Yoshinori Mizuno

ON CHARACTERIZATION OF SIEGEL CUSP FORMS OF DEGREE 2 BY THE HECKE BOUND

We give a new proof of a recent result of Kohnen–Martin on a characterization of degree 2 Siegel cusp forms by the growth of their Fourier coefficients. Our main tools are Koecher–Maass series, Imai’s converse theorem and the theory of singular modular forms.

Regularized theta lift and formulas of Katok–Sarnak type

Abstract. We study theta lifts for . The theta-lift is realized via an integral transform with a Siegel theta series as kernel function. Since this Siegel theta series fails to be square integrable, it has to be regularized. The regularization is obtained by applying a suitable differential operator built from the Laplacian. For the regularized theta series we compute the theta lift for cusp forms. The regularized lift also gives a correspondence for non-cusp forms such as Eisenstein series. Also we obtain the spectral expansion of the theta series in either of its variables. As an application we prove a three dimensional analogue of Katok–Sarnaks correspondence using the Selberg transform.

A p-ADIC LIMIT OF SIEGEL–EISENSTEIN SERIES OF PRIME LEVEL q

We show that a p-adic limit of a Siegel–Eisenstein series of prime level q becomes a Siegel modular form of level pq. This paper contains a simple formula for Fourier coefficients of a Siegel–Eisenstein series of degree two and prime levels.

The rankin-seiberg convolution for cohen’s eisenstein series of half integral weight

In this paper, we give a meromorphic continuation and a functional equation of the convolution product of certain Eisenstein series of half integral weight of one variable which were introduced by COHEN for a special case. We give also a precise description of the poles of the convolution product and their residues. These results have some applications to the Koecher-Maass series of Siegel-Eisenstein series. To prove our result, we use ZAGIER’S Rankin-Selberg method for automorphic functions which are not of rapid decay. In the final section, we present KUDLA’S version of ZAGIER’S Rankin-Selberg method. This is simpler than ZAGIER’S original one.