TWO KINDS OF THETA CONSTANTS AND PERIOD RELATIONS ON A RIEMANN SURFACE

Abstract

It was recognized in Riemann's work more than one hundred years ago and proved recently by Rauch (cf. Bull. Am. Math. Soc., 71, 1-39 (1965) 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 exhibited the one relation for g = 4 as the vanishing of an explicit homogeneous polynomial in the Riemann theta constants. Sixty years ago, Schottky and Jung 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 (these PROCEEDINGS, 62, 320 (1969)), 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 can do this for any genus). We do not prove that our relations imply all relations, but there are some indications that they do.