Exponential sums and polynomial congruences along p-adic submanifolds

Abstract

In this article, we consider the estimation of exponential sums along the points of the reduction mod p m of a p-adic analytic submanifold of Z p n . More precisely, we extend Igusaʼs stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod p m of a p-adic analytic submanifold of Z p n . In addition, we attach a Poincaré series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the corresponding polynomial congruences.