Hyperelliptic Prym Varieties and Integrable Systems

Abstract

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the generic fiber of the momentum map of the periodic Volterra lattice $${\dot{a}}_i=a_i(a_{i-1}-a_{i+1}),i=1,...,n,a_{n+1}=a_1,$$ is an affine part of a hyperelliptic Prym variety, obtained by removing $n$ translates of the theta divisor, and we conclude that this integrable system is algebraic completely integrable.

Final version. To appear in CMP