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)     

The extreme core

For a Siegel modular cusp formf of weightk letv(f) be the closure of the convex ray hull of the support of the Fourier series inside the cone of semidefinite forms. We show the existence of the extreme core,C _ext, which satisfiesv(f) ⊇k C_ext for all cusp forms. This is a generalization of the Valence Inequality to Siegel modular cusp forms. We give estimations of the extreme core for general n. For n ≤5 we use noble forms to improve these estimates. Forn = 2 we almost specify the extreme core but fall short. We supply improved estimates for all relevant constants and show optimality in some cases. The techniques are mainly from the geometry of numbers but we also use IGUSA’s generators for the ring of Siegel modular forms in degree two.     

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     

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.     

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.     

Congruences for Hecke eigenvalues of Siegel modular forms

We prove some congruences for Hecke eigenvalues of Klingen-Eisenstein series and those of cusp forms for Siegel modular groups modulo special values of automorphic L-functions.     

Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.     

Periods of mixed cusp forms

We define the periods of mixed cusp forms and establish generalized Eichler-Shimura relations for the periods of mixed cusp forms. We also construct modular symbols for mixed cusp forms and express the periods of mixed cusp forms in terms of these modular symbols.     

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.     

Fourier-Jacobi expansion and the Ikeda lift

In this article, we consider a Fourier-Jacobi expansion of Siegel modular forms generated by the Ikeda lift. There are two purposes of this article: first, to give an expression of L-function of certain Siegel modular forms of half-integral weight of odd degree; and secondly, to give a relation among Fourier-Jacobi coefficients of Siegel modular forms generated by the Ikeda lift.     

On the mean square of standard L-functions attached to Ikeda lifts

By using estimates on the frequency of large values of the Riemann zeta-function and modular L-functions attached to the full modular group SL(2, ℤ), we prove sharp upper and lower estimates of the mean square of standard L-functions attached to Siegel cusp forms which are Ikeda lifts, on boundaries and the central line of the critical strip. The mean square of spinor L-functions attached to Saito-Kurokawa lifts is also studied.     

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.      

On Saito-Kurokawa descent for congruence subgroups

The conjecture made by H. Saito and N. Kurokawa states the existence of a “lifting” from the space of elliptic modular forms of weight 2k?2 (for the full modular group) to the subspace of the space of Siegel modular forms of weightk (for the full Siegel modular group) which is compatible with the action of Hecke operators. (The subspace is the so called “Maaß spezialschar” defined by certain identities among Fourier coefficients). This conjecture was proved (in parts) by H. Maaß, A.N. Andrianov and D. Zagier. The purpose of this paper is to prove a generalised version of the conjecture for cusp forms of odd squarefree level.     

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.     

On Rankin-Cohen brackets for Siegel modular forms

We determine an explicit formula for a Rankin-Cohen bracket for Siegel modular forms of degree on a certain subgroup of the symplectic group. Moreover, we lift that bracket via a Poincaré series to a Siegel cusp form on the full symplectic group.

     

Maass relations in higher genus

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 generalizes the classical Maass relations in degree two to higher degrees.     

Modular forms for the $$A_{1}$$-tower

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct \(\chi _{5}\), the cusp form of lowest weight for the group \({\text {Sp}}(2,\mathbb {Z})\). In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice \(A_{1}\) and Igusa’s form \(\chi _{5}\) appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.     

Classical interpretation of the Ramanujan conjecture for Siegel cusp forms of genus <Emphasis Type="Italic">n</Emphasis>

We obtain a classical interpretation of the representation theoretic statement of the Generalized Ramanujan Conjecture for Siegel cusp forms of genus n in terms of estimates on Hecke eigenvalues.     

On convolutions of Siegel modular forms

In this article we study a Rankin‐Selberg convolution of n complex variables for pairs of degree n Siegel cusp forms. We establish its analytic continuation to ?ⁿ, determine its functional equations and find its singular curves. Also, we introduce and get similar results for a convolution of degree n Jacobi cusp forms. Furthermore, we show how the relation of a Siegel cusp form and its Fourier‐Jacobi coefficients is reflected in a particular relation connecting the two convolutions studied in this paper. As a consequence, the Dirichlet series introduced by Kalinin [7] and Yamazaki [19] are obtained as particular cases. As another application we generalize to any degree the estimate on the size of Fourier coefficients given in [14]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)