Hyperelliptic Prym Varieties and Integrable Systems

Details Hyperelliptic Prym Varieties and Integrable Systems Hyperelliptic Prym Varieties and Integrable Systems

2000-11-29

arxiv.org/abs/math-ph/0011051v2

Physics

Mathematical Physics

Final version. To appear in CMP

Scientific paper

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 $$\dot a_i=a_i(a_{i-1}-a_{i+1}), \qquad i=1,...,n,\quad a_{n+1}=a_1,$$ 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.

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