Morphisms and algebraic points on the quotients of Fermat curves and Hurwitz curves

Abstract

In this paper we determine rational morphisms between the Hurwitz curves of affine equation : $ u^{n}v^{l}+v^{n}+u^{l}=0$ and the quotients of Fermat curves of affine equation $v^{m}=u^{\lambda}(u-1)$ where the integers $n > l \geq 1$ are coprime and $m=n^{2}-ln+l^{2}$ and $\lambda \geq 1$. We also give a parametrization of the algebraic points of low degree on the quotient of Fermat curve : $v^{7}=u(u-1)^{2}$. Using these morphisms, we explicitly determine the algebraic points of degree at most $3$ on the Hurwitz curve $ u^{3}v^{2}+v^{3}+u^{2}=0$ birationally isomorphic to the quotient of Fermat curve $v^{7}=u^{2}(u-1)$.