Parametrization of Algebraic Points of Low Degrees on the Schaeffer Curve

Year 2021, Volume: 4 Issue: 2, 51 - 55, 31.08.2021

Moussa Fall

https://doi.org/10.33187/jmsm.931258

Cited By: 1

Abstract

In this paper, we give a parametrization of algebraic points of degree at most $4$ over $\mathbb{Q}$ on the schaeffer curve $\mathcal{C}$ of affine equation : $ y^{2}=x^{5}+1 $. The result extends our previous result which describes in [5] ( Afr. Mat 29:1151-1157, 2018) the set of algebraic points of degree at most $3$ over $\mathbb{Q}$ on this curve.

Keywords

Degree of algebraic points, Plan curve, Rational points

References

• [1] N. Bruin, M. Stoll, The Mordell-Weil sieve : proving the nonexistence of Rational points on curves, LMS J. Comp. Math., 13 (2010), 272 -306.
• [2] R. F. Coleman, Effective Chabauty Duke Math. J. 52(3) (1985), 765-770.
• [3] E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Mathematische Annalen, 310 (1998), 447–471.
• [4] M. J. Klassen, E. F. Schaefer, Arithmetic and geometry of the curve x4 = y3 +1, Acta Arithmetica LXXIV.3 (1996) 241-257.
• [5] M. Fall, O. Sall, Ponts alg´ebriques de petit degr´e sur la courbe d’´equation affine y2 = x5 +1, Afr. Mat. 29 (2018) 1151-1157.
• [6] O. Sall, Points alg´ebriques sur certains quotients de courbes de Fermat, C. R. Acad. Sci. Paris S´er I 336 (2003) 117-120.
• [7] S. Siksek, M. Stoll, Partial descent on hyper elliptic curves and the generalized Fermat equation x3 +y4 +z5 = 0, Bulletin of the LMS 44 (2012) 151 -166