Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 ? k. The operator ξ_2-k (resp. D ^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ_2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D ^k-1.     

Harmonic Maass forms and periods

According to Waldspurger’s theorem, the coefficients of half-integral weight eigenforms are given by central critical values of twisted Hecke $L$ -functions, and therefore by periods. Here we prove that the coefficients of the holomorphic parts of weight $1/2$ harmonic Maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves.     

Shimura lifts of half-integral weight modular forms arising from theta functions

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.      

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have “essentially the same” eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).     

Hecke actions on certain strongly modular genera of lattices

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.Received: 10 September 2004     

Eichler integrals and harmonic weak Maass forms

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 period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.     

Combinatorial formulas for certain sequences of multiple numbers

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.     

Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.     

On Shimura, Shintani and Eichler-Zagier correspondences

In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.

     

On shifted convolution of half-integral weight cusp forms

In this note, we present a simple approach for bounding the shifted convolution sum involving the Fourier coefficients of half-integral weight holomorphic cusp forms and Maass cusp forms.     

Linear relations among Poincaré series via harmonic weak Maass forms

We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan??s mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series.     

Nonvanishing of Koecher–Maass series attached to Siegel cusp forms

We prove a nonvanishing result for Koecher–Maass series attached to Siegel cusp forms of weight k and degree n   in certain strips on the complex plane. When n=2$n=2$, we prove such a result for forms orthogonal to the space of the Saito–Kurokawa lifts ‘up to finitely many exceptions’, in bounded regions.     

Hecke operators for weakly holomorphic modular forms and supersingular congruences

We consider the action of Hecke operators on weakly holomorphic modular forms and a Hecke-equivariant duality between the spaces of holomorphic and weakly holomorphic cusp forms. As an application, we obtain congruences modulo supersingular primes, which connect Hecke eigenvalues and certain singular moduli.

     

Some computational results on Hecke eigenvalues of modular forms on a unitary group

In this paper we present some computational results on Hecke eigenforms and eigenvalues for a unitary group in three variables. Our results are based on the work of Shiga [SHig], Holzapfel [Holz1,Holz2] and Feustel ]Feustel] which gives in a special case a generating system for the ring of (holomorphic) modular forms consisting of powers of theta constants. We compute all Hecke eigenforms in this ring for weights up to 12 and for each eigenform the first Hecke eigenvalues. Received: 25 July 1997 / Revised version: 7 January 1998     

The asymptotic distribution of traces of Maass–Poincaré series

We establish an asymptotic formula with a power savings in the error term for traces of CM values of a family of Maass–Poincaré series which contains the modular j-function as a special case. By work of Borcherds (1998) [2], Zagier (2002) [31], and Bringmann and Ono (2007) [4], these traces are Fourier coefficients of half-integral weight weakly holomorphic modular forms and Maass forms.     

Second moments of twisted Koecher-Maass series

Imai considered the twisted Koecher-Maass series for Siegel cusp forms of degree?2, twisted by Maass cusp forms and Eisenstein series, and used them to prove the converse theorem for Siegel modular forms. They do not have Euler products, and it is not even known whether they converge absolutely for Re(s)>1. Hence the standard convexity arguments do not apply to give bounds. In this paper, we obtain the average version of the second moments of the twisted Koecher-Maass series, using Titchmarsh??s method of Mellin inversion. When the Siegel modular form is a Saito Kurokawa lift of some half integral weight modular form, a theorem of Duke and Imamoglu says that the twisted Koecher Maass series is the Rankin-Selberg L-function of the half-integral weight form and Maass form of weight?1/2. Hence as a corollary, we obtain the average version of the second moment result for the Rankin-Selberg L-functions attached to half integral weight forms.     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen’s Γ₀(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.