Recently different authors have studied the conormal modules I/I2 of almost complete intersections in local Gorenstein rings (c. t. Aoyama [1], Herzog [8], Kunz [13], Matsuoka [16]). An essential tool in these papers is the theory of canonical modules and the fact that these modules are easy to handle in the case of almost complete intersections.

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

Several kinds of generalizations of classical Abel Theorem in algebraic curves are known, for example see [12] and [13]. It seems to the author these are all regarded as local relations among rational differential forms. In this article we shall try to generalize Abel Theorem for integrals of rational forms in some specific cases where these can be described in terms of hyper-logarithms (for the definition see [3] and [4], Theorem 2). Trigonometric functions have been generalized to higher dimensional cases by L. Schläfli who has obtained a very important variational formula related to the volume of a spherical simplex [16].

In the previous paper [3], we introduced new coordinates to the Schottky space, and defined the augmented Schottky spaces . Here, in § 1, we will define fiber spaces over the augmented Schottky spaces. In §2, we will consider a property of the new coordinates, namely, we will state a relation between limits of sequences of elements of the new coordinates and limits of sequences of length of loops on Riemann surfaces.

This is the second half of the paper entitled “On augmented Schottky spaces and automorphic forms”. In the first half, we introduced new coordinates to the Schottky space, and defined the augmented Schottky space by using the new coordinates. Furthermore we considered relations between the augmented Schottky space and Riemann surfaces with or without nodes.

Cobordism invariants have been applied to both real and complex categories. For example, the index of 4k-dimensional manifolds was treated by Hirzebruch for the generalization of the Riemann-Roch theorem [3]. He also considered, in relation with this, virtual genus or virtual characteristics. But many invariants, such as virtual characteristics, have their origin in complex analytic category. In view of this, we consider certain complex analytic cobordism, i.e., quasilinear cobordism among quasilinear subvarieties in complex manifolds (see for the definition of quasilinear structure [4].) Quasilinear body has very simple type of singularities, as well as its quasilinear boundaries. Therefore, the theory of quasilinear cobordism can be reduced, through σ-processes, to that of non-singular cobordism theory.

In the complex function theory, Picard’s Great Theorem plays an essential and important role. It is well-known as generalizations of this theorem that in a neighborhood of an isolated essential singularity, a meromorphic function cannot be exceptionally ramified (see W. Gross [2]) and that even it cannot be normal (see O. Lehto and K. I. Virtanen [7]). We are therefore interested in the behaviour of meromorphic functions with non-isolated essential singularities as well as in generalizations of the Gross’ result. Several approaches in this direction have been made by G. af Hällström [3], S. Kametani [4], K. Noshiro [13], K. Matsumoto [8], [9], [10], [11], [12], S. Toppila [15], etc..

Recently, A. Morimoto [5] proved that every isometry of a compact Riemannian manifold of positive dimension has not the pseudo-orbit tracing property, and that if a homeomorphism of a compact metric space has the pseudo-orbit tracing property then Eφ— 0φ(see § 1 for definition). The purpose of this paper is to show that every distal homeomorphism of a compact connected metric space has not the pseudo-orbit tracing property.

In a classical paper [C] Chevalley considered the invariants of a finite group H ⊂ GLk(S1) generated by pseudo-reflections, acting on the graded polynomial ring S = k[X1,…,Xn] over a field k of characteristic zero. He proved that S is free as a graded SH-module, hence SH is a graded polynomial ring (Theorem A), and that the natural representation of H in is equivalent to the regular representation (Theorem B). On the other hand, a theorem of Shephard and Todd shows that when SH is a polynomial ring, the (finite) group H is generated by pseudo-reflections. These results have been extended by Bourbaki [Bo2] to fields whose characteristic may be positive, but does not divide the order |H| of the group.

In this paper, we shall give explicit Z-basis of certain maximal orders of definite quaternion algebras over the rational number field Q (See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. Let D be a division quaternion algebra over Q.

Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].

This paper is a continuation of the two preceding papers [12], [13] where the classification of the de Jonquières type subgroups in the Cremona group of 3 variables is promised. However the classification of such subgroups is postponed until the article in preparation “On the maximal connected algebraic subgroups of the Cremona group II”. The purpose of this paper is to establish a general method to study algebraic subgroups in the Cremona group of n variables and to illustrate how it works and leads to the classification of Enriques (Theorem (2.25)) when applied to the 2 variable case. This method gives us also the classification of the maximal connected algebraic subgroups of the Cremona group of 3 variables.