Algebraic Points of Any Degree l with (l ≥ 9) over Q on the Affine Equation Curve C3 (11): y11 = x3(x-1)3

¹ Mathematics and Applications Laboratory (MAL), U.F.R of Sciences and Technologies, Université Assane Seck of Ziguinchor, Ziguinchor, Senegal

² Mathematics and Applications Laboratory (MAL), U.F.R of Sciences and Technologies, Université Assane Seck of Ziguinchor, Ziguinchor, Senegal

³ Mathematics and Applications Laboratory (MAL), U.F.R of Sciences and Technologies, Université Assane Seck of Ziguinchor, Ziguinchor, Senegal

Abstract

In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on curve C 3 ( 11): y 11 = x 3 ( x -1) 3 . This result is a special case of quotients of Fermat curves C r,s ( p ) : y p = x r ( x -1) s , 1 ≤ r , s , r + s ≤ p -1 for p = 11 and r = s = 3. The results obtained extend the work of Gross and Rohrlich who determined the set of algebraic points on C 1 (11)(K) of degree at most 2 on Q.

KeywordsMordell-Weil GroupJacobianGalois ConjugatesAlgebraic Extensionsthe Abel-Jacobi TheoremLinear Systems

Sall, O. (2003) Algebraic Points on Some Quotients of Fermat Curves. Comptes Rendus Mathematique, 336, 117-120. https://doi.org/10.1016/S1631-073X(02)00028-6
Gross, B.H. and Rohrlich, D.E. (1978) Some Results on the Mordell-Weil of the Jacobian of the Fermat Curve. Inventiones Mathematicae, 44, 201-224. https://doi.org/10.1007/BF01403161
Faddeev, D. (1961) On the Divisor Class Groups of Some Algebraic Curves. Soviet Mathematics, 2, 67-69.
Griffiths, P.A. (1989) Introduction to Algebraic Curves. Translations of Mathematical Monographs, Providence. https://doi.org/10.1090/mmono/076

Menu

Menu

Menu

Algebraic Points of Any Degree <i>l</i> with (<i>l</i> ≥ 9) over Q on the Affine Equation Curve <i>C</i><sub>3</sub> (11): <i>y</i><sup>11</sup> = <i>x</i><sup>3</sup>(<i>x</i>-1)<sup>3</sup>

Abstract

References (4)