Bilinear recurrences and addition formulae for hyperelliptic sigma functions

DOI

10.2991/jnmp.2005.12.s2.5How to use a DOI?

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. 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.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

TY  - JOUR
AU  - Harry W. Braden
AU  - Victor Z. Enolskii
AU  - Andrew N.W. Hone
PY  - 2005
DA  - 2005/12/01
TI  - Bilinear recurrences and addition formulae for hyperelliptic sigma functions
JO  - Journal of Nonlinear Mathematical Physics
SP  - 46
EP  - 62
VL  - 12
IS  - Supplement 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.s2.5
DO  - 10.2991/jnmp.2005.12.s2.5
ID  - Braden2005
ER  -