On the density of rational points on rational elliptic surfaces

Abstract

Let $\mathcal{E}\to {\mathbb{P}}_{\mathbb{Q}}^1$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base ${\mathbb{P}}_{\mathbb{Q}}^1$ (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense set of $\mathbb{Q}$-rational points. In this paper we work on solving the conjecture in case $\mathcal{E}$ is rational by means of geometric and analytic methods. First, we show that for $\mathcal{E}$ rational, the set $\mathcal{E}(\mathbb{Q})$ is Zariski-dense when $\mathcal{E}$ is isotrivial with non-zero $j$-invariant and when $\mathcal{E}$ is non-isotrivial with a fiber of type $I{I}^{\ast }$, $II{I}^{\ast }$, $I{V}^{\ast }$ or $I_m^{\ast }$ ($m\ge 0$). We also use the parity conjecture to prove analytically the density on a certain family of isotrivial rational elliptic surfaces with $j=0$, and specify cases for which neither of our methods leads to the proof of our conjecture.

arXiv comment32 pages. To appear in Acta Arithmetica