Binary quartic forms with vanishing J-invariant and the subvariety problem

Stan Xiao, University of Oxford

In a series of groundbreaking papers, M. Bhargava introduced powerful new geometry of numbers methods to arithmetic invariant theory. This enabled him to obtain several remarkable results, including the boundedness of the average rank of elliptic curves and density results for rings and fields of low degree. A limitation in his method is that so far it only works in the case of coregular spaces. In order to push the method further, one must solve the so-called subvariety problem. Perhaps the simplest examples of the subvariety problem are the cases of binary quartic forms with vanishing I or J-invariants. We shall address the latter in this talk.

MC 5417