In a recent paper [HI2] the first named author and S. Ikeda have investigated Gorenstein properties under blowing up of height two equimultiple ideals from the arithmetical point of view. The problem is to what extent does the Gorenstein property of the Rees algebra R(a) = of an equimultiple ideal a of a local ring (A, m) describe the structure of A and a. One result is that if A is a generalized Cohen-Mac aulay ring (i.e. a ring of finite local cohomology) with dim (A) ⊇ 4 and if there is an height two equimultiple prime ideal p such that R(p) is Gorenstein, then A is Gorenstein and p is generated by a regular sequence (see [HI2], Theorem 2.6). That has led to the question whether this result is still valid for dim (∧) = 3. ∧ partial answer was given in [HI2], Proposition 2.10, where “equimultiplicity” was replaced by the stronger condition “p/p2 is flat over A/p”

In [5], the author proved that the Gauss map of a nonflat complete minimal surface immersed in R3 can omit at most four points of the sphere, and in [7] he revealed some relations between this result and the defect relation in Nevanlinna theory on value distribution of meromorphic functions. Afterwards, Mo and Osserman obtained an improvement of these results in their paper [11], which asserts that if the Gauss map of a nonflat complete minimal surface M immersed in R3 takes on five distinct values only a finite number of times, then M has finite total curvature. The author also gave modified defect relations for holomorphic maps of a Riemann surface with a complete conformai metric into the n-dimensional complex projective space Pn(C) and, as its application, he showed that, if the (generalized) Gauss map G of a complete minimal surface M immersed in Rm is nondegenerate, namely, the image G(M) is not contained in any hyperplane in Pm − 1(C), then it can omit at most m(m + 1)/2 hyperplanes in general position ([8]). Here, the number m(m + 1)/2 is best-possible for arbitrary odd numbers and some small even numbers m (see [6]). Recently, Ru showed that the “nondegenerate” assumption of the above result can be dropped ([13]). In this paper, we shall introduce a new definition of modified defect and prove a refined Modified defect relation. As its application, we shall give some improvements of the above-mentioned results in [5], [7], [8], [11] and [13].

In this paper, we study 3-dimensional locally strongly convex affine hypersurfaces in ℝ4. Since the publication of Blaschke’s book [B] in the early twenties, it is well-known that on a nondegenerate affine hyper-surface M there exists a canonical transversal vector field called the affine normal. The second fundamental form associated to the affine normal is called the affine metric. In the special case that M is locally strongly convex, this affine metric is a Riemannian metric. Also, using the affine normal, by the Gauss formula one can introduce an affine connection on M, called the induced connection ∇. So on M, we can consider two connections, namely the induced affine connection ∇ and the Levi Civita connection of the affine metric h.

In this note we give a lattice path proof of the ribbon determinant formula for Schur functions ((1) below) which was originally formulated and proved in [2].

In discussing convergence of Feynman path integrals [2], we need a stationary phase method of oscillatory integrals over a space of large dimension. More precisely, we have to know how the remainder term behaves when the dimension of the space goes to ∞ (cf. [2], [3] and [5]). The aim of the present note is to give answer to this question under rather mild assumptions. Application to the Feynman path integrals is discussed in [3] and [5].

Let A be a regular local ring of dim A = 3 and p a prime ideal in A of dim A/p = 1. We put Rs(p) = (here t denotes an indeterminate over A) and call it the symbolic Rees algebra of p. With this notation the authors [5, 6] investigated the condition under which the A-algebra Rs(p) is Cohen-Macaulay and gave a criterion for Rs(p) to be a Gorenstein ring in terms of the elements f and g in Huneke’s condition [11, Theorem 3.1] of Rs(p) being Noetherian. They furthermore explored the prime ideals p = p(n1, n2, n3) in the formal power series ring A = k[X, Y, Z] over a field k defining space monomial curves and Z = with GCD(n1, n2, nz) = 1 and proved that Rs(p) are Gorenstein rings for certain prime ideals p = p(n1 n2, n3). In the present research, similarly as in [5, 6], we are interested in the ring-theoretic properties of Rs(p) mainly for p = p(n1 n2) nz) and the results of [5, 6] will play key roles in this paper.

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

F. van Engelen, K. Kunen, and A. W. Miller proved, in [EKM], that for every analytic set A on the plane, either A can be covered by a countable family of lines or else there is a perfect subset P of A such that no three points of P are collinear. In this paper, we present some generalizations of their result. In particular, a question which was raised by van Engelen et al. in the last paragraph of [EKM] is answered (see Section 3).

Thanks to K. Heegner [He], A. Baker [Ba] and H. Stark [S], we know that there are nine imaginary quadratic fields of class number one. Gauss conjectured that there are infinitely many real quadratic fields of class number one, but the conjecture is still open.

Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).