On periodicity of continued fractions in hyperelliptic function fields
type: Item
instance of: scholarly article
title: On periodicity of continued fractions in hyperelliptic function fields
zbMATH Open document ID: 0728.14027
DOI: 10.1007/BF01191166
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let \(F\in k[X]\) be a square-free polynomial of degree \(2p+2\) over a field k of odd characteristic, and let C be the nonsingular model of the curve \(y^ 2=F(x)\). Then C is a hyperelliptic curve of genus \(p.\) Consider the Pell equation of F, i.e. \(P^ 2-Q^ 2F=1\). The author gives necessary and sufficient conditions for the existence of a non-trivial solution \((P,Q)\in (k[X])^ 2\), \(Q\neq 0\), in terms of the quasi-periodicity of the continued fraction expansion of the form (\(\sqrt{F}+L)/M\) (where L, M are polynomials and M divides \(F-L^ 2)\).
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4202465
zbMATH Keywords: hyperelliptic function fields
zbMATH Keywords: hyperelliptic curve
zbMATH Keywords: Pell equation
zbMATH Keywords: continued fraction
author: T. G. Berry
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
cites work: Continued fractions and periodic Jacobi matrices
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cites work: Q5571530
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cites work: On some problems of the arithmetical theory of continued fractions II
description: scientific article; zbMATH DE number 4202465