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arXiv · 2204.12102

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

Remke Kloosterman
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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.

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BibTeXRIS

Remke Kloosterman. 2022-04-26. The average Mordell-Weil rank of elliptic surfaces over number fields. https://doi.org/10.2140/ant.2026.20.1

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