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 of f ‐1 {0} 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. 1991 Mathematics Subject Classification : 32S50 11S80 14E30 (14G20)