We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a harmonic lift of a given cusp form.
In this paper, we discuss the representation-theoretical Miyawaki lift for unitary groups in terms of the endoscopic classification. We give an explicit determination of Miyawaki lifts for U(1) and U(3) with respect to Hermitian Maass lifts, namely Ikeda lifts for U(4).
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.
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 consider pullbacks of hermitian Maass lifts of degree 2 to the diagonal matrices. By using the pullbacks, we give an explicit formura for central values of L-functions for GL(2)*GL(2).
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-…
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…
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.
We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \Z, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $ξ$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $ξ$-preimages of Zagier's weight $1/2$ generating…
We give an explicit formula for the twisted Koecher-Maass series of the Duke-Imamoglu-Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin-Selberg series at integers of half-integral weight modular forms, which was not treated by Shimura.
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…
Muto, Narita and Pitale construct counterexamples to the Generalized Ramanujan Conjecture for GL(2,B) over the division quaternion algebra B with discriminant two via a lift from SL(2). In this paper, we try to exactly characterize the image of this lift. The previous methods of Maass, Kohnen or Kojima do not apply here, hence we approach this problem via a combination of classical and representation theory techniques to identify the image. Crucially, we use the Jacquet Langlands correspondence described by Badulescu and Renard to characterize the representations.
We extend Borcherds' singular theta lift in signature $(1,2)$ to harmonic Maass forms of weight $1/2$ whose non-holomorphic part is allowed to be of exponential growth at $i\infty$. We determine the singularities of the lift and compute its Fourier expansion. It turns out that the lift is continuous but not differentiable along certain geodesics in the upper half-plane corresponding to the non-holomorphic principal part of the input. As an application, we obtain a generalization to higher level of the weight $2$ modular integral of Duke, Imamoglu and Tóth. Further, we construct automorphic pro…
For $K$ an imaginary quadratic field with discriminant $-D_K$ and associated quadratic Galois character $χ_K$, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level $D_K$ and nebentypus $χ_K$ via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group $U(2,2)$ split over $K$. We generalize this (under certain conditions on $K$ and $p$) to the case of $p$-oldforms of level $pD_K$ and character $χ_K$. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space o…
In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the $L$-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly hol…
In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important role in the construction by Kohnen and Zagier of a kernel function for the Shimura and Shintani lifts between half-integral and integral weight cusp forms. Although our functions share many properties in common with harmonic weak Maass forms, they also have some properties which strikingly contrast those exhibited by harmonic weak Maass forms. As a first application of the new theory developed in …
In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.
We investigate so-called "higher" Siegel theta lifts on Lorentzian lattices in the spirit of Bruinier-Ehlen-Yang and Bruinier-Schwagenscheidt. We give a series representation of the lift in terms of Gauss hypergeometric functions, and evaluate the lift as the constant term of a Fourier series involving the Rankin-Cohen bracket of harmonic Maass forms and theta functions. Using the higher Siegel lifts, we obtain a vector-valued analogue of Mertens' result stating that the Rankin--Cohen bracket of the holomorphic part of a harmonic Maass form of weight $\frac{3}{2}$ and a unary theta function, p…
For an imaginary quadratic field $K$ of discriminant $-D$, let $χ= χ_K$ be the associated quadratic character. We will show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $χ$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space of elliptic modular forms t…
We obtain resonances for short exponential sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(n,\mathbb Z)$. This involves deriving asymptotics for the integrals appearing in the $\mathrm{GL}(n)$ Voronoi summation formula. As an application, we also prove an $Ω$-result for short sums of Fourier coefficients.
Let K be an imaginary quadratic field with discriminant -D, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and level D with Neben character x, or a primitive form of weight 2k for SL(2,Z) according as m=2n, or m=2n+1. For such an f let I(m,f) be the lift of f to the space of Hermitian modular forms of degree m constructed by Ikeda. We then give an explicit formula of the Koecher-Maass series L(s,I(m,f)) of I(m,f). This is a generalization of [Y. Mizuno, Manuscripta Math. 119(2006), 175-216].
Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k-n/2+1/2, let f be the corresponding primitive form of weight 2k-n for SL(2,Z) under the Shimura correspondence, and I(h) the Duke-Imamoglu-Ikeda lift of h to the space of cusp forms of weight k of genus n. Moreover, let FJ(I(h),1) be the first Fourier-Jacobi coefficient of I(h) and s(FJ(I(h),1)) be the cusp form in the generalized Kohnen plus space of weight k-1/2 corresponding to FJ(I(h),1) under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series o…
It is known that among Siegel modular forms of degree $2$ and level $1$ the only functions that violate the Ramanujan conjecture are Saito-Kurokawa lifts of modular forms of level $1$. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for $\mathrm{GSp}_4$ is predicted to fail only in case of CAP representations. It is not known though whether the associated Siegel modular forms (of various levels) still satisfy a version of Maass relations. We show that this is indeed the case for the ones related to P-CAP representat…
Duke and the second author defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half integral weight and level 4 and showed that these lifts preserve the integrality of Fourier coefficients. We show that the generalization of these lifts to modular forms of genus 0 odd prime level also preserves the integrality of Fourier coefficients.
In this paper, we define certain symmetries for automorphic forms on O(2, m+2) and show that the space of automorphic forms satisfying these symmetries coincides with the Maass space, the image of Saito-Kurokawa lifting.
