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

Jef Laga

doi:10.2140/ant.2022.16.1161

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arXiv · 2008.13158

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

Jef Laga

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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.

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BibTeX RIS

Jef Laga. 2021-09-07. The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3. https://doi.org/10.2140/ant.2022.16.1161

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