The quotients of Fermat curves C_{r,s}(p) are studied by Oumar SALL. Among these studies are the cases C_{r,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 C_{r,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 C_{4,4}(11) which is a special case of Fermat quotient curves. Our study concerns the cases C_{r,s}(11) for r = s = 4. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J_{4,4}(11)(Q) is an essential condition. So to determine the algebraic points on the curve C_{4,4}(11) we need a finiteness of the Mordeill-Weill group of rational points of the Jacobien J_{4,4}(11)(Q). The Mordell-Weil group J_{4,4}(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 J_{4,4}(11)(Q) of the Jacobien of C_{4,4}(11) the Abel-Jacobi theorem and the study of linear systems on the curve C_{4,4}(11). The result obtained concerns some quotients of Fermat curves. Indeed, the curve C_{4,4}(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 C_{4,4}(11) on Q and the cubic points on the curve C_{4,4}(11) on Q.
Published in | American Journal of Applied Mathematics (Volume 10, Issue 4) |
DOI | 10.11648/j.ajam.20221004.15 |
Page(s) | 160-175 |
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Planes Curves, Degree of Algebraic Points, Jacobien
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APA Style
Mouhamadou Diaby Gassama, Oumar Sall. (2022). Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. American Journal of Applied Mathematics, 10(4), 160-175. https://doi.org/10.11648/j.ajam.20221004.15
ACS Style
Mouhamadou Diaby Gassama; Oumar Sall. Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4. Am. J. Appl. Math. 2022, 10(4), 160-175. doi: 10.11648/j.ajam.20221004.15
@article{10.11648/j.ajam.20221004.15, author = {Mouhamadou Diaby Gassama and Oumar Sall}, title = {Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4}, journal = {American Journal of Applied Mathematics}, volume = {10}, number = {4}, pages = {160-175}, doi = {10.11648/j.ajam.20221004.15}, url = {https://doi.org/10.11648/j.ajam.20221004.15}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221004.15}, abstract = {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.}, year = {2022} }
TY - JOUR T1 - Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11=x4(x-1)4 AU - Mouhamadou Diaby Gassama AU - Oumar Sall Y1 - 2022/08/18 PY - 2022 N1 - https://doi.org/10.11648/j.ajam.20221004.15 DO - 10.11648/j.ajam.20221004.15 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 160 EP - 175 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20221004.15 AB - 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. VL - 10 IS - 4 ER -