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Zeta functions for curves and log canonical models

Published online by Cambridge University Press:  01 March 1997

W Veys
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Abstract

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 $\Bbb A^2$. 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)

Keywords

Topological zeta functionIgusa's local zeta functioncurve singularitieslog canonical model
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 74 , Issue 2 , March 1997 , pp. 360 - 378
Copyright
London Mathematical Society 1997

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