Local Zeta Functions Supported on Analytic Submanifolds and Newton Polyhedra

The local zeta functions (also called Igusa’s zeta functions) over p-adic fields are connected with the number of solutions of congruences and exponential sums mod p^{m}. These zeta functions are defined as integrals over open and compact subsets with respect to the Haar measure. In this paper, we introduce new integrals defined over submanifolds, or more generally, over non-degenerate complete intersection varieties, and study their connections with some arithmetical problems such as estimation of exponential sums mod p^{m}. In particular we extend Igusa’s method for estimating exponential sums mod p^{m} to the case of exponential sums mod p^{m} along non-degenerate smooth varieties.