Zeta Functions for Curves and Log Canonical Models (original) (raw)

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial f and an arithmetical invariant associated to a polynomial f over a p{adic eld. When f is a polynomial in two variables we prove a formula for both zeta functions in terms of the so{called log canonical model of f ?1 f0g in A 2 . This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non{symmetric`q{deformation' of the intersection matrix of the minimal resolution of a Hirzebruch{Jung singularity.