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

Yuri N. Fedorov, Andrew N. W. Hone
2015 Journal of Integrable Systems  
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 sigmafunction 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 σ with two fixed points, and the Jacobian of X
more » ... the Jacobian of X arises as the Prym variety Prym(S. ( 1.2) Consequently, the relation (1.1) with generic coefficients is referred to as the Somos-6 recurrence, and the corresponding sequence (τ n ) as a Somos-6 sequence. The first proof that the original Somos-6 sequence (1.2) consists entirely of integers was an unpublished result of Hickerson (see [16] ); it relied on showing that the Somos-6
doi:10.1093/integr/xyw012 fatcat:ownq7af5sveufonfxux7mgup4i