Reflection theorems for number rings

Abstract

The Ohno-Nakagawa reflection theorem is an unexpectedly simple identity relating the number of $\GL_2 \ZZ$-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants $D$, $-27D$; it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we provide a framework for generalizing this theorem using a global and local step. The global step uses Fourier analysis on the adelic cohomology $H^1(\AA_K, M)$ of a finite Galois module, modeled after the celebrated Fourier analysis on $\AA_K$ used in Tate's thesis. The local step is combinatorial, more elementary but much more mysterious. We establish reflection theorems for binary quadratic forms over number fields of class number $1$, and for cubic and quartic rings over arbitrary number fields, as well as binary quartic forms over $\ZZ$; the quartic results are conditional on some computational algebraic identities that are probabilistically true. Along the way, we find elegant new results on Igusa zeta functions of conics and the average value of a quadratic character over a box in a local field.