Positive motivic measures are counting measures

Ellenberg, Jordan; Larsen, Michael

doi:10.2140/ant.2010.4.105

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Positive motivic measures are counting measures

Jordan S. Ellenberg and Michael Larsen

Vol. 4 (2010), No. 1, 105–109

DOI: 10.2140/ant.2010.4.105

Abstract

A motivic measure is a ring homomorphism from the Grothendieck group of a field $K$ (with multiplication coming from the fiber product over Spec $K$ ) to some field. We show that if a real-valued motivic measure $μ$ satisfies $μ ([V]) \geq 0$ for all $K$ -varieties $V$ , then $μ$ is a counting measure; that is, there exists a finite field $L$ containing $K$ such that $μ ([V]) = | V (L) |$ for all $K$ -varieties $V$ .

Keywords

motives, motivic measures, finite fields

Mathematical Subject Classification 2000

Primary: 14F43

Secondary: 14G15

Milestones

Received: 10 July 2009

Accepted: 10 August 2009

Published: 14 January 2010

Authors

Jordan S. Ellenberg
Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706 United States
http://math.wisc.edu/~ellenber
Michael Larsen
Department of Mathematics Indiana University Bloomington, IN 47405 United States
http://mlarsen.math.indiana.edu/~larsen/

Vol. 4, No. 1, 2010