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

TL;DR

This paper explicitly solves the general Somos-6 recurrence relation using hyperelliptic sigma-functions, linking the sequence's iteration to translations on a genus two Jacobian derived from a genus four spectral curve with involution.

Contribution

It introduces a novel explicit solution for Somos-6 sequences via Prym varieties and hyperelliptic functions, connecting recurrence dynamics to algebraic geometry.

Findings

Solution expressed in terms of Kleinian sigma-functions

Recurrence iteration corresponds to translation in Jacobian

Spectral curve with involution underpins the construction

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$(S,\sigma)$.