Manuscripta Mathematica

manuscripta math. 81, 161 - 182 (1993) manuSCripta

mathematlca

© Spnnger-Verlag 1993

ON SAITO-KUROKAWA DESCENT FOR CONGRUENCE SUBGROUPS

M . Manickam, B. Ramakrishnan and T. С Vasudevan

The conjecture made by H. Saito and N. Kurokawa states the existence of a "lifting" from the space of elliptic modular forms of weight 2fc — 2 (for the full modular group) to the subspace of the .space of Siegel modular forms of weight к (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.

1 . Introduction

The conjecture, made on the basis of numerical calculations of eigenvalues of Hecke operators, forimilated independently by H. Saito and by N. Kurokawa [12], asserted the existence of a 'lifting" from the space of elliptic modular