A K3 in ϕ4

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.