Towards a higher composition law of binary quartic forms

Composition identities play an important role in various questions about Diophantine equations such as representation of integers by binary forms and arithmetic of elliptic curves. The classic example is that of Gauss composition of binary quadratic forms which is significant also due to its connection with class groups of quadratic rings. In his PhD thesis, Manjul Bhargava proved a higher composition law on 2x2x2 cubes from which Gauss composition follows as a special case. The composition of binary cubic forms was also derived by Bhargava building on earlier work by Delone-Faddeev.

A natural question one might ask is - ''Is there a composition law for binary quartic forms?'' In this talk, we will formulate in more precise terms what a composition law of binary quartic forms might look like based on some known results. We will introduce the ideas from invariant theory that find application in this problem and if time permits, we will discuss the connection with the arithmetic of elliptic curves.