Exponential sums modulo p^m for Deligne polynomials

ALGEBRAIC GEOMETRY & NUMBER THEORY SEMINAR

Let f be a non-constant polynomial in n variables of degree d>1 with integer coefficients. Suppose that g is the homogeneous part of highest degree of f and the projective scheme V(g) associated with g is smooth. In the proof of Weil's conjecture, Deligne showed that if p is a large enough prime then p^{-n}|\sum_{x\in (\ZZ/p\ZZ)^n}\exp(2\pi if(x)/p)|\leq (d-1)^n p^{-n/2}. It is natural to ask about an analogue of Deligne's theorem for exponential sums modulo p^m. In this talk, I will introduce a conjecture on this question and my recent result in this direction.