On a set of equations characterizing Riemann matrices. (English) Zbl 0551.14016

Arbarello, Enrico; de Concini, Corrado

Ann. Math. (2) 120, 119-140 (1984).

In the first part of this paper it is shown that a criterion due to Welters and based on preceding work of Gunning which gives a characterization of Jacobians among irreducible principally polarized abelian varieties can be translated into an explicit finite number of equations involving theta constants and their derivatives.
In the second part it is shown that these equations are part of the K.P. hierarchy of partial differential equations expressed in Hirota bilinear form thus proving that the fact that the theta function satisfies a finite number of equations in the hierarchy characterizes Jacobians. Recently T. Shiota has proved that indeed it is sufficient that the fact that theta function satisfies the K.P. equation already suffices to characterizes Jacobians among irreducible principally polarized abelian varieties provided an extra mild hypothesis is satisfied, thus essentially giving a substantial improvement on the results of this paper.

MSC:

14K25	Theta functions and abelian varieties
14K30	Picard schemes, higher Jacobians
30F30	Differentials on Riemann surfaces
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
33E05	Elliptic functions and integrals

Keywords:

characterization of Jacobians; theta constants; K.P. equation