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
DOI:
https://doi.org/10.1090/S1061-0022-04-00824-6
Published electronically:
July 7, 2004
MathSciNet review:
2068984
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: For rational $A$, $B$, $C$, $D$, the period length for the continued fraction of the square root \[ {\sqrt {t^{4}+At^{3}+Bt^{2}+Ct+D}}\] can only take the values $1$, $2$, $3$, $4$, $5$, $6$, $8$, $10$, $14$, $18$, $22$, and perhaps $9$ and $11$.
- 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.
- P. L. Čebyšev, Izbrannye trudy, Izdat. Akad. Nauk SSSR, Moscow, 1955 (Russian). MR 0067792
- E. I. Zolotarev, Sur la méthode d’intégration de M. Tchébycheff, Complete Works. Issue 1, Akad. Nauk SSSR, Leningrad, 1931, pp. 161–360.
- B. Mazur, Rational points on modular curves, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 107–148. Lecture Notes in Math., Vol. 601. MR 0450283
- 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
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Additional Information
V. A. Malyshev
Affiliation:
Rybinsk State Aviation Technology Academy, Rybinsk, Russia
Email:
wmal@ryb.adm.yar.ru
Keywords:
Quadratic irrationalities,
elliptic curves
Received by editor(s):
March 10, 2003
Published electronically:
July 7, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
