Article dans une revue
Titre :
Algebraic Points of Degree at Most 3 on the Affine Equation Curve y^11=x^4(x-1)^4
Auteurs :
Mouhamadou Diaby Gassama, Oumar Sall
Résumé :
The quotients of Fermat curves Cr,s(p) are studied by Oumar SALL. Among these studies are the cases Cr,s(11) for r = s = 1. Mamina COLY and Oumar SALL have explicitly determined the algebraic points of degree at most 3 on Q for the cases Cr,s(11) for r = s = 2. Our work focuses on determining explicitly the algebraic points of degree at most 3 on Q on the curve C4,4(11) which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)(Q) is an essential condition. So to determine the algebraic points on the curve C4,4(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J4,4(11)(Q). The Mordell-Weil group J4,4(11)(Q) of rational points of the Jacobien is finite according to Faddev. Our note is in this framework. Our essential tools in this note are the Mordell-Weil group J4,4(11)(Q) of the Jacobien of C4,4(11) the Abel-Jacobi theorem and the study of linear systems on the curve C4,4(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C4,4(11) which is the subject of our study, the set of algebraic points of degree at most 3 on Q has been determined in an explicit way, to achieve this we have determined the quadratic points on the curve C4,4(11) on Q and the cubic points on the curve C4,4(11) on Q.
Journal :
American Journal of Applied Mathematics
Volume :
10
Issue :
4
Pages ou Numéro article :
160-175
Année :
2022
DOI :
https://doi.org/10.11648/j.ajam.20221004.15
Date de publication :
18 August 2022
Lien de la publication :
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