Speaker: Johannes Nicaise (Imperial)
Title: 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 and
Mirror Symmetry, 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.