Let S be a Noetherian local ring with maximal ideal J and k the residue field of S. Let G be a finite group of order n and suppose that G acts on S as automorphisms. Let R = SG and I = JG . We denote by S[G] (resp. R[G]) the twisted group ring of G over S (resp. the group algebra of G over R). Recall that the multiplication of S[G] is defined as follows : sg · th = sg(t) · gh for s, t ∊ S and g, h ∊ G. Let tG(S) = {Σg ∈ G g(s)/s ∈ S} and call it the trace ideal of S. Note that tG(S) = R if n is a unit of S. We say that S has a normal basis if S ≅ R[G] as R[G]-modules. This condition says that there is an element s of S so that {g(s)}g ∈ G forms an R-free basis of S.

As generalizations of the results in [5] and [4], the author gave some uniqueness theorems of meromorphic maps into PN(C) in previous papers [2] and [3].

In [7], R. Nevanlinna gave the following uniqueness theorem of meromorphic functions as an improvement of a result of G. Pólya ([8]).

Theorem A. Let f, g be non-constant meromorphic functions on C. If there are five mutually distinct values a1, …, a5 such that f−1(ai = g−1(ai) (1 ≦ i ≦ 5), then f ≡ g.

H. P. F. Swinnerton-Dyer determined the structure of the algebra of modular forms mod p for all prime numbers p in elliptic modular case (cf. [10]). Using his result, J.-P. Serre investigated the properties of p-adic modular forms and succeeded to construct the p-adic zeta functions for any totally real number fields (cf. [8]).

Let k be the rational number field, K/k be an Abelian extension defined mod whose degree is some power of a prime l, and let be the module of K belonging to in the sense of Fröhlich [1, p. 239].

Let Q2 be the 2-adic number field, T/Q2 be a finite unramified extension, ζv be a primitive 2v-th root of unity, and let Kv = T(ζv ). In a previous paper [1, Theorem 11], we stated the following theorem without its proof.

In this paper we shall study the global condition of curvatures of hypersurfaces in the complex Euclidean space. S. Bochner ([1]) and E. Calabi ([2]) have already investigated kähler embeddings (i.e. holomorphic and isometric embeddings) of a kähler manifold into the complex Euclidean space. However, it seems that they did not discuss the relation between the embeddability and the curvature.

Recently A. Morimoto [1] has proved the Takens conjecture in the tolerance stability by using the notion of pseudo-orbits and the stochastic stability. He also characterized group automorphisms of a torus to be stochastically stable and clarified the relations to other stabilities.

In this paper we shall give the condition for spherical or projective linear transformations to be stochastically stable.

We consider the decay of the local energy for the following equation in three dimension:

(0.1) utt + but − Δu = 0

u(0, x) = f(x) and ut (0, x) = g(x).

Recently the decay of the local energy for wave equations with a moving obstacle has been studied by Cooper [1] and Cooper and Strauss [2] etc. In their works it has been assumed that the obstacle is uniformly bounded in time t and that the origin is contained in for all t > 0 and is star-shaped with respect to the origin. (The second condition has been assumed implicitly in [2] (see Assumption (B), [2]).)

Similarly to the real quadratic field case by Doi and Naganuma ([3], [9]) there is a lifting from an elliptic modular form to an automorphic form on SL2(C) with respect to an arithmetic discrete subgroup relative to an imaginary quadratic field. This fact is contained in his general theory of Jacquet ([6]) as a special case. In this paper, we try to reproduce this lifting in its concrete form by using the theta function method developed first by Niwa ([10]); also Kudla ([7]) has treated the real quadratic field case on the same line. The theta function method will naturally lead to a theory of lifting to an orthogonal group of general signature (cf. Oda [11]), and the present note will give a prototype of non-holomorphic case.

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

Let D be a bounded domain in Rd with smooth boundary ∂D. We denote by Bt, t ≥ 0, a one-dimensional Brownian motion. We shall consider the initial-boundary value problem