Exponential Sums and Polynomial Congruences Along p-adic Submanifolds

Details Exponential Sums and Polynomial Congruences Along p-adic Submanifolds Exponential Sums and Polynomial Congruences Along p-adic Submanifolds

2009-10-10

arxiv.org/abs/0910.1887v2

Mathematics

Algebraic Geometry

Several typos were corrected.To Appear in Finite Fields and its Applications

Scientific paper

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.

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