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
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 $ℤ∕2ℤ$-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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