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Periods of quadratic irrationalities, and torsion of elliptic curves

Author(s): V. A. Malyshev
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), vypusk 4.
Journal: St. Petersburg Math. J. 15 (2004), 587-602.
MSC (2000): Primary 14K20, 11A55
Posted: July 7, 2004
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Abstract: For rational $A$, $B$, $C$, $D$, the period length for the continued fraction of the square root

\begin{displaymath}{\sqrt{t^{4}+At^{3}+Bt^{2}+Ct+D}}\end{displaymath}

can only take the values $1$, $2$, $3$, $4$, $5$, $6$, $8$, $10$, $14$, $18$, $22$, and perhaps $9$ and $11$.

References:

1.
N. H. Abel, Ueber die Integration der Differentialformel $ \frac{\rho dx}{\sqrt{R}}$, wenn $R$ und $\rho$ ganze Functionen sind, J. Reine Angew. Math. 1 (1826), 185-221.

2.
P. L. Chebyshev, On integration of irrational differentials, Selected Works, Akad. Nauk SSSR, Moscow, 1955, pp. 227-255. (Russian) MR 0067792 (16:781k)

3.
E. I. Zolotarev, Sur la méthode d'intégration de M. Tchébycheff, Complete Works. Issue 1, Akad. Nauk SSSR, Leningrad, 1931, pp. 161-360.

4.
B. Mazur, Rational points on modular curves, Modular Functions of One Variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., vol. 601, Springer, Berlin, 1977, pp. 107-148. MR 0450283 (56:8579)

5.
V. A. Malyshev, The Abel equation, Algebra i Analiz 13 (2001), no. 6, 1-55; English transl., St. Petersburg Math. J. 13 (2002), no. 6, 893-938. MR 1883839 (2003a:14064)

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Additional Information:

V. A. Malyshev
Affiliation: Rybinsk State Aviation Technology Academy, Rybinsk, Russia
Email: wmal@ryb.adm.yar.ru

DOI: 10.1090/S1061-0022-04-00824-6
PII: S 1061-0022(04)00824-6
Keywords: Quadratic irrationalities, elliptic curves
Received by editor(s): 10/MAR/2003
Posted: July 7, 2004
Copyright of article: Copyright 2004, American Mathematical Society

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