Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $P_0$, $P_1$ be two non-zero $\mathbb{F}_{\!q}$-points on $E$. Is it true that there is always a lift $\mathcal{E}$ of $E$ over the function field $\mathbb{F}_{\!q}(t)$ and there are independent $\mathbb{F}_{\!q}(t)$-points $P_0(t)$, $P_1(t)$ on $\mathcal{E}$ such that $P_0(\alpha) = P_0$ and $P_1(\alpha) = P_1$ for some $\alpha \in \mathbb{F}_{\!q}$? Of course, I require that $P_0(t)$, $P_1(t)$, and $\mathcal{E}$ are of little heights, i.e., they are independent of $q$.
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$\begingroup$ How could $P_0(t)$ etc. ever be independent of $q$ when all the data in your problem depends on it? $\endgroup$– WojowuCommented Jun 11, 2023 at 9:36
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$\begingroup$ Their coefficients may depend on $q$, but their heights are restricted above by a constant. $\endgroup$– Dimitri KoshelevCommented Jun 11, 2023 at 9:44
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$\begingroup$ You can just choose $\mathcal{E}$ to be the "constant" family, namely $E\times_{\text{Spec}\ \mathbb{F}_q} \text{Spec}\ \mathbb{F}_q(t)$. Then every $\mathbb{F}_q$-point of $E$ lifts to a "constant" $\mathbb{F}_q(t)$-point of $\mathcal{E}$. $\endgroup$– Jason StarrCommented Jun 11, 2023 at 11:46
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$\begingroup$ I require that the points $P_0(t)$, $P_1(t)$ are independent. In particular, they have to be non-torsion. Hence, your approach does not work. $\endgroup$– Dimitri KoshelevCommented Jun 11, 2023 at 11:51
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$\begingroup$ Fine: take the Weirrstrass model, and use a pencil of plane cubics containing the two points that includes the Weirrstrass model as one member of the pencil. The intersection matrix shows the lifts are independent. $\endgroup$– Jason StarrCommented Jun 11, 2023 at 19:21
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