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15 July 2012 A K3 in ϕ4
Francis Brown, Oliver Schnetz
Duke Math. J. 161(10): 1817-1862 (15 July 2012). DOI: 10.1215/00127094-1644201

Abstract

Inspired by Feynman integral computations in quantum field theory, Kontsevich conjectured in 1997 that the number of points of graph hypersurfaces over a finite field Fq is a (quasi-) polynomial in q. Stembridge verified this for all graphs with at most twelve edges, but in 2003 Belkale and Brosnan showed that the counting functions are of general type for large graphs. In this paper we give a sufficient combinatorial criterion for a graph to have polynomial point-counts and construct some explicit counterexamples to Kontsevich’s conjecture which are in ϕ4 theory. Their counting functions are given modulo pq2 (q=pn) by a modular form arising from a certain singular K3 surface.

Citation

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Francis Brown. Oliver Schnetz. "A K3 in ϕ4." Duke Math. J. 161 (10) 1817 - 1862, 15 July 2012. https://doi.org/10.1215/00127094-1644201

Information

Published: 15 July 2012
First available in Project Euclid: 27 June 2012

zbMATH: 1253.14024
MathSciNet: MR2954618
Digital Object Identifier: 10.1215/00127094-1644201

Subjects:
Primary: 14G10
Secondary: 05A15, 11G25, 14M12, 81T18

Rights: Copyright © 2012 Duke University Press

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Vol.161 • No. 10 • 15 July 2012
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