Cambridge Journal of Mathematics
Volume 4 (2016)
Coregular spaces and genus one curves
Pages: 1 – 119
A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is freely generated. We show that the orbits of many coregular irreducible representations of algebraic groups where the number of generating 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 Jacobian curves. In particular, we give explicit descriptions of certain moduli spaces of genus one curves with extra structure as quotients by algebraic groups of open subsets of affine spaces.
In most cases, we also describe how the generators of the invariant rings are geometrically manifested, often as coefficients for the Jacobian families of elliptic curves for the genus one curves. We also show how many of the correspondences, including their proofs, are special cases of two general constructions related to Hermitian matrices and cubic Jordan algebras.
In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.