Joint IAS/Princeton University Number Theory Seminar

Proving the equidistribution of roots of quadratic congruences, with strong estimates on the Weyl sums, is one of the most spectacular applications of the spectral theory of automorphic forms to arithmetic. See for example Duke, Friedlander and Iwaniec's proof of the equidistribution to prime moduli. Unfortunately the equidistribution of roots of cubic congruences has not seen the entrance of automorphic forms. In this talk I will indicate why I believe this is the case along the way towards deriving spacing results for roots of a cubic congruence, which lead to a large sieve type inequality analogous to the one found by Fouvry and Iwaniec in proving that there are infinitely many primes of the form $n^2 + p^2$.