In seminal work, Bhargava found many generalizations of Gauss's composition law on binary quadratic forms. These generalizations take the form of parametrizing the orbits of the integer points of a reductive group G on a lattice in a prehomogeneous vector space V for G. The orbits are parametrized by interesting arithmetic data. I will explain how one can obtain twisted versions of some of these results of Bhargava. The key idea involves "lifting" elements in the open orbit for the action of G on V to elements in the minimal nonzero orbit of another prehomogeneous vector space (G',V').
Speaker(s): Aaron Pollack (Stanford University)