Describing parametric families of algebraic points of degree at most ‘ on the superelliptic curve family Cm defined by y^2 = x^5 + m^2

Abstract

We give an explicit description of the set of algebraic points of arbitrary degree on the family of hyperelliptic curves C_m, studied in [21] by Jeong and al. and defined by the affine equation y² = x⁵ + m², where m is a square-free integer. This family is a special case of the superelliptic family of curves Cq;p;a defined by y^q = x^p + a, with q and p prime numbers and a an integer. This more general family was also studied in [22] by J¸edrzejak, who, using height theory, explicitly determined the set C⁽¹⁾_m (Q) of rational points (see Proposition 1.1). The purpose of this note is to extend those results to algebraic points of arbitrary degree. We first construct an explicit Q-basis of the vector spaces L(l infty) for l in N, and we provide an explicit description of the Mordell--Weil group of rational points on the Jacobian J_m(Q). We then apply a fundamental form of the Abel{Jacobi theorem [15, 17] to characterize principal divisors associated with rational functions under consideration. Finally, by introducing a parameter γ, we explicitly construct a union of families of algebraic points, denoted C_m^(l)(Q).