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

Abstract

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$(S,\sigma)$.