For smooth linear group schemes over ℤ, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Li (1 ≤ i ≤ n) be a positive definite quadratic form which is equivalent to one of Cartan matrices of Lie algebras of type An (n ≥ 2), Dn (n ≥ 4), E6, E7, E8 and assume that is positive definite quadratic forms and satisfies that rk Mt ≥ 2 and implies rk K or rk L = 1. Then we have n = m and Lt is equivalent to a constant multiple of Ms(i) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case.

We show that if E → X is a locally trivial holomorphic fibrations whose fiber is an open Riemann surface and X is a q-complete space, then E is q-complete.

Let Q be the set of symbols 1, 2,…, 3 p, where p is an prime number greater than 3. Let be a transitive permutation group on Ω, which is simple and in which the normalizer of a Sylow p-subgroup has order 2 p.

We classify the periodic monodromies which are realized as the monodromies of hyperelliptic families.

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion

We consider the continuous valuation of logic, where the certainty of a statement is measured with a number in the closed unit interval I =[0, 1].

The idea originates in continuous logics, which have been investigated from various standpoints, in [1] ~ [4] and [8] ~ [10], for example. More comprehensive information can be seen in [4] and others.

The purpose of this note is to prove the following proposition, which is made use of in the preceding paper by Mr. M. Kuranishi in this number. First we shall explain the notations. Let be a word with respect to two letters, and let G be a free group generated by two free generators x and y.

In this paper we investigate indices of umbilics of a closed surface in the euclidean space. Most part of the discussion is concerned with a symmetric tensor field of degree 2, or rather a direction field, on a Riemannian manifold of dimension 2.

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

In the following, for any group G and any g ∈ G, Ad[g] stands for the inner automorphism defined by Ad[g] (x) = gxg−1 for any x ∈ G.

Let K be a field of characteristic zero and let

be m generic n by n matrices over K. That is, xij(k) are independent commuting indeterminates over K. The K-subalgebra generated by X1…, Xm is called a ring of n by n generic matrices and is denoted by R(n, m).

The main object of this paper is to develop techniques for investigating the local properties of actions of an algebraic group on an algebraic variety. Our main tools are certain schemes which may be associated to Lie algebras.

Let ⌈ be a subgroup of the modular group PSL(2,Z) then ⌈ acts on the upper half plane H = {zЄC : Im z > 0} and we can form the Riemann surface M = H/⌈, see [3]. The complex line bundles on a Riemann surface M form a group H1(M, *), see [4], and whenever we raise a line bundle to a power it will be in this group. Let κ denote the canonical bundle on M then a modular form of weight ν is a section of the bundle . A modular n-jet is then a section of Jn the n-th jet bundle, see [7]. We can reformulate these ideas in the following terms. A modular form can be viewed as a function Φ: H → C and a modular n-jet as a vector valued function Φ: H → Cn+1 both of which satisfy a transformation law under the elements of Γ.

In [10] and [11], the second author proved the following theorem by using Synge’s formula:

THEOREM I. Let M be a kähler manifold with positive holomorphic bi-sectional curvature. Then every pseudoconvex domain in M is a Stein manifold.

Let be a transitive group of degree n, and let be the stabilizer of a symbol in ®. Then we owe to J.S. Frame the following remarkable relations between the lengths ni of the orbits of and the degrees fi of the absolutely irreducible components of the permutation matrix representation * of :

(A) If the irreducible constituents of * are all different, then the rational number

is an integer, where k is the number of the orbits of

The rotational part of the holonomy group of a Riemann space is called its homogeneous holonomy group. A Riemann space, whose homogeneous holonomy group is one-parametric, was investigated by Liber and an alternative treatment of the same problem was given by S. Sasaki [1]. I will treat here a Riemann space with two-parametric homogeneous holonomy group and prove the following theorem by the method analogous to that of [1]. I thank Prof. T. Ootuki for his kind advice.

The purpose of this paper is to define minimality of surfaces in an infinite dimensional space E by probabilistic methods with the description of the relation between minimal surfaces and harmonic functions on the space E, and to analyze purely analytic properties of a certain class of quadratic forms on the space E.

The metrics to which the title of the present paper refers are expressed in the form of elements of arc length as follows:

Let $I$ and $J$ be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of $I$ and $g(J)$, where $g$ is a general change of coordinates. Our main result gives a generalization of Green’s hyperplane section theorem.