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

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 \((S,\sigma)\).