Help Privacy Policy Disclaimer
  Advanced SearchBrowse




Journal Article

On the density of rational points on rational elliptic surfaces


Desjardins,  Julie
Max Planck Institute for Mathematics, Max Planck Society;

External Resource
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)

(Preprint), 395KB

Supplementary Material (public)
There is no public supplementary material available

Desjardins, J. (2019). On the density of rational points on rational elliptic surfaces. Acta Arithmetica, 189(2), 109-146. doi:10.4064/aa170220-23-7.

Cite as: https://hdl.handle.net/21.11116/0000-0005-A737-1
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 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 onjecture.