Bilinear recurrences and addition formulae for hyperelliptic sigma functions

Abstract

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model y 2 = 4x 5 + c 4 x 4 + . . . + c 1 x + c 0 . We show that the sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves the genus two Kleinian sigma function with its argument shifted by the Abelian image of the reduced divisor of a single point on the curve. The genus two recurrences are related to a Bäcklund transformation (BT) for an integrable Hamiltonian system, namely the discrete case (ii) Hénon-Heiles system.

Hyperelliptic curves, continued fractions, and Somos sequences

“…Indeed, the readily checked assertion that given y ∈ C there are constants α and β so that A program of this genre is elegantly carried out by Andy Hone in [12] for Somos 4 sequences. In [5], the ideas of [12] are shown to allow a Somos 8 recursion to be associated with adding a divisor on a genus 2 curve. Incidentally, that result coheres with the guess mooted at Comment 11 above that there always then is a cubic relation of width 6 .…”

Sigma function solution of the initial value problem for Somos 5 sequences

“…By using the continued fraction expansion of the square root of a sextic, van der Poorten [21] has generated a certain class of Somos 6 sequences associated with genus 2 curves. With a different approach [2] based on the addition formulae for Kleinian sigma functions (see [6] and references), we have constructed Somos 8 sequences derived from genus 2 curves, taking a different (quintic) affine model compared with van der Poorten. The recurrences in [2] generalise the genus 2 case of Cantor's hyperelliptic division polynomials [7] (for analytic formulae, see [17]), and they are related to a family of integrable symplectic maps (discrete Hénon-Heiles systems).…”

“…With a different approach [2] based on the addition formulae for Kleinian sigma functions (see [6] and references), we have constructed Somos 8 sequences derived from genus 2 curves, taking a different (quintic) affine model compared with van der Poorten. The recurrences in [2] generalise the genus 2 case of Cantor's hyperelliptic division polynomials [7] (for analytic formulae, see [17]), and they are related to a family of integrable symplectic maps (discrete Hénon-Heiles systems). We should also remark that Buchstaber and Krichever have derived bilinear addition formulae for Riemann theta functions [4], which have exactly N + 2 terms in genus N .…”

Laurent Polynomials and Superintegrable Maps

“…Nevertheless, in [1] it was shown that a particular family of solutions of Somos-8 recurrences can be described in terms of the Kleinian sigma function for a genus two curve (which is equivalent to an expression in theta functions), and these solutions are related to the BT for the Hénon-Heiles system that was found in [23,24]. The author has also found that the Somos-6 and Somos-7 recurrences correspond to a rational map in C 4 with two independent conserved quantities, and there is a similar expression for the solutions in terms of genus two sigma functions.…”