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If S is a compact Riemann surface of genus g>2 together with a canonical homology basis (F, A) r = at, *.., y, A = al .**, 3g and if de1, **,9 denotes the basis of abelian differentials of first kind on S dual to (F, A); i.e., 4 there must be these relations. Another way of looking at the problem is the following: The totality of g x g matrices which are symmetric and have positive definite imaginary part is called the Siegel upper half plane of degree g, denoted by ?Rg. Not all elements of g9 are Riemann matrices for some (S, r, A). As a matter of fact for g > 4 the elements of 9g which are Riemannmat rices for (S, F, A) form a set of positive codimension. The problem we are considering is to determine this set. The first person to make a break-through in this direction was F. Schottky [12]. In the case g = 4, Schottky showed that for the set in question in e, the associated even theta constants satisfy a special relation of the form 1/ r, ?+ Vr2 + V r3 = 0 where ri is a product of 8 theta constants. Rationalizing this expression we have an explicit homogeneous polynomial in the Riemann theta constants which of course gives the one relation for g = 4 among the 10 periods. Subsequently, Schottky and Jung in a joint note [13] indicated a way of re-deriving the genus 4 result and generalizing it to arbitrary g. Their idea was to establish certain relations between the Riemann theta constants and what was referred to in [10] as the Schottky theta constants; however, to our knowledge they never establish these relations. These relations were established for the case g = 2 by Rauch and the author in [10, 11] and the relations for g = 2 were seen to be a consequence of the vanishing

the theory of complex analytic mappings of complex manifolds and the classical study of value distributions is the study of the "size" of the image of a complex analytic mapping. We give in this paper a treatment, from a purely differential-geometric viewpoint, of complex analytic mappings of a Riemann surface (= one-dimensional complex analytic manifold) into a compact Riemann surface. In the case when the first Riemann surface is a compact one with a finite number of points deleted, we derive defect relations which generalize the classical relations of Nevanlinna-Ahlfors. In a subsequent paper we will consider the case when the first Riemann surface is a compact one with a finite number of points and a finite number of disks deleted. The paper is written for differential geometers, so that concepts currently in use in differential geometry are freely used and a minimum of function theory will be required. The explicit models of the Gaussian plane or the unit disk are avoided.

It was recognized in Riemann's work more than one hundred years ago and proved recently by Rauch, cf. [R2], that the g(g + 1)/2 unnormalized periods of the normal differentials of first kind on a compact Riemann surface S of genus g > 2 with respect to a canonical homology basis are holomorphic functions of 3g 3 complex variables, "the" moduli, which parametrize the space of Riemann surfaces near S and, hence, that there are (g 2)(g 3)/2 holomorphic relations among those periods. Eighty years ago, Schottky [S1] exhibited the relation for g = 4 as the vanishing of an explicit homogeneous polynomial in the Riemann theta constants. Sixty years ago, Schottky and Jung [SJ] conjectured a result which implies Schottky's earlier one and some generalizations for higher genera. Here, we formulate Schottky and Jung's conjecture precisely and, on the basis of a recent result of Farkas [F3], [F4], prove it. We then derive Schottky's result (we believe for the first time correctly) and exhibit a typical relation of this kind for g = 5 (we show how to do this for any genus). We do not prove that our relations imply all relations, but there are some indications that they do, indications to be dealt with in subsequent publications. The present paper is a consolidation and expansion of the notes [RF1] and [FR]. Our main result is formulated in ? 3 as Theorem 1 which asserts the proportionality of the squares of one set of theta constants, the Schottky constants, to certain two-term products of another set, the Riemann theta constants, both sets attached to S with a definite canonical homology basis and both defined in ? 2. Section 2 also contains other essential preliminary definitions and lemmata. In ? 5 we attain the principal object of the whole investigation by showing how, for g ? 4, the substitution of the proportionalities of Theorem 1 into suitable identities for general (g 1)-theta constants, in particular, for the Schottky theta constants leads immediately to relations among the Riemann

Since every compact Riemann surface can be uniformized by a Schottky group, it is natural to look at the space of Schottky groups, or Schottky space, as a space of moduli for Riemann surfaces. The main purpose of this paper is to investigate Schottky groups and the boundary of Schottky space. In the course of this investigation, we prove that a group of Mobius transformations, which is a limit of a sequence of Schottky groups of genus g, is a free group on g generators without elliptic transformations. We generalize this result to groups which are limits of general kleinian groups. The author would like to thank Professor L.Bers for his patient guidance and encouragement in the preparation of this manuscript. These results have been announced in the Bulletin of the American Mathematical Society.

In two papers [6; 7], Hurwitz dealt with unbounded coverings of compact Riemann surfaces Y. He was able to determine the number of ("geometrically") different coverings in case all resp. all but one ramified points of Ysplit in a single point of order two and points of order one. In the present paper we take up this question and ask for upper and lower bounds of the number of different unbounded coverings of Y which have a prescribed ramification type. In case the degree of the covering (= number of sheets) equals n we find a lower bound that is very roughly (n !)2,+r-3 and an upper bound that is very roughly (n !)2g+r-1 where g is the genus of the compact Riemann surface Y and r is the number of ramified points of Y. The tools used in getting these estimates are some results on permutation groups like: (i) Every element of the alternating group An of n elements is a commutator. (ii) Let n # 4 and let a be an element of the symmetric group Sn of n elements that is not the identity. If oc E An (o An), then every element of An (Sn) can be written as a product of at most n conjugates of a.

Introduction. In this paper we show how to finite-dimensional spaces of differentials on a Riemann surface can be associated a finite collection of Riemannian metrics whose curvatures are closely related to the zeros of the differentials on the surface. In particular, we show that the Weierstrass points of a compact Riemann surface can be characterized as points of zero curvature of certain "naturally defined" metrics on the surface. The main theorem and application are in Part II; in Part I we have collected all the preliminary details. In the conclusion we point out the origin of this study and a possible direction for further research.

k=1 defined by p(ak) = gk, p (bk) = gk-'- If K is the kernel, then G is isomorphic to F/K. F is the fundamental group of a compact Riemann surface . of genus n. - is covered by the half-plane R = { (x, y) I y > 0}, and the group of covering transformations is a discrete group of linear fractional transformations, isomorphic to F. If we identify points of .) which are congruent under K, we obtain a Riemann surface C- = /K. (E is compact, since it is a finite covering space of - == $/F (the index [F: K] = order (G) is finite). A conformal transformation c of e can be lifted to a coniformal transformation

Let S be a compact Riemann surface of genus g > 2, let X be a holomorphic line bundle of positive Chern class, and suppose that X admits holomorphic sections which have no common zero. Let F(S, x) = the space of holomorphic sections of x, and let ry(X) = the complex dimension of F(S, x). Let %n = ... X A) X* n times. Define W.(X) to be the set of points p of S for which there exists an element a of F(S, \n) such that the order of c at p is not less than y(Xn) Let W(X) be the union of the W,(X). Our main result was conjectured by Lipman Bers for the case X = a, the canonical bundle