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)     

Congruences between Siegel modular forms

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.     

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.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

Restricting Hecke-Siegel operators to Jacobi modular forms

We compute the action of Hecke operators on Jacobi forms of “Siegel degree” n and m×m index M, provided 1?j?n−m. We find they are restrictions of Hecke operators on Siegel modular forms, and we compute their action on Fourier coefficients. Then we restrict the Hecke-Siegel operators T(p), Tj(p²) (n−m<j?n) to Jacobi forms of Siegel degree n, compute their action on Fourier coefficients and on indices, and produce lifts from Jacobi forms of index M to Jacobi forms of index M^′ where detM|detM^′. Finally, we present an explicit choice of matrices for the action of the Hecke operators on Siegel modular forms, and for their restrictions to Jacobi modular forms.     

Kohnen’s limit process for real-analytic Siegel modular forms

Kohnen introduced a limit process for Siegel modular forms that produces Jacobi forms. He asked if there is a space of real-analytic Siegel modular forms such that skew-holomorphic Jacobi forms arise via this limit process. In this paper, we initiate the study of harmonic skew-Maass–Jacobi forms and harmonic Siegel–Maass forms. We improve a result of Maass on the Fourier coefficients of harmonic Siegel–Maass forms, which allows us to establish a connection to harmonic skew-Maass–Jacobi forms. In particular, we answer Kohnen’s question in the affirmative.     

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.     

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 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.     

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.     

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.     

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 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.     

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     

Continuation of siegel modular forms to the Siegel-Satake semiplane

With the aid of the continuation of the Siegel modular forms to the Siegel-Satake half-plane one establishes the relation between the Siegel operators and Hecke operators for modular forms with respect to subgroups of finite index of the group Sp_n(Z). Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 99–109, 1987.     

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.     

On special L-values attached to metaplectic modular forms

Mathematische Zeitschrift - In this paper we establish some algebraic properties of special L-values attached to Siegel modular forms of half-integral weight, often called metaplectic modular...     

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.     

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.