On the density of rational points on rational elliptic surfaces

Julie Desjardins

Published 2017-02-06Version 1

Let $\mathscr{E}\rightarrow\mathbb{P}^1_\mathbb{Q}$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base $\mathbb{P}^1_\mathbb{Q}$ (with a section). We show the Zariski-density of the rational points of $\mathscr{E}$ when $\mathscr{E}$ is isotrivial with non-zero $j$-invariant and when $\mathscr{E}$ is non-isotrivial with a fiber of type $II^*$, $III^*$, $IV^*$ or $I^*_m$ ($m\geq0$). Moreover, we produce examples of non-trivial elliptic surfaces whose rational points might not be dense. However, given the result at our disposal, we conjecture that any non-trivial elliptic surface has a dense set of $\mathbb{Q}$-rational points.