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

By H. M. Farkas and Harry E. Rauch

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

Topics: Physical Sciences: Mathematics
Year: 1969
OAI identifier: oai:pubmedcentral.nih.gov:223651
Provided by: PubMed Central
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