Quantum field theory over F_q

We consider the number \bar N(q) of points in the projective complement of graph hypersurfaces over \F_q and show that the smallest graphs with non-polynomial \bar N(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 \bar N(q) depends on the number of cube roots of unity in \F_q. At graphs with 16 edges we find examples where \bar N(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 ¶^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.

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