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Zeta Functions for Curves and Log Canonical Models
Author(s) -
Veys W
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000130
Subject(s) - mathematics , arithmetic zeta function , riemann zeta function , prime zeta function , polynomial , invariant (physics) , pure mathematics , combinatorics , mathematical analysis , mathematical physics
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)