Zeta Functions For Curves And Log Canonical Models

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 field. 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 f0g in A . 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 lcal zeta function appears a remarkable non--symmetric `q--deformation' of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity