A K3 in phi4

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 $\F_q$ is a (quasi-) polynomial in $q$. Stembridge verified this for all graphs with $\le 12$ 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 counter-examples to Kontsevich's conjecture which are in ${\varphi }^4$ theory. Their counting functions are given modulo $p{q}^2$ ($q=p^n$) by a modular form arising from a certain singular K3 surface.

Final version

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