论文标题
在功能场的各向同性椭圆曲线上的积分点上
On integral points on isotrivial elliptic curves over function field
论文作者
论文摘要
令$ k $为有限字段，$ l $是曲线$ c/k $ g \ geq 1 $的功能字段。 在本说明的第一部分中，我们表明，在恒定的椭圆曲线$ e/l $上可分开的$ s $ integral点的数量仅以$ g $，大小为$ s $和mordell-weil group $ e（l）$的排名。 在第二部分中，我们假设$ l $是过纤维曲线的功能字段$ c_a：s^2 = a（t）$，其中$ a（t）$是无正方形的$ k $ - 奇数奇数。如果$ \ infty $是$ c_a $的点相关的$ l $的位置，那么我们证明一组可分离的$ \ {\ {\ infty \} $ - 点可以仅以$ g $的限制，并且似乎并不依赖于mordell-weil组$ e（l）$。这是通过在$ e_a的椭圆曲线上限制$ k（t）$上可分离积分的数量来完成的：a（t）y^2 = f（x）$，其中$ f（x）$是$ k $的多项式。此外，我们表明，在$ a（t）$的额外条件下，存在椭圆曲线上的“小”高度的可分离积分$ e_a/k（t）$确定椭圆形曲线的同构类别$ y^2 = f（x）$。
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 $y^2=f(x)$.
