Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(\mathbb F_p)$. Has there been any work on the asympotitcs of $n_p$ as $p \to \infty$?

More generally, suppose $x,y \in E(\mathbb Q)$ are two linearly independent sections and let them generate subgroups $G_x(p),G_y(p) \subset E(\mathbb F_p)$ for a prime of good reduction. Have the asymptotics of $G_x(p)\cap G_y(p)$ been studied?

This question seems tangentially related.

Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(\mathbb F_p)$. Has there been any work on the asympotitcs of the average of $n_p$ for $p < X$ as $X \to \infty$?

More generally, suppose $x,y \in E(\mathbb Q)$ are two linearly independent sections and let them generate subgroups $G_x(p),G_y(p) \subset E(\mathbb F_p)$ for a prime of good reduction. Have the asymptotics of the average of $G_x(p)\cap G_y(p)$ been studied?

This question seems tangentially related.