The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

Laga, Jef

doi:10.2140/ant.2022.16.1161

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The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3

Jef Laga

Vol. 16 (2022), No. 5, 1161–1212

DOI: 10.2140/ant.2022.16.1161

Abstract

We show that the average size of the $2$ -Selmer group of the family of Jacobians of nonhyperelliptic genus- $3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$ . We achieve this by interpreting $2$ -Selmer elements as integral orbits of a representation associated with a stable $ℤ ∕ 2 ℤ$ -grading on the Lie algebra of type $E_{6}$ and using Bhargava’s orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_{6}$ , and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

Keywords

arithmetic statistics, non-hyperelliptic curves, rational points, Selmer groups, geometry of numbers, Mumford theta groups

Mathematical Subject Classification

Primary: 14G25, 14H45

Secondary: 11E72, 14G05, 14H40

Milestones

Received: 23 December 2020

Revised: 15 July 2021

Accepted: 26 August 2021

Published: 16 August 2022

Authors

Jef Laga
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom

Vol. 16, No. 5, 2022