# Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Fedorov, Yuri N., Hone, Andrew N.W. (2016) Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties. Journal of Integrable Systems, 1 (1). ISSN 2058-5985. (doi:10.1093/integr/xyw012) (KAR id:53527)

We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve $$X$$, such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of $$X$$. The construction is based on a Lax pair with a spectral curve $$S$$ of genus four admitting an involution $$\sigma$$ with two fixed points, and the Jacobian of $$X$$ arises as the Prym variety Prym $$(S,\sigma)$$.