Throughout this paper, A denotes a noetherian local ring with maximal ideal m and M a finitely generated A-module with d: = dim M≥1.

In [8], H. Maass introduced the ‘Spezialschar’ which is now called the Maass space. It is defined by the relation of the Fourier coefficients of modular forms as follows. Let f be a Siegel modular form on Sp(2,Z) of weight k, and let be its Fourier expansion, where . Then f belongs to the Maass space if and only if

In [2, § 9] there is a general result of Fulton and Lazarsfeld relating the homotopy groups of a subvariety of in a certain range of dimensions with those of its pullback under a holomorphic map in the corresponding range of dimensions. It is asked in [2, § 10] whether here is a corresponding result with replaced by a general rational homogeneous manifold, Y, and with the range of dimensions alluded to above shifted by the ampleness of the holomorphic tangent bundle of Y in the sense of [4]. In this paper we use the techniques of [4, 5, 6, 7] to answer this question in the affirmative.

It has been proved by A. Baker [1] and H. M. Stark [7] that there exist exactly 9 imaginary quadratic fields of class-number one. On the other hand, G.F. Gauss has conjectured that there exist infinitely many real quadratic fields of class-number one, and the conjecture is now still unsolved.

Let f: E→ P′(C) be an elliptic fibration with non-constant j-invariant, and possessing a section σ0. The group of sections of f is then naturally identified with the group of points defined over a function field C(u) of the generic fibre of f.

In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

In this paper, we improve an inequality given in [3]. Let Lp (1 ≦ p ≦ ∞) and BMO denote the Lp space and the BMO space on the real line. Their norms are denoted by ‖ · ‖p and ‖ · ‖BMO, respectively.

Let (A, m, k) be a Noetherian local ring and I an ideal of A. We set and call this graded A-algebra the Rees algebra of I. The importance of the Rees algebra R(I) is in the fact that Proj R(I) is the blowing up of Spec A with center in V(I). The Cohen-Macaulayness of Rees algebras was studied by many mathematicians. In [GS] S. Goto any Y. Shimoda gave a criterion for R(m) to be Cohen-Macaulay under the assumption that A is Cohen-Macaulay. Their results have been generalized to R(I) in [HI].

In a recent paper [5], it was shown that the tensor product of a finite number of fields over a common subfield satisfies the property that each localization at a prime ideal is a primary ring (in the sense that a zero-divisor is in fact a nilpotent element).

In the first section of this paper, we exploit the properties of associated primes and of flat extensions so as to generalize the above result to zero-dimensional algebras; a simple example shows that this is the best one can hope for. The converse situation is also investigated.

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

Let X be a locally compact and σ-compact abelian group and let denote the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure associated with ξ. In [2], we show the following

Theorem. Let (αt)t≧0 be a sub-Markov convolution semi-group on X and let ψ be the negative definite function associated with (αt)t≧0. Then (αt)t≧0 is transient if and only if Re (1/ψ) is locally -summable.