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Non divisible point on two-parameters family of elliptic curves

publié le

Vendredi 23 octobre 2020
Valentin Petit
(Université de Franche-Comté)

This work is focused on a family of two-parameters elliptic curves. The family considered is a generalization of the Washington’s family. For a point to be among the generator(s) of an elliptic curve, it must be non divisible. In this presentation, we exhibit a point on the curve for which this requirement is satisfied.
More precisly, let $n \in \mathbb{N}^{*}$ and $t \in \mathbb{Z}_{\neq 0}$, we consider the elliptic curve $E : y^2=x^3+tx^2-n^2(t+3n^2)x+n^6$ defined over $ \mathbb{Q}$. The element $(0,n^3)$ is a point of $E(\mathbb{Q})$ of infinite order for all $n \in \mathbb{N}^{*}, \ t \in \mathbb{Z}_{\neq 0}$. Under mild conditions, we proved that this point is not divisible on $E(\mathbb{Q})$.