Non-divisible point on a two-parameter family of elliptic curves

Abstract

Let n be a positive integer and t be a non-zero integer. We consider the two-parameter family of elliptic curves over Q given byWe prove a result of non-divisibility of the point (0, n 3 ) ∈ E n (t)(Q) whenever t is sufficiently large compared to n and t 2 + 3n 2 t + 9n 4 is squarefree. Our work extends to this family of elliptic curves a previous study of Duquesne mainly stated for n = 1 and t > 0.