Canonical heights on Pell conics over number fields
DOI:
https://doi.org/10.52737/18291163-2020.12.5-1-9Keywords:
canonical height, Pell conicAbstract
In "Higher descent on Pell conics. III. The first 2-descent", Lemmermeyer introduced the canonical heights on the groups of rational points on Pell conics, which are analogues of the canonical heights on elliptic curves. In this paper, we generalize this: We introduce the canonical heights on the groups of $\overline{\mathbb{Q}}$-rational points on Pell conics over number fields.
References
E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Mathematical Monographs, 4. Cambridge University Press, Cambridge, 2006.
F. Lemmermeyer, textit{Higher descent on Pell conics. III. The first 2-descent}, preprint, available at: https://arxiv.org/abs/math/0311310
P. Shastri, textit{Integral points on the unit circle}, J. Number Theory textbf{91} (2001), no. 1, 67--70.
S. A. Shirali, textit{Groups associated with conics}, Math. Gaz. textbf{93} (2009), no. 526, 27--41.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986.
L. Tan, textit{The group of rational points on the unit circle}, Math. Mag. textbf{69} (1996), no. 3, 163--171.
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