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The average size of the
2-Selmer group of a family of non-hyperelliptic curves of
genus 3
Jef Laga |
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Vol. 16 (2022), No. 5, 1161–1212
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Abstract |
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We show that the average size of the -Selmer group of the family of Jacobians of nonhyperelliptic genus- curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by . We achieve this by interpreting -Selmer elements as integral orbits of a representation associated with a stable -grading on the Lie algebra of type 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 , and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves. |
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Keywords
arithmetic statistics, non-hyperelliptic curves, rational
points, Selmer groups, geometry of numbers, Mumford theta
groups
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Mathematical Subject Classification
Primary: 14G25, 14H45
Secondary: 11E72, 14G05, 14H40
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Milestones
Received: 23 December 2020
Revised: 15 July 2021
Accepted: 26 August 2021
Published: 16 August 2022
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Authors |
| Jef Laga | |
| Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge United Kingdom |
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