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

TL;DR

This paper investigates the non-divisibility of a specific rational point on a two-parameter family of elliptic curves, extending previous work and providing new insights into their arithmetic properties.

Contribution

It introduces new results on the non-divisibility of a rational point on a generalized two-parameter elliptic curve family, expanding prior studies.

Findings

Established conditions for non-divisibility of the point (0, n^3)

Extended previous results to a broader two-parameter family

Connected the work to recent studies on elliptic surfaces

Abstract

Let n be a positive integer and t a non-zero integer. We consider the elliptic curve over Q given by E : y 2 = x 3 + tx 2 -- n 2 (t + 3n 2)x + n 6. It is a special case of an elliptic surface studied recently by Bettin, David and Delaunay [2] and it generalizes Washington's family. The point (0, n 3) belongs to E(Q) and we obtain some results about its nondivisibility in E(Q). Our work extends to this two-parameter family of elliptic curves a previous study of Duquesne (mainly stated for n = 1 and t > 0).