Vol. 4, No. 1, 2010

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Positive motivic measures are counting measures

Jordan S. Ellenberg and Michael Larsen

Vol. 4 (2010), No. 1, 105–109

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 .

motives, motivic measures, finite fields
Mathematical Subject Classification 2000
Primary: 14F43
Secondary: 14G15
Received: 10 July 2009
Accepted: 10 August 2009
Published: 14 January 2010
Jordan S. Ellenberg
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706
United States
Michael Larsen
Department of Mathematics
Indiana University
Bloomington, IN 47405
United States