Morphisms and Algebraic Points on the Quotients of the Fermat Quintic - Fall, Moussa (Indonesian Journal of Mathematics and Applications, 2024)
The purpose of this paper is to determine morphisms and algebraic points of low degree on the quotients of the Fermat quintic. An algebraic point of degree at most $3$ is called an algebraic point of low degree. The quotients of the Fermat quintic is a family curves $\mathcal{C}_{rs}(5) : v^5 =u^r(u-1)^s$ where $r$ and $s$ are integers such that: $1< r, s, r+s< 5$. We determine explicitly the morphisms between the curves $\mathcal{C}_{rs}(5)$. By applying Abel Jacobi's theorem and using Riemann-Roch spaces, we give a parametrization of algebraic points of low degree on the special curve $\mathcal{C}_{1,1}(5)$. Birational morphisms allow us to determine the set of algebraic points of low degree on each curve of the family curves $\mathcal{C}_{rs}(5)$.