Duke Mathematical Journal

A K3 in ϕ4

Francis Brown and Oliver Schnetz

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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.

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Duke Math. J., Volume 161, Number 10 (2012), 1817-1862.

First available in Project Euclid: 27 June 2012

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Zentralblatt MATH identifier

Primary: 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 11G25: Varieties over finite and local fields [See also 14G15, 14G20] 14M12: Determinantal varieties [See also 13C40] 81T18: Feynman diagrams


Brown, Francis; Schnetz, Oliver. A K3 in $\phi^{4}$. Duke Math. J. 161 (2012), no. 10, 1817--1862. doi:10.1215/00127094-1644201. https://projecteuclid.org/euclid.dmj/1340801625

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