Volume 12, Issue Supplement 2, December 2005, Pages 46 - 62
Bilinear recurrences and addition formulae for hyperelliptic sigma functions
- Harry W BRADEN, Victor Z ENOLSKII, Andrew N W HONE
- Corresponding Author
- Harry W BRADEN
Available Online 2 December 2006.
- https://doi.org/10.2991/jnmp.2005.12.s2.5How to use a DOI?
- 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.
- Open Access
- This is an open access article distributed under the CC BY-NC license.
Cite this article
TY - JOUR AU - Harry W BRADEN AU - Victor Z ENOLSKII AU - Andrew N W HONE PY - 2006 DA - 2006/12 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 - https://doi.org/10.2991/jnmp.2005.12.s2.5 ID - BRADEN2006 ER -