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. 
Shamgar Gurevich (UC Berkeley), Ronny Hadani (U of Chicago), Roger   Howe (Yale)