Vol. 15, No. 3, 2021

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The geometric average size of Selmer groups over function fields

Aaron Landesman

Vol. 15 (2021), No. 3, 673–709
Abstract

We show, in the large q limit, that the average size of n-Selmer groups of elliptic curves of bounded height over 𝔽q(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 𝔽q(t) have rank 0 or 1.

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