Vol. 16, No. 5, 2022

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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

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 2-grading on the Lie algebra of type E6 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 E6, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.

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
Received: 23 December 2020
Revised: 15 July 2021
Accepted: 26 August 2021
Published: 16 August 2022
Jef Laga
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom