On the even-order vanishing of Jacobian theta functions
The Schottky-relations on the thetanullwerte for Jacobi varieties have an extensive history beginning with Riemann's formulas for the ''Prym'' theta- functions, and culminating in the recent work of \textit{H. M. Farkas} and \textit{H. E. Rauch} [Ann. Math., II. Ser. 92, 434-461 (1970; Zbl 0204.096)], \textit{D. Mumford} [''Prym varieties. II'', Contribut. to Analysis, Collect. Papers dedic. L. Bers, 325-350 (1974; Zbl 0299.14018)] and \textit{J. Igusa} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 531-545 (1981; Zbl 0501.14026)]. These relations are established from an analysis of the configuration of Jacobian-Prym theta-functions on smooth double coverings. In this paper, we obtain these results directly from a special even-order vanishing property of Jacobian theta-functions: this phenomenon enables one to construct certain linear systems on an (intrinsic) Abelian covering leading to irrational identities on theta- functions of type first observed in genus 3 by Frobenius and Schottky. The vanishing property can naturally also be explained in terms of the Prym theta-functions for general double coverings; this is indicated explicitly in {\S} 3 with a version of the Schottky-Jung-Farkas-Rauch proportionality theorem for a class of generalized theta-functions on singular curves.
scientific article; zbMATH DE number 3933225
scientific article