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Rationality criteria for motivic zeta functions

Published online by Cambridge University Press:  15 October 2004

Michael Larsen  and
Valery A. Lunts
Michael Larsen
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USAlarsen@math.indiana.edu, vlunts@indiana.edu
Valery A. Lunts
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USAlarsen@math.indiana.edu, vlunts@indiana.edu
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Abstract

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The zeta function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta function of a surface is rational if and only if the Kodaira dimension of the surface is negative.

Keywords

Grothendieck groupmotivic zeta functionmotivic measureλ-ringalgebraic surfaceDwork's theorem14F4214G10
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 6 , November 2004 , pp. 1537 - 1560
Copyright
Foundation Compositio Mathematica 2004