Quantum Field Theory over Fq

We consider the number (N) over bar (q) of points in the projective complement of graph hypersurfaces over F-q and show that the smallest graphs with non-polynomial (N) over bar (q) have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class (N) over bar (q) depends on the number of cube roots of unity in F-q. At graphs with 16 edges we find examples where (N) over bar (q) is given by a polynomial in q plus q(2) times the number of points in the projective complement of a singular K3 in P-3. In the second part of the paper we show that applying momentum space Feynman-rules over F-q lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.