Quadratic Reciprocity and the Sign of the Gauss Sum via the Finite Weil Representation

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. 0. Introduction Two basic results due to Gauss are the quadratic reciprocity law and the sign of the Gauss sum [8]. The first concerns the identity (0.1) ( p q )( q p ) = (−1) p−1 2 q−1 2 , where p, q are two distinct odd prime numbers and ( · p ) (respectively ( · q ) ) is the Legendre symbol modulo p (respectively q), i.e., ( x p ) = 1 if x is a square modulo p and −1 otherwise. The latter result asserts that