ARIZONA WINTER SCHOOL LECTURE NOTES ON p-ADIC AND MOTIVIC INTEGRATION

1.1. The p-adic measure. Let p be a prime number. We consider a field K with a valuation ord : K× → Z, extended to K by ord(0) = ∞. We denote by OK the valuation ring OK = {x ∈ K|ord(x) ≥ 0} and we fix an uniformizing parameter $, that is, an element of valuation 1 in OK . The ring OK is a local ring with maximal ideal MK of OK generated by $. We shall assume the residue field k := OK/MK is finite with q = p elements. We endow K with a norm by setting |x| := q−ord(x) for x in K. We shall furthermore assume K is complete for | |. It follows in particular that the abelian groups (K,+) are locally compact, hence they have a canonical Haar measure μn, unique up to multiplication by a non zero constant, so we may assume μn(O K) = 1. The measure μn is the unique R-valued Borel measure on K which is invariant by translation and such that μn(O K) = 1. For instance the measure of a + $O K is q−mn. For any measurable subset A of K and any λ in K, μn(λA) = |λ|μn(A). More generally, for every g in GLn(K), (1.1.1) μn(gA) = |detg|μn(A). If f is, say, a K-analytic function on A, we set ∫