The geometric average size of Selmer groups over function fields

Landesman, Aaron

doi:10.2140/ant.2021.15.673

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The geometric average size of Selmer groups over function fields

Aaron Landesman

Vol. 15 (2021), No. 3, 673–709

DOI: 10.2140/ant.2021.15.673

Abstract

We show, in the large $q$ limit, that the average size of $n$ -Selmer groups of elliptic curves of bounded height over $𝔽_{q} (t)$ is the sum of the divisors of $n$ . As a corollary, again in the large $q$ limit, we deduce that $100 %$ of elliptic curves of bounded height over $𝔽_{q} (t)$ have rank $0$ or $1$ .

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Keywords

Selmer groups, arithmetic statistics, function fields, moduli stacks

Mathematical Subject Classification 2010

Primary: 11G05

Milestones

Received: 5 August 2019

Revised: 5 July 2020

Accepted: 7 September 2020

Published: 20 May 2021

Authors

Aaron Landesman
Department of Mathematics Stanford University Stanford, CA United States

Vol. 15, No. 3, 2021