Geometry-of-numbers methods over global fields II: Coregular representations

TL;DR

This paper develops geometry-of-numbers techniques to count orbits in coregular representations over global fields, applying them to analyze average ranks and Selmer groups of elliptic and hyperelliptic curves.

Contribution

It introduces new geometry-of-numbers methods for counting orbits in coregular representations over global fields and applies these to arithmetic statistics of curves.

Findings

Bounded average ranks of elliptic curves over global fields

Determined average sizes of Selmer groups for hyperelliptic Jacobians

Extended counting techniques to fields of characteristic not 2, 3, or 5

Abstract

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field $F$ of characteristic not $2$, $3$ or $5$.