Poles of maximal order of Igusa zeta functions

Abstract:

Igusa’s p-adic zeta function Z(s) attached to a polynomial f in N variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation f=0 modulo powers of a prime p. It is expressed as a p-adic integral, and Igusa proved that it is rational in p^{-s} using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of Z(s) is at most N, the number of variables in f. In 1999, Wim Veys conjectured that the only possible pole of order N of the so-called topological zeta function of f is minus the log canonical threshold of f. I will explain a proof of this conjecture, which also applies to the p-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry (Berkovich spaces) but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.