Exponential Sums and Polynomial Congruences Along p-adic Submanifolds

TL;DR

This paper extends Igusa's stationary phase method to estimate exponential sums and count solutions of polynomial congruences along p-adic submanifolds, establishing rationality of associated Poincare series and providing geometric bounds.

Contribution

It introduces a novel extension of Igusa's method to p-adic submanifolds and analyzes polynomial congruences in this context, including rationality results.

Findings

Extended Igusa's stationary phase method to p-adic submanifolds

Established rationality of Poincare series for solution counts

Derived geometric bounds for polynomial congruence solutions

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 $ \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.