## Duke Mathematical Journal

### A K3 in $\phi^{4}$

#### 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 $\mathbb {F}_{q}$ 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 $\phi^{4}$ theory. Their counting functions are given modulo $pq^{2}$ ($q=p^{n}$) by a modular form arising from a certain singular K3 surface.

#### Article information

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

Dates
First available in Project Euclid: 27 June 2012

https://projecteuclid.org/euclid.dmj/1340801625

Digital Object Identifier
doi:10.1215/00127094-1644201

Mathematical Reviews number (MathSciNet)
MR2954618

Zentralblatt MATH identifier
1253.14024

#### Citation

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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