Rational points on certain elliptic surfaces

Abstract

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 $t\mapsto \varphi (t)$ such that on the surface ${\mathcal{E}}_{f\circ \varphi }$ 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 ${\mathcal{E}}_f$. Next, we consider a surface ${\mathcal{E}}^g:y^2=x^3+g(t)$, 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 $t\mapsto \psi (t)$ such that on the surface ${\mathcal{E}}^{g\circ \psi }$ 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.

16 pages. Submitted for publication

Cited by in corpus (2)

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