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On the density of rational points on rational elliptic surfaces

Volume 189 / 2019

Julie Desjardins Acta Arithmetica 189 (2019), 109-146 MSC: 14G05, 14J27, 14J26. DOI: 10.4064/aa170220-23-7 Published online: 17 May 2019

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 towards 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 $\mathit{II}^*$, $\mathit{III}^*$, $\mathit{IV}^*$ or $\mathit{I}^*_m$ ($m\geq0$). 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.

Authors

  • Julie DesjardinsUniversity of Toronto Mississauga
    Mississauga, ON, Canada L5L 1C6
    e-mail

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