Jacobi Maaß forms

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J _k,m ^nh ) of weight k∈? and index m∈? as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\). We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J _k,m ^nh . We construct new examples of cuspidal Jacobi Maaß forms F _f of weight k∈2? and index 1 from weight k?1/2 Maaß forms f with respect to Γ₀(4) and show that the map f ? F _f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J _k,m ^nh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.     

Koecher-maaß-series for modular forms of quaternions

In this paper we demonstrate the meromorphic continuation and functional equation for Koecher-Maaß-series attached to modular forms of quaternions. The Koecher-Maaß-series associated with modular forms in the Maaß space, which are simultaneous eigenforms under all Hecke operators, possess Euler product expansions.     

Ramanujan type congruences for modular forms of several variables

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.     

Siegel modular forms of degree 2 over rings

Based on moduli theory of abelian varieties, extending Igusa's result on Siegel modular forms over C, we describe the ring of Siegel full modular forms of degree 2 over any Z-algebra in which 6 is invertible.     

Galois representations for holomorphic Siegel modular forms

We prove local–global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert–Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local–global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case.     

Cuspidality and the growth of Fourier coefficients: small weights

We characterize Siegel cusp forms in the space of Siegel modular forms of small weight \(k \ge n+4\) on the congruence subgroups \(\Gamma ^n_0(N)\) of any degree n and any level N, by a suitable growth of their Fourier coefficients (e.g., by the well known Hecke bound) at any one of the cusps. For this, we use the formalism of Jacobi forms and the ‘Witt-operator’ on modular forms.     

The dimension of the space of Siegel–Eisenstein series of weight one

In general, it is difficult to determine the dimension of the space of Siegel modular forms of low weights. In particular, the dimensions of the spaces of cusp forms are known in only a few cases. In this paper, we calculate the dimension of the space of Siegel–Eisenstein series of weight 1, which is a certain subspace of a complement of the space of cusp forms.      

Garrett's pullback formula for vector valued Siegel modular forms

Let be the Siegel Eisenstein series of degree n and weight k. Garrett showed a formula of on Hp×Hq, where Hn is the Siegel upper half space of degree n. This formula was useful for investigating the Fourier coefficients of the Klingen Eisenstein series and the algebraicity of the space of Siegel modular forms and of special values of the standard L-functions. We would like to generalize this formula in the case of vector valued Siegel modular forms. In this paper, using a differential operator D by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of and investigate the Fourier coefficients of the Klingen Eisenstein series.     

On p-adic quaternionic Eisenstein series

We show that certain p-adic Eisenstein series for quaternionic modular groups of degree 2 become “real” modular forms of level p in almost all cases. To prove this, we introduce a U(p) type operator. We also show that there exists a p-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of p-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).     

On zeros of quasi-modular forms

Several authors have studied the nature and location of zeros of modular forms for the full modular group Γ and other congruence subgroups. In this paper, we investigate the zeros of certain quasi-modular forms for Γ. In particular, we study the transcendence and existence of infinitely many Γ-inequivalent zeros of these quasi-modular forms. We also estimate the number of such zeros in Siegel sets, motivated by a recent work of Ghosh and Sarnak.     

On the proof of the reciprocity law for arithmetic Siegel modular functions

Earlier we obtained a new proof of Shimura’s reciprocity law for the special values of arithmetic Hilbert modular functions. In this note we show how from this result one may derive Shimura’s reciprocity law for special values of arithmetic Siegel modular functions. To achieve this we use Shimura’s classification of the special points of the Siegel space, Satake’s classification of the equivariant holomorphic imbeddings of Hilbert-Siegel modular spaces into a larger Siegel space, and, finally, a corrected version of some of Karel’s results giving an action of the Galois group Gal(Q_ab/Q) on arithmetic Siegel modular forms. Research supported in part by the NSF Grant No. DMS-8601130.     

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.     

Pullbacks of Siegel Eisenstein series and weighted averages of critical L-values

In this paper we obtain a weighted average formula for special values of L-functions attached to normalized elliptic modular forms of weight k and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition.     

Maass relations for Saito–Kurokawa lifts of higher levels

The Ramanujan Journal - 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...     

On the Ramanujan-Petersson conjecture for modular forms of half-integral weight

It is shown that a “character twist” of a growth estimate for certain weighted infinite sums of Kloosterman sums which is equivalent to the Ramanujan-Petersson conjecture for modular forms of half-integral weight, can easily be proved using Deligne’s theorem (previously the Ramanujan-Petersson conjecture for modular forms of integral weight).     

Mixed Siegel Modular Forms and Special Valuesof Certain Dirichlet Series

 Let be a Siegel modular form of weight ?, and let be an Eichler embedding, where denotes the Siegel upper half space of degree n. We use the notion of mixed Siegel modular forms to construct the linear map of the spaces of Siegel cusp forms for the congruence subgroup and express the Fourier coefficients of the image of an element under in terms of special values of a certain Dirichlet series. We also discuss connections between mixed Siegel cusp forms and holomorphic forms on a family of abelian varieties. (Received 28 February 2000; in revised form 11 July 2000)     

Estimates of Fourier Coefficients of Siegel Cusp Forms for Subgroups and in the Case of Small Weight

Here we develop estimates for Fourier coefficients of Siegelcusp forms. First we consider the case of Siegel modular formsfor the full modular group     

On Siegel modular forms of level p and their properties mod p

Using theta series we construct Siegel modular forms of level p which behave well modulo p in all cusps. This construction allows us to show (under a mild condition) that all Siegel modular forms of level p and weight 2 are congruent mod p to level one modular forms of weight p + 1; in particular, this is true for Yoshida lifts of level p.     

Explicit formulas for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift and their applications

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 (J Math Soc Japan 33:649–672, 1981).