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Journal of the London Mathematical Society Advance Access originally published online on February 27, 2008
Journal of the London Mathematical Society 2008 77(3):593-606; doi:10.1112/jlms/jdm125
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© 2008 London Mathematical Society

Periodic continued fractions and hyperelliptic curves

M.-P. Grosset

Department of Mathematical Sciences
Loughborough University
Loughborough
Leicestershire LE11 3TU
United Kingdom
Lycèe Multilingue Ombrosa
95 quai Clémenceau
69300 Caluire
France
mpgrosset@googlemail.com

A. P. Veselov

Department of Mathematical Sciences
Loughborough University
Loughborough
Leicestershire LE11 3TU
United Kingdom
Landau Institute for Theoretical Physics
ul. Kosygina 2
Moscow 119334
Russia

We investigate when an algebraic function of the form {phi}({lambda})=(–B({lambda})+{surd}R({lambda}))/A({lambda}), where R({lambda}) is a polynomial of odd degree N=2g+1 with coefficients in C, can be written as a periodic {alpha}-fraction of the form


Formula

for some fixed sequence {alpha}i. We show that this problem has a natural answer given by the classical theory of hyperelliptic curves and their Jacobi varieties. We also consider pure periodic {alpha}-fraction expansions corresponding to the special case when bN=b0.

2000 Mathematics Subject Classification 14H05, 14H40, 30B70.

Received July 18, 2007; revised November 23, 2007; published online February 27, 2008.


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