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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Periods of quadratic irrationalities, and torsion of elliptic curves


Author: V. A. Malyshev
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), nomer 4.
Journal: St. Petersburg Math. J. 15 (2004), 587-602
MSC (2000): Primary 14K20, 11A55
Published electronically: July 7, 2004
MathSciNet review: 2068984
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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.
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  • 5. V. A. Malyshev, The Abel equation, Algebra i Analiz 13 (2001), no. 6, 1–55 (Russian, with Russian summary); 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: http://dx.doi.org/10.1090/S1061-0022-04-00824-6
PII: S 1061-0022(04)00824-6
Keywords: Quadratic irrationalities, elliptic curves
Received by editor(s): March 10, 2003
Published electronically: July 7, 2004
Article copyright: © Copyright 2004 American Mathematical Society