# On the density of rational points on rational elliptic surfaces

## TL;DR

This paper investigates the density of rational points on rational elliptic surfaces, proving density in specific cases using geometric and analytic methods, and discusses the conjecture's validity more generally.

## Contribution

It establishes Zariski-density of rational points on certain rational elliptic surfaces through geometric and analytic techniques, advancing understanding of the conjecture.

## Key findings

- Density proven for isotrivial surfaces with non-zero j-invariant
- Density proven for non-isotrivial surfaces with specific fiber types
- Analytic proof of density for surfaces with j=0 in certain families

## Abstract

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 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 $\mathscr{E}$ is rational by means of geometric and analytic methods. First, we show that for $\mathscr{E}$ rational, the set $\mathscr{E}(\mathbb{Q})$ is Zariski-dense 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 $I^*_m$ ($m\geq0$). 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.