On explicit construction of Hilbert-Siegel modular forms of degree two

Introduction. Several authors have developed the theory of lifting from the space of modular forms of one variable to that of modular forms on the orthogonal groups attached to quadratic forms over Q (cf. [1, 4–6, 8]). Shimura [9], [10] dealt with the problem of construction of arithmetic modular forms on orthogonal groups over totally real algebraic number fields. However, he did not take up the explicit calculation of the Fourier coefficients of lifted modular forms. On the other hand, in [3], [4] we have established a correspondence Ψ k between the space S(2k−1)/2(M,χ) of modular cusp forms of half integral weight (2k − 1)/2 of level M to the space M (2) k (M,χ) of Maass forms of Siegel modular cusp forms of degree two of weight k of level M in such a way that it commutes with the actions of Hecke operators. We evaluated explicitly the Fourier coefficients of Ψ k (f) with a form f in S(2k−1)/2(M,χ), and made clear a coincidence with Shimura’s zeta functions attached to f and Andrianov’s zeta functions attached to Ψ k (f). We note that these results are closely related to Saito–Kurokawa’s conjecture concerning Siegel modular forms of degree two. Using the technique in the theory of group representation of Jacquet and Langlands, PiatetskiShapiro [7] discussed Saito–Kurokawa’s conjecture in the case of Siegel modular forms on GpSp(2, AF ) where AF is the adele ring of an arbitrary number field F . Unfortunately, it seems that his approach is difficult to use for an explicit calculation of the Fourier coefficients of the lifted forms. The first purpose of the present note is to show the existence of a correspondence ΨN ′ between Hilbert modular forms f of half integral weight with respect to the principal congruence group and Hilbert–Siegel modular forms ΨN ′(f) of degree two attached to totally real number fields. The second one