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

Kloosterman, Remke

doi:10.2140/ant.2026.20.1

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

Remke Kloosterman

Vol. 20 (2026), No. 1, 1–16

DOI: 10.2140/ant.2026.20.1

Abstract

Let $K$ be a finitely generated field over $ℚ$ . Let $𝒳 \to 𝒞 \to ℬ$ be a family of nontrivial elliptic surfaces over $K$ such that the configuration of singular fibers of $𝒳_{b} \to 𝒞_{b}$ is the same for each closed point $b \in | ℬ |$ . Let $r$ be the minimum of the Mordell–Weil rank in this family. Then we show that the locus inside $| ℬ |$ 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 $ℚ$ and prove a similar result for elliptic surfaces over arbitrary number fields.

Keywords

Mordell–Weil rank of elliptic surfaces, families of elliptic surfaces

Mathematical Subject Classification

Primary: 11G05

Secondary: 14G05, 14J27

Milestones

Received: 29 April 2022

Revised: 29 July 2024

Accepted: 20 January 2025

Published: 28 November 2025

Authors

Remke Kloosterman
Dipartimento di Matematica “Tullio Levi-Civita” Università degli Studi di Padova 35121 Padova Italy

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Vol. 20, No. 1, 2026