Reflection theorems for number rings

In 1997, Y. Ohno discovered (quite by accident) a beautiful reflection identity relating the number of cubic rings, equivalently binary cubic forms, of discriminants D and -3D. In the case that D is squarefree, this corresponds to Scholz's 1932 reflection principle comparing the 3-class groups of the quadratic fields Q(√D), Q(√(-3D)). Ohno’s conjectured identity was proved in 1998 by Nakagawa. In my talk, I will present a new and more illuminating method for proving reflection identities of this type, based on Poisson summation on adelic cohomology (in the style of Tate’s thesis). Results from this method include reflection theorems for cubic forms over a general global field, as well as quadratic forms and quartic forms and rings. An application (joint with Brandon Alberts) is to count coclasses sigma in H¹(K, T) of a Galois module T by a quite general discriminant-like invariant.