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.
In this paper the image of the Saito-Kurokawa lift of level $N$ with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level $N$ by symmetries involving Hecke operators related to $Γ_0(N)$. We finally obtain for all prime numbers $p$ local Maass relations. This generalizes known results for level $N=1$.
We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in term…
We study the sum of additively twisted Fourier coefficients of a symmetric-square lift of a Maass form invariant under the full modular group. Our bounds are uniform in terms of the spectral parameter of the Maass form, as well as in terms of the additive twist.
For an arbitrary even genus $2n$ we show that the subspace of Siegel cusp forms of degree $2n$ generated by Ikeda lifts of elliptic cusp forms can be characterized by certain linear relations among Fourier coefficients. This generatizes the work of Kohnen and Kojima.
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.
We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maas…
In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke--Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the twisted first and second moments of the central values $L\big(\frac 1 2, f\big)$, which can be applied to prove that at least $33 \%$ of $L\big(\frac 1 2, f\big)$ with $t_f \leqslant T$ are non-vanishing as $T \rightarrow \infty$. Our main tools are the spherical Kuznetsov trace formula and the Voronoï summation formula over imaginary quadratic fields.
Recently, Bruinier and Ono proved that the coefficients of certain weight -1/2 harmonic weak Maaß forms are given as "traces" of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight $3/2+k$, $k$ even, and weight $1/2-k$, $k$ odd, by extending the theta lift of Bruinier-Funke and Bruinier-Ono. Moreover, we generalize their result to include \textit{twisted} traces of singular moduli using earlier work of the author and Ehlen. Employing a duality result between weight $k$ and $2-k$, we are able to cover …
Let $f$ be a Hecke-Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $λ_f(Δ)=1/4+μ^2$ and let $λ_f(n)$ be its $n$-th normalized Fourier coefficient. It is proved that, uniformly in $α, β\in \mathbb{R}$, $$ \sum_{n \leq X}λ_f(n)e\left(αn^2+βn\right) \ll X^{7/8+\varepsilon}λ_f(Δ)^{1/2+\varepsilon}, $$ where the implied constant depends only on $\varepsilon$. We also consider the summation function of $λ_f(n)$ and under the Ramanujan conjecture we are able to prove $$ \sum_{n \leq X}λ_f(n)\ll X^{1/3+\varepsilon}λ_f(Δ)^{4/9+\varepsilon} $$ with the implied constant depending only …
We study a certain theta lift which maps weight $-2k$ to weight $1/2-k$ harmonic weak Maass forms for $k \in \mathbb{Z}, k \geq 0$, and which is closely related to the classical Shintani lift from weight $2k+2$ to weight $k+3/2$ cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we obtain a criterion for the non-vanishing of the central $L$-value of an integral weight newform $G$ in terms of the holomorphicity of the theta lift of a certain harmonic weak Maas…
We present some applications of the Kudla-Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight $3/2$ and $1/2$, respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions $f(q)$ and $ω(q)$ in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficien…
In this paper, we use regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and Imamoglu on cycle integrals of the modular j-invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, we give the `central value' of the (non-existing) `L-function' for j. The key to the proofs is the construction of …
Let $f$ be a primitive Maass cusp form for a congruence subgroup $Γ_0(D) \subset $ SL($2,\mathbb{Z}$) and $λ_f(n)$ its $n$-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many $λ_f(n)$ one can often solve for the level $D$, and in some cases, estimate the Laplace eigenvalue to arbitrarily high precision. This is done by analyzing the resonance and rapid decay of smoothly weighted sums of $λ_f(n)e(αn^β)$ for $X \leq n \leq 2X$ and any choice of $α\in \mathbb{R}$, and $β>0$. The methods include the Voronoi summation formula, asymptotic expansions of Bessel …
Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan's mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automorphy method due to Hejhal and Stark. As explicit examples we consider harmonic weak Maass forms of weight 1/2 associated to the elliptic curves 11a1, 37a1, 37b1. We made extensive numerical computations …
We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vector-valued harmonic weak Maass forms on Mp_2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie…
Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a smoothly weighted average of $A_f(n,1,...,1)$ against $e(α|n|^β)$ is proved to be rapidly decayed when $0<β<1/m$. When $β=1/m$ and $α$ equals or approaches $\pm mq^{1/m}$ for a positive integer $q$, this smooth average has a main term of the size of $|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$, which is a manifestation of resonance of oscillation exhibited by …
While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generat…
Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ with square free level. Although our approaches differ, where the two theories…
We present completions of mock theta functions to harmonic weak Maass forms of weight $1/2$ and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight $1/2$ that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.
We describe the construction and properties of a singular theta lift for the orthogonal group $\SO(2,1)$. We obtain locally harmonic Maass forms in the sense of Bringmann-Kane-Kohnen with singular sets along geodesics in the upper half plane. We consider these forms as currents and derive properties of the Shimura-Shintani correspondence. This work provides extensions of the theta lifts considered by Hoevel and Bringann-Kane-Viazovska.
The quaternionic Maass Spezialschar on split $\mathrm{SO}(8)$
University of Idaho（爱达荷大学）；The University of Oklahoma（俄克拉荷马大学）；University of Toronto（多伦多大学）；The University of California, San Diego（加州大学圣地亚哥分校）；Lafayette College（拉斐特学院）
Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular $L$-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivativ…
