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 ...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<sub>3 </sub>(11): y<sup>11</sup> = x<sup>3</sup> (x-1)<sup>3</sup>. This result is a special case of quotients of Fermat curves C<sub>r,s </sub>(p) : y<sup>p</sup> = x<sup>r</sup>(x-1)<sup>s</sup>, 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<sub>1</sub>(11)(K) of degree at most 2 on Q.展开更多
文摘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<sub>3 </sub>(11): y<sup>11</sup> = x<sup>3</sup> (x-1)<sup>3</sup>. This result is a special case of quotients of Fermat curves C<sub>r,s </sub>(p) : y<sup>p</sup> = x<sup>r</sup>(x-1)<sup>s</sup>, 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<sub>1</sub>(11)(K) of degree at most 2 on Q.
