  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 $% p^{m}$ 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.
