Theta constants, moduli, and compact riemann surfaces.

One of the classical tools for the study of moduli of compact Riemann surfaces is the Riemann theta function. Preliminary results were announced earlier(1) which established relations between two kinds of theta constants on a compact Riemann surface of genus 2. In this note we generalize the results there obtained. The main theorem is as follows:A sufficient condition for [Formula: see text] to be independent of [Formula: see text] is that [Formula: see text] for the 2(g-2)(2(g-1) - 1) characteristics [Formula: see text] where [Formula: see text] ranges over all odd g - 1 characteristics.