Exponential Sums and Polynomial Congruences Along p-adic Submanifolds

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 ${\mathbb{Z}}_p^n$. More precisely, we extend Igusa’s 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 $$ of a $p$-adic analytic submanifold of ${\mathbb{Z}}_p^n$. In addition, we attach a Poincare 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.