Let A be a noetherian ring and let I be an ideal of A contained in the Jacobson radical of A: Rad (A). We assume that the form ring of A with respect to the ideal I: G = Gr (A, I), is a normal integral domain. Hence A is a normal integral domain and one can ask for the links between Cl(A) and Cl(G).

We denote by D the unit disc {z; |z| < 1} and by the totality of one to one conformal mappings z′ = s(z) of D onto itself. A meromorphic function f(z) in D is normal if and only if the family is a normal family in D in the sense of Montel. We denote by the totality of the normal meromorphic functions in D.

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs.

In their paper [12], Toda and the author have concerned themselves in the following

Theorem of Kuramochi. Let R be a hyperbolic Riemann surface of the class OHB(OHD, resp.). Then, for any compact subset K of R such that R−K is connected, R−K as an open Riemann surface belongs to the class OAB(OAD, resp.) (Kuramochi [4]).

They have raised there the question as to whether there exists a hyperbolic Riemann surface, which has no Martin or Royden boundary point with positive harmonic measure and has yet the same property as stated in Theorem of Kuramochi, and given a positive answer to the Martin part of this question.

Unfortunately, there is a mistake in [PS, Lemma 3.10] which invalidates [PS, Theorem 3.12]. We show that the theorem still holds if the ring is assumed to be Gorenstein.

Soit X un espace localement compact et dénombrable à l’infini, et supposons qu’il existe une mesure de Radon positive ξ partout dense dans X. Dans toute la suite X et ξ seront fixés.

Let M be an n-dimensional Hadamard manifold, that is, a complete simply connected C∞ Riemannian manifold with nonpositive sectional curvatures. Making use of geodesic rays, Eberlein and O’Neill [11] constructed a compactification = MS(∞) of M which gives a homeomorphism of (M, S(∞)) with the Euclidean pair (Bn, Sn-1). In this paper we shall study the asymptotic Dirichlet problem for the Laplace-Beltrami operator, which is stated as follows:

The device of representing a predicate by a universal sentence gives rise to logics of higher order. To represent an object constructed by a series of certain steps, we usually use terms. To represent a predicate constructed by a series of certain steps, we now use sentences in general. We can now substitute special predicates represented by sentences for predicate variables of a sentence just as we have been able to substitute special objects represented by terms for object variables. It is remarkable that we can always distinguish predicates from objects, proceeding in this way.

In the book “Foundations of algebraic geometry” A. Weil proposed the following problem ; does every differential form of the first kind on a complete variety U determine on every subvariety V of U a differential form of the first kind? This problem was solved affirmatively by S. Koizumi when U is a complete variety without multiple point. In this note we answer this problem in affirmative in the case where V is a simple subvariety of a complete variety U (in §1). When the characteristic is 0 we may extend our result to a more general case but this does not hold for the case characteristic p≠0 (in §2).

In order to extend Nevanlinna’s first and second fundamental theorems to arbitrary analytic mappings between Riemann surfaces, Sario [8, 9] introduced a kernel function on an arbitrary Riemann surface generalizing the elliptic kernel on the Riemann sphere. Because of the importance of the potential theoretic method in the value distribution theory, we discussed potentials of Sario’s kernel in [4]. In that paper the validty of Frostman’s maximum principle for Sario’s potentials was left unsettled. The main object of this paper is to resolve this question (Theorem 1). As a consequence the fundamental theorem of the potential theory is obtained in its complete form for Sario’s potentials (Theorem 2).

Let M = M(D π) be a (real analytic) fibered manifold of dimension n over a manifold D of dimension p, with projection π. We denote by M′ (D, α) the prolonged fibered manifold of M(D, π). Every point of M′ is a p-dimensional contact element of M, and a p-dimensional contact element X of M belongs to M′ if and only if π*X = Tπ(x)(D), where x is the origin of X. We write x = β(X) and α = π°β. We also denote by M″(D, α′) the prolonged fibered manifold of M′(D, α), where α′ = α°β′ and β′ is the projection of each X′ ∈ M″ to its origin in M′.

In a Green space Ω we can introduce Martin’s topology and make it the Martin space Ω, Ω is a dense open subset of and the kernel

can be extended continuously to , where G(p, x) is a Green function in Ω and y0 the fixed point of Ω. is a metric space. is divided into two disjoint subsets Δ0, Δ1 and s ∊ Δ1 is characterized by the fact that K(s, x) is a minimal positive harmonic function in x∊Ω.

L. Carleson ([6]) proved the following theorem:

Let u be a finite continuous function in the unit open ball B with center zero in the complex plane.

Let L = K be the composite of two imaginary quadratic fields and K. Suppose that the discriminants of and K are relatively prime. For any primitive ray class character χ of L, we shall compute L(1, χ) for the Hecke L-function in L. We write for the conductor of χ and C for the ray class modulo . Let c ε C be any integral ideal prime to . We write as g-module where g, n and ϑL are, respectively, the ring of integers in k, an ideal in k and the differente of L. Let where T(χ) is the Gaussian sum and, as in (3.2),

Let M, N be connected complex submanifolds of a neighborhood of the origin 0 ∈ Cd, the space of d complex variables, such that 0 ∈ M ∩ N. We shall suppose throughout that M ⊄ N and N ⊄ M in any neighborhood of 0. Let X = M ∪ N. X is an analytic subvariety with the irreducible branches M and N. Let Δ be a neighborhood of 0 in Cd.

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

An important problem in the theory of arithmetic automorphic functions is to construct, for a reductive algebraic group over Q which defines a bounded symmetric domain, a system of canonical models [2], [6], [18]. For the similitude group of a hermitian form over a quaternion algebra whose center is a totally real field, this is solved by Shimura [17], and for the similitude group of a hermitian form with respect to an involution of the second kind of a central division algebra over a CM-field, by Miyake [8], In this paper, we show that this also can be done for the special Clifford group of a quadratic form Q over a totally real algebraic number field. (We have to impose certain conditions on the signature of Q in order that G defines a bounded symmetric domain, see 1.1.)

We compute cones of effective cycles on some blowups of projective spaces in general sets of lines.

Let f(z) be a k-valued algebroid function in |z| <. ∞ and

(1) F(z, f) ≡ A0(z)fk + A1(z)fk-1 + · · · Ak(z) = 0

be its defining equation such that the coefficients Ai(z) (i = 0, 1, · · ·, k) are entire functions without any common zero and the left hand side is irreducible.

In this section we shall present some facts in the background of the problem under our next consideration. Let (X, A, μ) be a finite measure space, and let B be a Banach space with a norm ‖ • ‖. Let M(μ) denote the linear space of all μ-measurable functions from X into R, and let T be a linear operator from B into M (μ).