American Journal of Applied Mathematics

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Algebraic Points of Any Given Degree on the Affine Equation Curve y11= x4(x − 1)4

Received: Sep. 29, 2022    Accepted: Jul. 24, 2023    Published: Nov. 01, 2023
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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.

DOI 10.11648/j.ajam.20231105.12
Published in American Journal of Applied Mathematics ( Volume 11, Issue 5, October 2023 )
Page(s) 89-94
Creative Commons

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.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Mordell-Weil Group, Jacobian, Galois Conjugates

References
[1] C. M. Coly, O. Sall, points algébriques de degré au-plus 3 sur la courbe d’équation affine y11= x2(x − 1)2, Annales Mathématiques Africaines Volume 8 (2020) pp. 27-32.
[2] O. Debarre, R. Fahlaoui, Abelian varieties and curves in W (C) and points of bounded degree on algebraic curves, Compositio Math. 88 (1993) 235-249.
[3] O. Debarre, M. Klassen, Points of low degree on smooth plane curves, J. Reine Angew. Math. 446 (1994) 81-87.
[4] Diophantine approximation on Abelian varieties, Ann. Math. 133 (1991) 549-576.
[5] D. Faddeev, On the divisor class groups of some algebraic curves, Dokl. Akad. Nauk SSSR 136 (1961) 296-298. English translation : Soviet Math. Dokl. 2 (1) (1961) 67- 69.
[6] G. Faltings, Endlichkeitsätze für abelsch Varietäten über Zahlkörpen, Invent. Math. 73 (1983) 349-366.
[7] B. Gross, D. Rohrlich, Some results on the Mordell-Weil group of the Jacobian of the Fermat curve, Invent. Math. 44 (1978) 201-224.
[8] M. Klassen, P. Tzermias, Algebraic points of low degree on the Fermat quintic, Acta Arith. 82 (4) (1997) 393-401.
[9] Mouhamadou Diaby Gassama, Oumar Sall. Algebraic Points of Degree at Most 3 on the Affine Equation Curve y11 = x4(x − 1)4. American Journal of Applied Mathematics. Vol. 10, No. 4, 2022, pp. 160-175. doi: 10.11648/j.ajam.20221004.15.
[10] O. Sall, Points algébriques de petit degré sur les courbes de Fermat, C. R. Acad. Sci. Paris Sér. I 330 (2000) 67-70.
[11] O. Sall, Points cubiques sur la quartique de Klein, C. R. Acad. Sci. Paris Sér. I 333 (2001) 931-934.
[12] O. Sall, algebraic points on some Fermat curves and some quotients of Fermat curves : Progress, African Journal of Mathematical Physics Volume 8(2010) 79-83.
[13] O. Sall, M. Fall, points algébriques de petits degrés sur les courbes d’équations affines y3n= x(x−1)(x−2)(x−3), Annales Mathématiques Africaines (2015).
[14] J. P. Serre, Lecture of Mordell weil theorem. the translated from the french and edited by martin brows from notes by michel Waldschmidt. Aspects of mathematics, E15 (1989).
[15] P. Tzermias, Algebraic points of low degree on the Fermat curve of degree seven, Manuscriptc Math. 97 (4) (1998) 483-488.
[16] P. Tzermias, Torsion parts of Mordell-Weil groups of Fermat Jacobians, Internat. Math. Res. Notices 7 (1998) 359-369.
[17] Siegel’s theorem in the compact case, Ann. Math. 133 (1991) 509-548.
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    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

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    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

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    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

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  • @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}
    }
    

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    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
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    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
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    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  - 

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Author Information
  • Mathematics and Applications Laboratory, Faculty of Science and Technology, Assane Seck University in Ziguinchor, Ziguinchor, Senegal

  • Mathematics and Applications Laboratory, Faculty of Science and Technology, Assane Seck University in Ziguinchor, Ziguinchor, Senegal

  • Mathematics and Applications Laboratory, Faculty of Science and Technology, Assane Seck University in Ziguinchor, Ziguinchor, Senegal

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