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 $g$ and the size of $S$. 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 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 $y^{2}=f(x)$.

中文翻译：

---

关于函数域上等微椭圆曲线的积分点

让$k$是一个有限域并且$L$是曲线的函数场$C/千美元属$g\geq 1$. 在本说明的第一部分中，我们展示了可分离的数量$新元- 恒定椭圆曲线上的积分点$E/L$仅在以下方面有界$g$和大小$新元. 在第二部分中，我们假设$L$是超椭​​圆曲线的函数场$C_{A}:s^{2}=A(t)$， 在哪里$A(t)$是一个无平方的$k$-奇数次多项式。如果$\infty$是地方$L$与无穷远点相关联$C_{A}$，那么我们证明可分离的集合$\{\infty \}$-点可以仅根据$g$并且不依赖于 Mordell-Weil 群$E(L)$. 这是通过限制可分离积分点的数量来完成的$k(t)$在形式的椭圆曲线上$E_{A}:A(t)y^{2}=f(x)$， 在哪里$f(x)$是一个多项式$k$. 此外，我们表明，在一个额外的条件下$A(t)$，椭圆曲线上存在一个“小”高度的可分离积分点$E_{A}/k(t)$确定椭圆曲线的同构类$y^{2}=f(x)$.