# 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)

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## 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 $\upsigma$ with two fixed points, and the Jacobian of $X$ arises as the Prym variety Prym$(S,\upsigma)$.

Item Type: Article 10.1093/integr/xyw012 Q Science > QA Mathematics (inc Computing science) > QA150 Algebra > QA241 Number theoryQ Science > QA Mathematics (inc Computing science) > QA351 Special functionsQ Science > QA Mathematics (inc Computing science) > QA564 Algebraic Geometry Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Andrew N W Hone 18 Dec 2015 12:52 UTC 29 May 2019 16:49 UTC https://kar.kent.ac.uk/id/eprint/53527 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-9780-7369