Coregular spaces and genus one curves

TL;DR

This paper establishes a correspondence between orbits of certain coregular representations and genus one curves with additional geometric data, providing a new perspective on parametrizations related to elliptic curves.

Contribution

It introduces a novel link between coregular space orbits and genus one curves with bundles or points on Jacobians, expanding the understanding of orbit classifications.

Findings

Orbit parametrizations correspond to genus one curves with line bundles and points.

Framework applicable over arbitrary fields, not just algebraically closed.

Lays groundwork for future work on Selmer group averages.

Abstract

A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is free. In this paper, we show that the orbits of many coregular irreducible representations where the number of invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobians. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.