Gauss composition, Bhargava cubes, and more

Sameera Vemulapalli, Princeton University
Fine Hall 110

A classical fact in number theory states that the narrow class group of a quadratic number field of discriminant D is in bijection with the SL_2(Z) equivalence classes of integral binary quadratic forms of discriminant D. In this talk, I will explain a generalization of this presented in Bhargava's dissertation. We define a composition law on 2 by 2 by 2 cubes of integers, and from this recover the above fact. Additionally, a similar composition law generalizes the above fact to cubic fields. Time permitting, we will also discuss the parametrization of rings of integers of cubic, quartic, and quintic number fields, as this plays an essential role in generalizing Gauss composition.