PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image