Vol. 16, No. 5, 2022

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 6, 1327–1546
Issue 5, 1025–1326
Issue 4, 777–1024
Issue 3, 521–775
Issue 2, 231–519
Issue 1, 1–230

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
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
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 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.

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