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Hyperelliptic Prym Varieties and Integrable Systems

Rui Loja Fernandes·Pol Vanhaecke
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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 $$\dot a_i=a_i(a_{i-1}-a_{i+1}), \qquad i=1,...,n,\quad a_{n+1}=a_1,$$ 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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Rui Loja Fernandes, Pol Vanhaecke. 2001-03-22. Hyperelliptic Prym Varieties and Integrable Systems. https://arxiv.org/abs/math-ph/0011051

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