Eisenstein series on quaternion half-space

Kaufhold [5] calculated the Fourier coefficients of the Siegel's Eisenstein series of degree 2 and obtained its analytic continuation and functional equation. In this paper, we follow his procedure to obtain the analytic continuation and a functional equation of the Eisenstein series on quaternion half-space defined by Krieg [7]. S. Nagaoka has announced a similar result (see below). The author wishes to express his gratitude to Prof. Walter Baily Jr., for suggesting the topic of this research and for his encourgement and advice and to Dr. A. Krieg and the referee for helpful suggestions and corrections.     

Über die Fourierkoeffizienten der Siegelschen Eisensteinreihen

It is well known that the Fourier coefficients a _n ^k (T) of Siegel's Eisenstein series of degree n and weight k are rational numbers with bounded denominators [14], [15]. In this paper we introduce a number b _n ^k (T), which is equal to a _n ^k (T) in many cases. This number can be computed in an elementary way. From our explicit formulas for b _n ^k (T) we can easily get results about the denominators of the Fourier coefficients a _n ^k (T), which are better than those obtained by Siegel in his difficult paper [15].     

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.     

p-adic Siegel–Eisenstein series of degree two

We prove an explicit formula for Fourier coefficients of Siegel–Eisenstein series of degree two with a primitive character of any conductor. Moreover, we prove that there exists the p-adic analytic family which consists of Siegel–Eisenstein series of degree two and a certain p-adic limit of Siegel–Eisenstein series of degree two is actually a Siegel–Eisenstein series of degree two.     

Poles of Eisenstein series on quaternion groups

We show that the normalized Siegel Eisenstein series of quaternion groups have at most simple poles at certain integers and half integers. These Eisenstein series play an important role of Rankin-Selberg integral representations of Langlands L-functions for quaternion groups.     

Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

We find nice representatives for the 0-dimensional cusps of the degree n Siegel upper half-space under the action of \(\Gamma _0(\mathcal N )\). To each of these, we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level \(\mathcal N \) and Dirichlet character \(\chi _{_L}\) modulo \(\mathcal N \) as a linear combination of Siegel Eisenstein series.     

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)     

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 integrality of Eisenstein liftings

Letf be a holomorphic Siegel modular form of integral weightk for Sp_2r (Z). Forn≥r, let[f] _r ⁿ be the lift off to Sp_2n (Z) via the Klingen type Eisenstein series, which is defined under some conditions onk. We study an integrality property of the Fourier coefficients of[f] _r ⁿ . A common denominator for them is described in terms of a critical value of the standardL-function attached tof, some Bernoulli numbers, and a certain ideal depending only onf. The result specialized to the caser=0 coincides with the Siegel-Böcherer theorem on the Siegel type Eisenstein series.     

On the Cauchy–Szegö Kernel for Quaternion Siegel Upper Half-Space

The work is dedicated to the construction of the Cauchy–Szegö kernel for the Cauchy–Szegö projection integral operator from the space of $L^2$ -integrable functions defined on the boundary of the quaternion Siegel upper half-space to the space of boundary values of the quaternion regular functions of the Hardy space over the quaternion Siegel upper half-space.     

Jacobi–Hecke algebras and a rationality theorem for a formal Hecke series

It is well known that the L-function associated to a Siegel eigenform f is equal to a Rankin-Selberg type zeta-integral involving f and a restricted Eisenstein series ([3], [14]). At some point in the proof one has to show the equality of a certain Dirichlet series and the L-function, which follows from a rationality theorem for a certain formal power series over the Hecke algebra. The main purpose of this paper is to develop a Hecke theory for the Jacobi group and to prove such a rationality theorem. Received: 17 August 1998 / Revised version: 17 February 1999     

L-functions for Jacobi forms and the basis problem

This paper deals with Jacobi forms Φ on ?×ℂ. The Rankin–Selberg doubling method is employed to study properties of the standard L-function of Hecke–Jacobi eigenforms. It is shown that every analytic Klingen–Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke–Jacobi cusp eigenforms of weight k > 4 and k≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved. Received: 12 March 2000 / Revised version: 17 September 2001     

Le prolongement analytique de la solution du problème de Cauchy pour un certain opérateur différentiel à coefficients polynomiaux III

In part II of this series [To appear in J. Fixed Point Theory Appl. 14], we have studied analytic continuation and global structure of singularities of the solution of the Cauchy problem with meromorphic data for certain differential operators with principal part of polynomial coefficients in the complex domain. On the other hand, by a classical result of asymptotic development, we obtain a local expression of this solution. In this paper, we study an analytic continuation of this local solution. This gives a complement of part II.     

On differential modular forms and some analytic relations between Eisenstein series

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the moduli of polarized Hodge structures in the case h ¹⁰=h ⁰¹=1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.      

On the Siegel-Eisenstein measure and its applications

An Eisenstein measure on the symplectic group over rational number field is constructed which interpolatesp-adically the Fourier expansion of Siegel-Eisenstein series. The proof is based on explicit computation of the Fourier expansions by Siegel, Shimura and Feit. As an application of this result ap-adic family of Siegel modular forms is given which interpolates Klingen-Eisenstein series of degree two using Boecherer’s integral representation for the Klingen-Eisenstein series in terms of the Siegel-Eisenstein series.     

OnL-functions and intertwining operators for unitary groups

The ingredients of an “L-function machine” for the quasi-split groupU _{n, n} +1 × Res GL _n are treated here, following similar theories of P. Shapiro and S. Gelbart. We start with a known Rankin-Selberg type integral having an Euler product. In section 2 we compute the local integral to get a localL function. This is done by working with an “L group” related to ^L G and the relative root system. All computations are carried out for the split and the non-split case. In section 3 we address the problem of analytic continuation of the Eisenstein series. This involves computation of poles of intertwining operators.     

The Rankin-Selberg convolution for real analytic Cohen's Eisenstein series of half-integral weight

We give a meromorphic continuation and a functional equationfor the Rankin–Selberg convolution of certain real analyticEisenstein series of half-integral weight. Our result and methodhave several applications to the Koecher–Maass seriesassociated with the real analytic Siegel–Eisenstein series.     

On the diophantine equation x 2 + p 2k = y n

Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23]. Received: 25 April 2008