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Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be an ordinary isotrivial curve of $j$-invariant $\neq 0, 1728$.

Also, we are explicitly given two non-torsion $\mathbb{F}_{q}(t)$-points on $E$ denoted by $P_0(t)$, $P_1(t)$. By assumption, their canonical heights $\hat{h}(P_0)$, $\hat{h}(P_1)$ are very small and known.

Since $E$ is ordinary, its endomorphism ring is an order $\mathcal{O}$ in some imaginary quadratic field. Assume that $P_0(t)$, $P_1(t)$ are linearly independent in the Mordell--Weil group $E(\mathbb{F}_{q}(t))$. However, the latter is also an $\mathcal{O}$-module with the natural action.

Is there a method to check if $P_0(t)$, $P_1(t)$ are dependent over $\mathcal{O}$ or not? And if they are, how to find two endomorphisms $\varphi_0, \varphi_1 \in \mathcal{O}$ such that $\varphi_0 \circ P_0(t) + \varphi_1 \circ P_1(t)$ is the identical zero on $E$?

Thanks in advance!

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Let $\mathcal O=\mathbb Z[\tau]=\mathbb Z+\mathbb Z\tau$. Then you're asking if the four points $$ P_0(t),\; \tau\bigl(P_0(t)\bigr),\; P_1(t),\; \tau\bigl(P_1(t)\bigr) $$ are $\mathbb Z$-linearly independent. So you can compute the canonical height regulator matrix for these four points to check for dependence.

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  • $\begingroup$ I am grateful to you. However, I forgot to say that I am interested in the case when $\deg(\tau)$ is known, but huge and not smooth (e.g., prime) in contrast to the little $\hat{h}(P_0)$ and $\hat{h}(P_1)$. Do you have an idea how to compute the canonical heights $\hat{h}(P_0 + \tau(P_1))$ and $\hat{h}(P_1 + \tau(P_0))$? I guess that this is the only obstacle to determine the regulator matrix. $\endgroup$
    – Dimitri Koshelev
    Commented May 8, 2023 at 16:19
  • $\begingroup$ If you can't compute $\tau(P_i)$ explicitly, that makes it difficult. In any case, what you need is the intersection of the images of the sections $P_i$ and $\tau(P_j)$, and also information on how they intersect the singular fibers. Maybe you can do those computations using the geometry without explicitly writing down the sections. $\endgroup$
    – Joe Silverman
    Commented May 8, 2023 at 17:05
  • $\begingroup$ Do you know an example of a set $\{P_i\}_{i=0}^{n-1}$ (where $n \in \mathbb{N}$) of independent points over $\mathbb{Z}$ that are dependent over $\mathbb{Z}[\tau]$ at the same time? As before, I assume that the elliptic $\mathbb{F}_q(t)$-curve $E$ is ordinary isotrivial, $\deg(\tau)$ is huge, but the heights of the $\mathbb{F}_q(t)$-points $P_i$ are small. $\endgroup$
    – Dimitri Koshelev
    Commented May 30, 2023 at 18:55
  • $\begingroup$ @DimitriKoshelev Can't you just take $P_1$ to be pretty much any point of infinite order, and take $P_2=\tau P_1$? Then $P_1$ and $P_2$ are $\mathbb Z$-independent, but they are $\mathbb Z[\tau]$-dependent. $\endgroup$
    – Joe Silverman
    Commented May 31, 2023 at 2:36
  • $\begingroup$ In this case, the canonical height of $P_2$ is equal to $\deg(\tau)$ times the canonical height of $P_1$. This contradicts my assumptions that $\deg(\tau)$ is huge, but the points $P_i$ are short. $\endgroup$
    – Dimitri Koshelev
    Commented May 31, 2023 at 8:19

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