On integral points on isotrivial elliptic curves over function field

Let $k$ be a finite field and $L$ be the function field of a curve $C/k$ of genus $g\geq 1$. In the first part of this note, we show that the number of separable $S$-integral points on a constant elliptic curve $E/L$ is bounded solely in terms of $g$, the size of $S$ and the rank of the Mordell-Weil group $E(L)$. In the second part, we assume that $L$ is the function field of a hyperelliptic curve $C_A:s^2=A(t)$, where $A(t)$ is a square-free $k$-polynomial of odd degree. If $\infty$ is the place of $L$ associated to the point at infinity of $C_A$, then we prove that the set of separable $\{\infty\}$-points can be bounded solely in terms of $g$ and does not seem to depend on the Mordell-Weil group $E(L)$. This is done by bounding the number of separable integral points over $k(t)$ on elliptic curves of the form $E_A:A(t)y^2=f(x)$, where $f(x)$ is a polynomial over $k$. Additionally, we show that, under an extra condition on $A(t)$, the existence of a separable integral point of "small" height on the elliptic curve $E_A/k(t)$ determines the isomorphism class of the elliptic curve $y^2=f(x)$.

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