Elliptic curves and continued fractions

2004/03/14 by Alfred J. van der Poorten, van der Poorten, Alfred J. · 2 citations
Computer Science · Mathematics · #11A55 #11G05 (primary) #14H05 #14H52 (secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT #msc:11A55 #msc:11G05 #msc:14H05 #msc:14H52

paper · pdf · doi:10.48550/arxiv.math/0403225

v1; final -- I hope

openalex publication_date 2004/03/14 · arxiv created 2004/07/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general expansion. In that way we obtain `elliptic sequences' satisfying Somos relations. I mention several new results on such sequences. The paper includes a detailed `reminder exposition' on continued fractions of quadratic irrationals in function fields.

Cited by

• Sigma function solution of the initial value problem for Somos 5 sequences
• Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials

Related

• Reduction and specialization of hyperelliptic continued fractions
• Genus 2 curves, continued fractions, and Somos sequences
• Hyperelliptic continued fractions in the singular case of genus zero
• The complete evaluation of Rogers Ramanujan and other continued fractions with elliptic functions
• On a quartic equation and two families of hyperquadratic continued fractions in power series fields