Maass cusp form of weight zero for the group $SL_2(\mathbb Z)$. This article presents several quantitative results about the distribution of Hecke eigenvalues of $f$. Applications to the $Ω_{\pm}$-results for the Hecke eigenvalues of $f$ and its symmetric square sym$^2(f)$ are also given.
Let $L$ be a degree-$2$ $L$-function associated to a Maass cusp form. We explore an algorithm that evaluates $t$ values of $L$ on the critical line in time $O(t^{1+\varepsilon})$. We use this algorithm to rigorously compute an abundance of consecutive zeros and investigate their distribution.
