arXiv 0705.29552007-05-21math.NT

某些椭圆曲面上的有理点

Rational points on certain elliptic surfaces

Maciej Ulas

备注：: 16 pages. Submitted for publication
DOI：: 10.4064/aa129-2-3

摘要：

令$\mathcal{E}_{f}:y^2=x^3+f(t)x$，其中$f\in\Q[t]\setminus\Q$，让我们假设$\op{deg}f\leq 4$。本文证明了若$\op{deg}f\leq 3$，则存在有理基变换$t\mapstoϕ(t)$，使得曲面$\cal{E}_{f\circϕ}$上存在无挠截面.一个类似的定理是有效的情况下，当$\op{deg}f=4$和存在$t_{0}\in\Q$使得无穷多个有理点位于曲线$E_{t_{0}}:y^2=x^3+f(t_{0})x$。特别地，我们证明了，如果$\op{deg}f=4$和$f$是不是一个甚至多项式，那么有一个合理的点$\cal{E}_{f}$。接下来，我们考虑一个表面$\cal{E}^{g}:y^2=x^3+g(t)$，其中$g\in\Q[t]$是一个monic多项式的次数为6。证明了若多项式$g$不是偶数，则存在有理基变换$t\mapstoψ(t)$使得曲面$\cal{E}^{g\circψ}$上存在无挠截面.此外，如果存在$t_{0}\in\Q$，使得在曲线$E^{t_{0}}:y^2=x^3+g(t_{0})$上有无限多个有理点，则这些$t_{0}$的集合是无限的。我们还提出了一些结果的丢番图方程的形式$x^2-y^3-g(z)=t$，其中$t$是一个变量。

英文摘要：

Let $\mathcal{E}_{f}:y^2=x^3+f(t)x$, where $f\in\Q[t]\setminus\Q$, and let us assume that $\op{deg}f\leq 4$. In this paper we prove that if $\op{deg}f\leq 3$, then there exists a rational base change $t\mapstoϕ(t)$ such that on the surface $\cal{E}_{f\circϕ}$ there is a non-torsion section. A similar theorem is valid in case when $\op{deg}f=4$ and there exists $t_{0}\in\Q$ such that infinitely many rational points lie on the curve $E_{t_{0}}:y^2=x^3+f(t_{0})x$. In particular, we prove that if $\op{deg}f=4$ and $f$ is not an even polynomial, then there is a rational point on $\cal{E}_{f}$. Next, we consider a surface $\cal{E}^{g}:y^2=x^3+g(t)$, where $g\in\Q[t]$ is a monic polynomial of degree six. We prove that if the polynomial $g$ is not even, there is a rational base change $t\mapstoψ(t)$ such that on the surface $\cal{E}^{g\circψ}$ there is a non-torsion section. Furthermore, if there exists $t_{0}\in\Q$ such that on the curve $E^{t_{0}}:y^2=x^3+g(t_{0})$ there are infinitely many rational points, then the set of these $t_{0}$ is infinite. We also present some results concerning diophantine equation of the form $x^2-y^3-g(z)=t$, where $t$ is a variable.

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英文摘要：

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