Let G be a connected complex Lie group. Then there exists the smallest closed complex subgroup G0 of G such that G/G0 is a Stein group (Morimoto). Moreover G0 is a connected abelian Lie group and every holomorphic function on G0 is a constant. G0 is called an (H, C)-group or a toroidal group. Every connected complex abelian Lie group is isomorphic to the direct product G0 × Cm × C*n, where G0 is an (H,C)-group.

In this note, we will construct simple examples of isospectral surfaces. In what follows, we will use the term “surface” to mean a surface endowed with a Riemannian metric, while the term “Riemann surface” will be reserved for a surface endowed with a metric of constant curvature. We will show:

Let M be a quadratic lattice with positive definite quadratic form over the ring of rational integers, M’ a submodule of finite index, S a finite set of primes containing all prime divisors of 2[M: M’] and such that Mp is unimodular for p ∉ S. In [2] we showed that there is a constant c such that for every lattice N with positive definite quadratic form and every collection (fp)p∊s of isometries fp: NP → MP there is an isometry f: N → M satisfying

f ≡ fp mod M′p for every p |[M: M],

f(Np) is private in Mp for every p ∉ S,

provided the minimum of N ≥ c and rank M ≥ 3 rank N + 3.

Let k be a -adic number field with the ring 0 of all integers in k, and K be a finite normal extension with Galois group G. ∏ denotes a prime element of the ring of all integers in K. Then, an ideal (∏a) of is an 0G-module. E. Noether showed that if K/k is tamely ramified, is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows: is relatively projective with respect to a subgroup S of G if and only if S ⊇ G1 where G1 is the first ramification group of K/k. Now we define the vertex V(∏a) of (∏a) as the minimal normal subgroup S of G such that (∏a) is relatively projective with respect to a subgroup S of G (cf. § 1). Then, the above generalization by A. Fröhlich implies V() = G1. In the previous paper, we proved G1 ⊇ V(∏a) ⊇ G2, (where G2 is the second ramification group of K/k (cf. Theorem 5). Further, we dealt with the case where G = G1 is of order p2, and proved that if V(∏a) ≠ G1 then a ≡ 1(p2) and t2 ≡ 1(p2) for the second ramification number t2 of K/k (cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degree pn (Theorem 7).

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

Let Ω be a bounded domain in Rn with the boundary of class C0,1 and E be a compact subset (resp. a compact subset on an (n —dimensional hypersurface of class C0,1) in Ω. We assume that the usual function spaces and are known.

By means of simple exact sequences in commutative algebra, we can derive some effective criteria for Cohen-Macaulay property of finite partially ordered sets.

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

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.

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X