Igusa zeta functions and motivic generating series

Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen december 2004 ii Introduction Let f be a polynomial in Z[x], x = (x 1 ,. .. , x m), defining a hypersurface X 0 in A m Z. Igusa's Monodromy Conjecture predicts an intriguing relationship between the arithmetic properties of f , and the topology of the (not necessarily reduced) complex hypersurface X 0 = X 0 × Spec Z Spec C in A m C , defined by f = 0. More specifically, the conjecture links the asymptotic behaviour of the number of solutions of f ≡ 0 mod p n as n tends to ∞, to the topology of the singularities of X 0 , for almost all prime numbers p. The best-known instance of interplay between the arithmetic properties of a variety X over Z, and its complex topology, is probably given by the Weil Conjectures [108]: one of the assertions says that, if X is smooth and proper over a non-empty open subscheme U of Spec Z, and (p) ∈ U , the Betti numbers of X × Spec Z Spec C are closely related to the number of F p n-rational points on X (see for instance [56][85]). Although counting rational points over F p n and over Z/p n are a priori two completely different operations, we will explain in a future project that there exists a tighter connection than one might expect at first glance. A sketch of this project is given in the final chapter 7. To lift a tip of the veil: intuitively, the second operation amounts to counting rational points over field extensions, not on X 0 itself, but on the nearby fiber. Let us give an exact statement of the Monodromy Conjecture. Fix a prime number p. The arithmetic properties of f are encoded in the Igusa Poincaré series Q p (T), as follows: for each integer n ≥ 0, we define N n to be |X 0 (Z/p n+1)|, i.e. the number of solutions of f = 0 over the ring Z/p n+1. Then the Igusa Poincaré series is defined as Q p (T) = n≥0 N n T n ∈ Z[[T ]]. Writing Q p (T) in terms of a p-adic integral, and applying resolution of singularities to compute it, Igusa showed that Q p (T) is actually an element of the subring …