Introduction to Model Theory of Valued Fields

These are lecture notes from a graduate course on p-adic and motivic integration (given at BGU). Themain topics are: Quantifier elimination in the p-adics, rationality of p-adic zeta functions and their motivic analogues, basic model theory of algebraically closed valued fields, motivic 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 systemX 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/pZ) of solutions of X in the finite ring Z/pZ, 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 by understanding the behaviour of the sequence ak = #X(Z/pZ), which we 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 equation x1 = 0. There are then ak = p(n−1)k solutions in Z/pZ, and PX(T) = ∑ (pn−1T)k = 1 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 mod p. Each such sequence has the property that πk,l(xk) = xl, where πk,l : Z/pZ −→ Z/pZ for k ⩾ l is the residue map. The set of sequences (xk)with this property forms a ring (with pointwise operations), called the ring of p-adic integers, Zp. If the first k entries of such a sequence are 0, one p-adic integers may view this sequence as “close to 0, up to the k-th approximation”. This notion of closeness can be formalised by defining the absolute value |x| of an absolute value element x = (xi) to be p−k, where k is the smallest i for which xi ̸= 0 (and |0| = 0). The number k is called the valuation vp(x) of x. Thus, x is k-close valuation to 0 if |x| ⩽ p−k (one could choose a different base instead of p to obtain the same topology; the motivation for choosing p will soon become apparent.) It is easy to check that Zp is a local ring whose maximal ideal is the set of 1