Hyperelliptic curves, continued fractions, and Somos sequences. (English) Zbl 1221.11015

van der Poorten, Alfred J.

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 212-224 (2006).

The author details the continued fraction expansion of the square root of monic polynomials of even degree. It is well known that each step of the continued fraction expansion corresponds to addition of the divisor at infinity on the relevent elliptic or hyperelliptic curve. In the quartic and sextic cases he observes explicitly that the parameters appearing in the expansion yield integer sequences defined by relations including and generalizing that of the sequence \(\dots, 3,2,1,1,1,1,1,2,3,5,11,37,83,\dots\) produced by the recursive definition \(B_{h+3}=(B_{h-1}B_{h+2}+B_h B_{h+1})/B_{h-2}\).
For the entire collection see [Zbl 1113.60008].

Reviewer: Takao Komatsu (Hirosaki)

MSC:

11A55	Continued fractions
11G05	Elliptic curves over global fields
14H05	Algebraic functions and function fields in algebraic geometry
14H52	Elliptic curves

Keywords:

continued fraction expansion; function field of characteristic zero; hyperelliptic curve; Somos sequence

Software:

OEIS