Two kinds of theta constants and period relations on a riemann surface.
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.
STATE UNIVERSITY OF NEW YORK AT STONY BROOK AND THE CITY COLLEGE OF THE CITY UNIVERSITY OF NEW YORK.
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HM
Farkas
H E
HE
Rauch