Siegel modular forms of degree two

Introduction.In this note, we discuss correspondence between the space of modular orms of half integral weight and the space of Siegel modular orms of degree two, and its application to Maass spaces, in close relation with Saito-Kurokawa's conjecture (cf.[2], [3], [4], [8]).Let M be any positive integer, Z a character mod M, M=l.c.m (4, M), and k an even integer.In our previous paper [3], we constructed a linear mapping ff, of ._(M,Z) into S(P)(M), ).In this note, we construct another linear mapping Y o (R)_(4N, ) into S(F)(2N), ;), k being an even integer and ; a character mod 2N.It will be seen that : is more useful than .in several points and serves to gener- alize our results in [3].For example, Theorem 4 in [3] is generalized in the sense that the assumption (5.1) in [3] can be dropped.1. We denote by Z, R and C the ring of rational integers, the field of real numbers and the field of complex numbers.For a ring