Coregular spaces and genus one curves
Algebraic Geometry
2013-06-20 v1 Number Theory
Representation Theory
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.
Cite
@article{arxiv.1306.4424,
title = {Coregular spaces and genus one curves},
author = {Manjul Bhargava and Wei Ho},
journal= {arXiv preprint arXiv:1306.4424},
year = {2013}
}
Comments
Sections 2 and 3 form an extended introduction containing the statements of the main theorems and some of the ideas behind the constructions