Co-regular Spaces and Genus One Curves

Some examples of coregular spaces were mentioned last time binary quadratic forms, binary cubic forms etc. Clearly, any pre-homogeneous space is a vector space. Coregular spaces were first classified by Littelman (1989) for semisimple irreducible representations; there are around 50 of them including infinite families. A lot of these spaces come from Vinberg theory, something that’ll be discussed over the next couple of weeks, but there are others as well. In particular, several coregular spaces are related to genus 1 curves as the title of this talk suggests. We are interested in the K-orbits of these spaces, VK/GK where K is any field of characteristic different from 2, 3. Our prototypical example will be Q. We see that GK\VK 1–1 ↔ {Genus 1 curves + extra data}