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 pm of a p-adic analytic submanifold of Zp . 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 pm of a p-adic analytic submanifold of Zp . In addition, we attach a Poincaré 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. 1. Introduction LetK be a p-adic …eld, i.e. [K : Qp] <1. Let RK be the valuation ring ofK, PK the maximal ideal of RK , andK = RK=PK the residue …eld ofK. The cardinality of the residue …eld of K is denoted by q, thus K = Fq. For z 2 K, ord(z) 2 Z[f+1g denotes the valuation, jzjK = q ord(z) the p-adic norm, and ac z = z ord(z) the angular component of z, where is a …xed uniformizing parameter of RK . Let fi 2 K[[x1; : : : ; xn]] be a formal power series for i = 1; : : : ; l, with l 2, and put x = (x1; : : : ; xn). Let U be an open and compact subset of K. Assume that each series fi converges on U . We set V (l 1) := V (l 1) (K) := fx 2 U j f1(x) = = fl 1(x) = 0g and assume that V (l 1) is a non-empty closed submanifold of U , with dimension m := n l + 1 1, which implies that n l. We assume that fl is not identically zero on V (l 1) and that fl has a zero on V (l .We consider on V (l 1) an analytic di¤erential form of degree m, and denote the measure induced on V (l 1) as j j. Later on, we specialize to a Gel’fand-Leray form GL on V (l . Let : K ! C be a Bruhat-Schwartz function with support in U . Let ! be a quasicharacter of K . To these data we associate the following local zeta function: Z (!; V (l ; fl) := Z (!; f1; : : : ; fl; )