Periods of quadratic irrationalities, and torsion of elliptic curves

Malyshev, V.

doi:10.1090/S1061-0022-04-00824-6

Periods of quadratic irrationalities, and torsion of elliptic curves
HTML articles powered by AMS MathViewer

by V. A. Malyshev
Translated by: the author

St. Petersburg Math. J. 15 (2004), 587-602

DOI: https://doi.org/10.1090/S1061-0022-04-00824-6

Published electronically: July 7, 2004

PDF | Request permission

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 $11$.

References

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) Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977, pp. 107–148. 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