Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation

TL;DR

This paper presents new proofs of quadratic reciprocity and the Gauss sum sign by linking them to the Weil representation and Fourier transform actions in finite fields.

Contribution

It introduces a novel approach connecting classical number theory results to the finite Weil representation and Fourier analysis.

Findings

Quadratic reciprocity law derived via Weil representation

Sign of Gauss sum explained through Fourier transform and Weyl element

New proofs simplify understanding of fundamental number theory results

Abstract

We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p,q and pq.