  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 $\leq12$ 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 $\phi^4$ theory. Their counting functions are given
modulo $pq^2$ ($q=p^n$) by a modular form arising from a certain singular K3
surface.
