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 $t\mapsto\phi(t)$ such that on the surface $\cal{E}_{f\circ\phi}$ 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 $t\mapsto\psi(t)$ such that on the surface $\cal{E}^{g\circ\psi}$ 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.