Let X be a hyperbolic complex space in the sense of S. Kobayashi [2]. We write Aut(X)(resp. Bim (X)) for the group of all biholomorphic (resp. bimeromorphic) automorphisms of X.

Let Q be a bounded open set of Rn (n ≧ 2) with a C∞ boundary Γ and assume that be functions in such that where s1, …, sN are integers ≧ and are n-tuples of non-negative integers with

Soit (ℒ, M) un pseudogroupe de Lie infinitesimal [4] transitif d’ordre h sur une variété M différentiate, Jhℒ son équation de définition et L son algèbre formelle au point a de M.

Let k be an algebraic number field of finite degree and be a prime ideal of k, lying above a rational prime p. We denote by G () the multiplicative group of residue classes modulo (N ≧ 0) which are relatively prime to . The structure of G () is well-known, when N = 0, or k is the rational number field Q. If k is a quadratic number field, then the direct decomposition of G () is determined by A. Ranum [6] and F.H-Koch [4] who gives a basis of a group of principal units in the local quadratic number field according to H. Hasse [2]. In [5, Theorem 6.2], W. Narkiewicz obtains necessary and sufficient conditions so that G () is cyclic, in connection with a group of units in the -adic completion of k.

Let p be a prime. Let k be a -adic number field and be the ring of all integers of k. Let K/k be a cyclic totally ramified extension of degree pn with Galois group G. Clealy the ring of all integers of K is an [G]-module, and the purpose of this paper is to give a necessary and sufficient condition for the [G]-module to be indecomposable.

In this paper we consider nonlinear evolution equations of the form

(E) u″(t) + Au(t) + B(t)u′(t) = f(t), 0 ≦ t < ∞,

(u′(t) = d2u(t)/dt2, u′(t) = du(t)/dt), where A and B(t) are (possibly) nonlinear operators. Various examples of equations of type (E) arise in physics; for instance, if Au = –Δu and B(t)u′ = | u′ | u′, the equation represents a classical vibrating membrane with the resistance proportional to the velocity.

At first, we fix the notation. Let k = C and we shall work in the category of schemes over k. For an algebraic variety V of dimension n, we have the following numerical invariants:

• Pm(V) = the m-genus of V,

• q(V) = the irregularity of V,

• k(V) = the Kodaira dimension of V;

• = the logarithmic m-genus of V,

• = the logarithmic irregularity of V,

• = the logarithmic Kodaira dimension of V.

In 1938, C. L. Siegel studied zeta functions of indefinite quadratic forms ([6], c). On the other hand, M. Sato and T. Shintani constructed the general theory of zeta functions of one complex variable associated with prehomogeneous vector spaces in 1974 ([1]). Moreover T. Shintani studied several zeta functions of prehomogeneous vector spaces, especially, “Dirichlet series whose coefficients are class-numbers of integral binary cubic forms” ([3]) and “Zeta functions associated with the vector space of quadratic forms” ([2]).

Let L, M, N be positive definite quadratic lattices over Z. We treated the following problem in [5], [6]:

If L ⊗ M is isometric to L ⊗ N, then is M isometric to N?

We gave a condition (**) in [6] such that the answer is affirmative for an indecomposable lattice L satisfying (**), and we gave some examples. In this paper we introduce a certain apparently weaker condition (A) than the condition (**), and we show that the condition (A) implies the condition (**) and more on integral orthogonal groups than a result in [6].

In [1], Maass proved that the Dirichlet series associated with Siegel modular form satisfies a function equation. In this paper, we try to generalize the above fact to the indefinite case.

In my recent paper [1], I associated a projectively invariant pseudo-distance dM to every affinely connected manifold M and proved the following

Theorem 1. Let M be a Riemannian manifold with metric and Ricci tensor RicM such that RicM ≤–c2. Let δM be the Riemannian distance defined by .

In his book [2], Pyatetskii-Shapiro describes representations of classical domains as certain “fibrations” over their boundary components. The fibers are quasi-symmetric Siegel domains of the second kind [3]. Professor Kobayashi asked “how symmetric” these fibers are, or more precisely, he asked for totally geodesic directions in the fiber. The object of this paper is to determine at least a totally geodesic sub-manifold of the fiber, and it turns out to be complex.