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

TL;DR

This paper establishes an upper bound of 3 for the average size of the 2-Selmer group of Jacobians of non-hyperelliptic genus-3 curves with a marked hyperflex point, using orbit-counting and representation theory.

Contribution

It introduces a novel approach linking 2-Selmer groups to integral orbits of an E6 Lie algebra representation, with new constructions of integral representatives and a representation-theoretic interpretation of the Mumford theta group.

Findings

Average 2-Selmer group size is bounded above by 3.

Marked points are the only rational points for a positive proportion of curves.

New methods connect Selmer groups to orbit counts via Lie algebra representations.

Abstract

We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic 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 $\mathbb{Z}/2\mathbb{Z}$-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.