In this article we want to analyze the structure of a 4 dimensional projective variety X which has a smooth ample divisor A that is a P1 bundle π : A→S over a smooth surface S.

In [Fa+So], as a consequence of a more general result, the first and third authors determined the structure of X in the case the base S of the P1 bundle A has a cover with h2,0()≠0. Here we look at the remaining cases except for those surfaces which are the projectivization of a stable rank two vector bundle over a curve (the result is obviously true for S rational).

In the study of elliptic differential operators of second order, we found that the automorphism pseudogroups are of finite type ([4]). However this fact takes a complete change in elliptic differential operators of first order.

So as to make the objects which can be dealt with clear, we introduce the concept of pseudoellipticity of first order differential operators (Definition 1.1), which is naturally satisfied by first order elliptic differential operators.

In this paper, we shall give a new characterization of hyperfunctions without algebraic method and apply to give simpler proofs to problems discussed in [3], Chapter 9. In [3], the spaces of hyperfunctions A′(K) with compact support in K ⊂ Rn (n ≧ 1) is considered as the dual of the space A(K) of functions which are real analytic near K. Each element u of A′(K) is characterized as a density of a double layer potential in Rn × R.

The class of normal functionals

is, as is well known, adapted to the domain of Lévy’s Laplacian and plays important roles in the works by P. Lévy and T. Hida (cf. [1], [2] and [8]), where Bx denotes one-dimensional parameter white noise and denotes the renormalization of

Let Sm(r) be an m-sphere of constant sectional curvature 1/r2 and M an n-dimensional compact minimal submanifold of Sm(r). If Sm(r) is imbedded in Em+1 by its first standard imbedding, then, by a well-known result of Takahashi [11], the Euclidean coordinate functions restricted to M are eigenfunctions of Δ on M with the same eigenvalue n/r2. Moreover, the center of mass of M in Em+1 coincides with the center of the hypersphere Sm(r) in Em+1. Thus, M is mass-symmetric in Sm(r) ⊂ Em+\ Consequently, we see that if one wants to study the spectral geometry of a submanifold of Sm(r), it is natural to immerse Sm(r) by its k-th standard immersion, in particular, by its second standard immersion.

In this paper we shall show a relation between a special value of an automorphic wave form and the inner product of the automorphic wave form with the pullback of an Eisenstein series on the upper half space. The main theorem is Theorem 3 in the end of this paper. As is shown in P. B. Garrett [13], pullbacks of Eisenstein series on Siegel upper half spaces have interesting properties as a kernel function of an integral operator. It is natural to try to investigate pullbacks of Eisenstein series of Hilbert type. We can say that Theorem 3 clarifies a property of such pullbacks in a special case. The idea of the proof is a lifting of automorphic forms by theta functions. We discuss a lifting of automorphic wave forms in 1, 2 and 3, and obtain Theorem 2 in the end of 3 as a result. We can prove Theorem 3 without much difficulty by using Theorem 2.

The Lévy Brownian motion with multidimensional parameter was introduced and discussed in his book [1] and it is known as the most important random field. Many approaches have been made to the investigation of the Lévy Brownian motion by H.P. McKean [7], Yu. A. Rozanov and others, by using various techniques.

Ankeny, Artin and Chowla [1] showed that there are congruences between class numbers of real quadratic fields and generalized Bernoulli numbers. Recently, Ito [3] has extended their results to the case of pure cubic fields using generalized Hurwitz numbers of Lichtenbaum [4]. In his paper, he suggested that similar results would be obtained for pure quartic and sectic fields. In this paper, we carry out this by following his idea. To give a congruence in an exact form, we need an idea due to Matthews [5]. As the argument in the sectic case is quite parallel to that in the quartic case, we shall discuss the former case briefly in the last two sections.

Soient X un espace localement compact et non-compact à base dénombrable, G une noyau-fonction continue sur X et M (resp. MK) l’ensemble des toutes mesures de Radon positives sur X (resp. des toutes mesures de Radon positives sur X à support compact).