I will begin with an overview of work on Feynman residues, graph hypersurfaces and their point counts over finite fields. Afterwards I will discuss recent work with Erik Panzer on point counts modulo $p^n$ and their relationship with Mellin-Feynman integrals.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/13203