The average Mordell-Weil rank of elliptic surfaces over number fields

TL;DR

This paper proves Cowan's conjecture by showing that the set of elliptic surfaces with higher Mordell-Weil rank is sparse, establishing that the average rank over number fields is minimal.

Contribution

It confirms Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over number fields and extends the result to arbitrary number fields.

Findings

The locus with rank at least r+1 is sparse.

The average Mordell-Weil rank over number fields is minimal.

Extension of Cowan's conjecture to arbitrary number fields.

Abstract

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside $|\mathcal{B}|$ where the Mordell-Weil rank is at least $r+1$ is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over $\mathbb{Q}$ and prove a similar result for elliptic surfaces over arbitrary number fields.