Vol. 4, No. 1, 2010

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Positive motivic measures are counting measures

Jordan S. Ellenberg and Michael Larsen

Vol. 4 (2010), No. 1, 105–109
Abstract

A motivic measure is a ring homomorphism from the Grothendieck group of a field K (with multiplication coming from the fiber product over SpecK) to some field. We show that if a real-valued motivic measure μ satisfies μ([V ]) 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/