## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 48, 2006, 212-224

### Hyperelliptic curves, continued fractions, and Somos sequences

#### Abstract

We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.

#### Chapter information

**Source***Dynamics & Stochastics: Festschrift in honor of M. S. Keane* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196285822

**Digital Object Identifier**

doi:10.1214/074921706000000239

**Mathematical Reviews number (MathSciNet)**

MR2306202

**Zentralblatt MATH identifier**

1221.11015

**Subjects**

Primary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11G05: Elliptic curves over global fields [See also 14H52]

Secondary: 14H05: Algebraic functions; function fields [See also 11R58] 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

**Keywords**

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

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

van der Poorten, Alfred J. Hyperelliptic curves, continued fractions, and Somos sequences. Dynamics & Stochastics, 212--224, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000239. https://projecteuclid.org/euclid.lnms/1196285822