In this paper, we extend the Doi-Naganuma lifting as suggested by Kudla [4], on the lines of Zagier's work [6]. For each fundamental discriminant D associated with a real quadratic field, we prove that there exists a Hecke-equivarient map ι_D which maps the mth Poincare series of weight k, level M and character χ_D = (./D) into a Hilbert cusp form of weight k, level M/D associated with the real quadratic field of discriminant D of class number one. Through this, we get its adjoint ι^*_D with respect to the Petersson inner product.

Salvati Manni showed that the difference of the Siegel theta series of degree 4 associated with the two even unimodular 48-dimensional extremal lattices is a constant multiple of the cube J³ of the Schottky modular form J, which is a Siegel cusp form of degree 4 and weight 8. His result implies that the Siegel theta series of degree up to 3 is unique. But apparently his method does not supply us the process to compute the Fourier coefficients of these series.

In the present paper we show that the Fourier coefficients of the Siegel theta series associated with the even unimodular 48-dimensional extremal lattices of degrees 2 and 3 can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree 4 for those lattices are computed almost explicitly.

Let X be a continuum. Let C(X) be the hyperspace of all closed, connected and nonempty subsets of X, with the Hausdorff metric. For a mapping f : X → Y between continua, let C(f) : C(X) → C(Y) be the induced mapping by f, given by C(f)(A) = f(A). In this paper we study the hyperspace ℭ(X) = {C(A) : A ∈ C(X)} as a subspace of C(C(X)), and define an induced function ℭ(f) between ℭ(X) and ℭ(Y). We prove some relationships between the functions f, C(f) and ℭ(f) for the following classes of mapping: confluent, light, monotone and weakly confluent.