On the Schottky Relation and Its Generalization to Arbitrary Genus

Abstract

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