Describing families of algebraic points of given degree on a hyperelliptic curve

Articles in Press

Document Type : Original Article

Author

Department of Mathematics, Assane Seck University of Ziguinchor Ziguinchor, Senegal

Abstract

In this paper, we give a parametrization of the set of algebraic points of degree at most l over Q on the hyperelliptic curve
y2 = x(x2 + x - 4)(x2 - x + 45)
This curve was studied by Bruin and Flynn in [4] where the heights explicitly described the set of rational points, i.e. C(1)(Q). Drawing on the work of Arnth-Jensen and Flynn based on [2] and one of Abel-Jacobi's fundamental theorems in [1, 8], we extend the results of [4] to algebraic points of any degree which we denote C(l)(Q).

Keywords

References

 1. E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris. The Basic Results of the BrillNoether Theory, Geometry of Algebraic Curves, 133 (3) (1985), 203{224.
2. A. Arnth-Jensen and E. V. Flynn, Non-trivial in the Jacobian of an infinite family of curves of genus 2, Journal de th´eorie des nombres de Bordeaux, 21 (1) (2009), 1-13.
3. A. Borel and J. P. Serre, Le th´eor´eme de Riemann-Roch, Bultin de la Soci´et´e math´ematiques de france, 86 (1958), 97-136.
4. N. Bruin and E. V. Flynn, Exhibiting SHA [2] on hyperelliptic Jacobians, Journal of Number Theory, 118 (2) (2006), 266-291.
5. M. Coppens and G. Martens, Secant spaces and Clifford’s theorem, Compositio Mathematica, 78 (2) (1991), 193-212.
6. M. Fall, M. M. D.Diallo and C. M. Coly, Algebraic Points of any Given Degree on the Affine Curves y2 = x(x + 2p)(x + 4p)(x2 - 8p2), Journal of Contemporary Applied Mathematics, 13 (1) (2023), 11-23.
7. G. Faltings, Finiteness Theorems for Abelian Varieties over Number Fields, Arithmetic Geometry, 1986.
8. P. A. Griffiths, Introduction to algebraic curves: Translations of mathematical monographs, American Mathematical Society, Providence, RI, 76 (1989).
9. M. Gromov and M. A. Shubin, The Riemann-Roch theorem for elliptic operators, IM Gelfand Seminar part, 1 (1993).
10. M. M. D. Diallo, Explicit family of algebraic points of given degree of the curve family Cq, Journal of Advanced Mathematical Studies, 17 (4) (2024), 444-450.
11. P. Vojta, Siegel’s theorem in the compact case, Annals of Mathematics, 133 (3) (1991),509-548. 
Journal of Finsler Geometry and its Applications

Articles in Press, Accepted Manuscript
Available Online from 18 April 2025

Files

Share

How to cite

Statistics