Group, Lie and Number Theory

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)