The quotients of Fermat curves C_{r,s}(p) are studied by SALL who extends the work of Gross and Rohlich. Among these studies are the cases C_{r,s}(11) for r = s = 1. COLY and Sall have explicitly determined the algebraic points of degree at most 3 on for the cases C_{r,s}(11) for r = s = 2. Our work focuses we determine explicitly the algebraic points of a given degree over on the curve C_{4,4}(11) of affine equation y^{11}= x^{4}(x − 1)^{4} which is a special case of Fermat quotient curves. Our study concerns the cases C_{r,s}(11) for r = s = 4. This note completes pevious work of Gassama and Sall who explicitly determined the algebraic points of degree at most three on the even curve. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J_{4,4}(11)() 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)(). The Mordell-Weil group J_{4,4}(11)() 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)() 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 of affine equation y^{11}= x^{4}(x − 1)^{4}, we made an extension of the work of work of Gassama and Sall by explicitly determining the algebraic points of given degree on the curve C_{4,4}(11) and this is what makes this note very interesting.
DOI | 10.11648/j.ajam.20231105.12 |
Published in | American Journal of Applied Mathematics ( Volume 11, Issue 5, October 2023 ) |
Page(s) | 89-94 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Mordell-Weil Group, Jacobian, Galois Conjugates
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APA Style
Mouhamadou Diaby Gassama, Chérif Mamina Coly, Oumar Sall. (2023). Algebraic Points of Any Given Degree on the Affine Equation Curve y11= x4(x − 1)4. American Journal of Applied Mathematics, 11(5), 89-94. https://doi.org/10.11648/j.ajam.20231105.12
ACS Style
Mouhamadou Diaby Gassama; Chérif Mamina Coly; Oumar Sall. Algebraic Points of Any Given Degree on the Affine Equation Curve y11= x4(x − 1)4. Am. J. Appl. Math. 2023, 11(5), 89-94. doi: 10.11648/j.ajam.20231105.12
AMA Style
Mouhamadou Diaby Gassama, Chérif Mamina Coly, Oumar Sall. Algebraic Points of Any Given Degree on the Affine Equation Curve y11= x4(x − 1)4. Am J Appl Math. 2023;11(5):89-94. doi: 10.11648/j.ajam.20231105.12
@article{10.11648/j.ajam.20231105.12, author = {Mouhamadou Diaby Gassama and Chérif Mamina Coly and Oumar Sall}, title = {Algebraic Points of Any Given Degree on the Affine Equation Curve y11= x4(x − 1)4}, journal = {American Journal of Applied Mathematics}, volume = {11}, number = {5}, pages = {89-94}, doi = {10.11648/j.ajam.20231105.12}, url = {https://doi.org/10.11648/j.ajam.20231105.12}, eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajam.20231105.12}, abstract = {The quotients of Fermat curves Cr,s(p) are studied by SALL who extends the work of Gross and Rohlich. Among these studies are the cases Cr,s(11) for r = s = 1. COLY and Sall have explicitly determined the algebraic points of degree at most 3 on for the cases Cr,s(11) for r = s = 2. Our work focuses we determine explicitly the algebraic points of a given degree over on the curve C4,4(11) of affine equation y11= x4(x − 1)4 which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. This note completes pevious work of Gassama and Sall who explicitly determined the algebraic points of degree at most three on the even curve. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)() 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)(). The Mordell-Weil group J4,4(11)() 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)() 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 of affine equation y11= x4(x − 1)4, we made an extension of the work of work of Gassama and Sall by explicitly determining the algebraic points of given degree on the curve C4,4(11) and this is what makes this note very interesting.}, year = {2023} }
TY - JOUR T1 - Algebraic Points of Any Given Degree on the Affine Equation Curve y11= x4(x − 1)4 AU - Mouhamadou Diaby Gassama AU - Chérif Mamina Coly AU - Oumar Sall Y1 - 2023/11/01 PY - 2023 N1 - https://doi.org/10.11648/j.ajam.20231105.12 DO - 10.11648/j.ajam.20231105.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 89 EP - 94 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20231105.12 AB - The quotients of Fermat curves Cr,s(p) are studied by SALL who extends the work of Gross and Rohlich. Among these studies are the cases Cr,s(11) for r = s = 1. COLY and Sall have explicitly determined the algebraic points of degree at most 3 on for the cases Cr,s(11) for r = s = 2. Our work focuses we determine explicitly the algebraic points of a given degree over on the curve C4,4(11) of affine equation y11= x4(x − 1)4 which is a special case of Fermat quotient curves. Our study concerns the cases Cr,s(11) for r = s = 4. This note completes pevious work of Gassama and Sall who explicitly determined the algebraic points of degree at most three on the even curve. It seems that the finiteness of the Mordell-Weil group of rational points of the Jacobien J4,4(11)() 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)(). The Mordell-Weil group J4,4(11)() 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)() 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 of affine equation y11= x4(x − 1)4, we made an extension of the work of work of Gassama and Sall by explicitly determining the algebraic points of given degree on the curve C4,4(11) and this is what makes this note very interesting. VL - 11 IS - 5 ER -