INTRODUCTION to MODEL THEORY of VALUED FIELDS 1. P-Adic Integration 1.1. Counting Solutions in Finite Rings. Consider a System
INTRODUCTION TO MODEL THEORY OF VALUED FIELDS MOSHE KAMENSKY Abstract. These are lecture notes from a graduate course on p-adic and motivic integration (given at BGU). The main topics are: Quantifier elimi- nation in the p-adics, rationality of p-adic zeta functions and their motivic analogues, basic model theory of algebraically closed valued fields, mo- tivic integration following Hrushovski and Kazhdan, application to the Milnor fibration. Background: basic model theory and a bit of algebraic geometry 1. p-adic integration 1.1. Counting solutions in finite rings. Consider a system X of polynomial equations in n variables over Z. Understanding the set X(Z) is, generally, very difficult, so we make (rough) estimates: Given a prime number p > 0, we consider the set X(Z=pkZ) of solutions of X in the finite ring Z=pkZ, viewing them as finer and finer approximations, as k increases. Each such set is finite, and we may hope to gain insight on the set of solutions byun- Z kZ derstanding the behaviour of the sequencePak = #X( =p ), which we 1 k organise into a formal power series PX(T) = k=0 akT , called the Poincaré series for X. Igusa proved: Poincaré series Theorem 1.1.1. The series PX(T) is a rational function of T Example 1.1.2. Assume that X is the equationP x1 = 0. There are then ak = (n-1)k Z kZ n-1 k 1 □ p solutions in =p , and PX(T) = (p T) = 1-pn-1T To outline the proof, we first note that the contribution of a given integer x to the sequence (ai) is determined by the sequence (xk) of its residues k mod p .
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