From plane waves to quadratic reciprocity: what is Gauss's theorem really about?

Gauss's law of quadratic reciprocity is an arithmetic identity that usually emerges at the end of ingenious manipulations of finite sums/congruences/lattice counts. This talk will explain quadratic reciprocity conceptually. The starting point is the Fourier transform and Schrödinger propagation. From this perspective, the finite Fourier-transform heroics that are the engine of the classical proof are replaced by geometry: when one transports an oscillatory state through caustics, the state picks up an accompanying Maslov phase correction.

This phase correction is already present in the simplest wave packet solutions of the Schrödinger equation in one spatial dimension. I will first describe a classical real-Fourier picture in which quadratic reciprocity emerges from the behavior of such wave packets under propagation. This gives the theorem an *optical interpretation*: reciprocity reflects a consistency law for quadratic phase transport. But philosophically this interpretation is not yet fully satisfactory: one still seems to need to compute the value of a certain quadratic phase constant, and this threatens to collapse the story back into the traditional Gauss-sum computation.

I will then explain the way out of this trap. In the global Weyl representation, the local phase constants are no longer numerically isolated, but part of a single rigid local-to-global structure. Reciprocity is then recovered by the compatibility of Fourier propagation across all places at once, without needing to calculate any non-trivial constants directly from finite Fourier analysis.