JP Journal of Geometry and Topology

The JP Journal of Geometry and Topology publishes articles in all branches of geometry and topology, with applications to physics. It covers areas such as differential geometry, algebraic topology, and geometric aspects of mathematical physics. Survey articles are also welcome.

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EXPLICIT GEOMETRIC PARAMETRIZATION OF ALGEBRAIC POINTS OF GIVEN DEGREE ON A QUOTIENT OF THE FERMAT CURVE  $C_5(11)$

Authors

  • Mohamadou Mor Diogou DIALLO
  • Cherif Mamina COLY

Keywords:

Mordell-Weil group, Jacobian, linear system

DOI:

https://doi.org/10.17654/0972415X25006

Abstract

We give a geometric description of all algebraic points of arbitrary degree over $\Theta$ on the affine curve defined by the equation

$$
y^{11}=x^5(x-1)^5 .
$$


This curve is a special case of the quotients of Fermat curves of the form

$$
y=x(x-1) \text { with } 1 \leq r, s, r+s \leq p-1,
$$

as studied by Sall (see [6]), completing earlier works of Gross and Rohrlich (cf. [4]).

Received: June 2, 2025
Revised: June 25, 2025
Accepted: July 1, 2025

References

[1] D. Faddeev, On the divisor class groups of some algebraic curves, Dokl. Akad. Dauk SSSR 136(2) (1961), 67-69.

[2] M. Fall and O. Sall, Points algébriques de degré donné sur la courbe de Picard, A Collection of Papers in Mathematics and Related Sciences 1 (2018), 33-42.

[3] P. Griffiths, Introduction to Algebraic Curves, Translations of Mathematical Monographs 76, Springer-Verlag, 1989.

[4] B. Gross and D. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Inventiones Mathematicae 44 (1978), 201-224.

[5] M. Klassen and E. F. Schaefer, Arithmetic and geometry of the curve Acta Arithmetica 74(3) (1996), 241-257.

[6] O. Sall, Points algébriques sur certains quotients de courbes de Fermat, Comptes Rendus. Mathématique Sci. Paris 2(336) (2003), 117-120.

[7] M. Coppens and G. Martens, Secant spaces and Clifford’s theorem, Compositio Mathematica 2(78) (1991), 193-212.

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Published

2025-08-22

Issue

Vol. 31 No. 2 (2025)

Section

Articles

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How to Cite

EXPLICIT GEOMETRIC PARAMETRIZATION OF ALGEBRAIC POINTS OF GIVEN DEGREE ON A QUOTIENT OF THE FERMAT CURVE  $C_5(11)$. (2025). JP Journal of Geometry and Topology, 31(2), 55-61. https://doi.org/10.17654/0972415X25006

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