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ALGEBRAIC POINTS OF LOW DEGREES ON A QUOTIENT OF FERMAT CURVES
Keywords:
degree of algebraic points, rational points, algebraic extensions.DOI:
https://doi.org/10.17654/0972555524027Abstract
In this paper, we determine the set of algebraic points of degree at most 2 over $\mathbb{Q}$ on the curve $C_{5,5}(11)$ given by the affine equation $y^{11}=x^5(x-1)^5$. Indeed, we consider a particular case of quotients of Fermat curves $C_{r, s}(p): y^p=x^r(x-1)^s$ with the conditions $1 \leq r, s$ and $r+s \leq p-1$. These curves are described by Sall in [4] who extended the work of Gross and Rohrlich in [2].
Received: February 5, 2024
Revised: June 27, 2024
Accepted: July 11, 2024
References
D. Faddeev, On the divisor class groups of some algebraic curves, Dokl. Akad. Nauk SSSR 136 (1961), 296-298. English translation: Soviet Math. Dokl. 2(1) (1961), 67-69.
B. Gross and D. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978), 201-224.
P. A. Griffiths, Introduction to Algebraic Curves, Translations of Mathematical Monographs, Volume 76, American Mathematical Society, Providence, 1989.
O. Sall, Points algébriques sur certains quotients de courbes de Fermat, C. R. Acad. Sci. Paris Ser. I 336 (2003), 117-120.
P. Tzermias, Torsion parts of Mordell-Weil groups of Fermat Jacobians, Internat. Math. Res. Notices 7 (1998), 359-369.
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