  In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the
local topological zeta function of a non-degenerated surface singularity. We
start from their work and obtain the same result for Igusa's p-adic and the
motivic zeta function. In the p-adic case, this is, for a polynomial f in
Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newton
polyhedron, we show that every pole of the local p-adic zeta function of f
induces an eigenvalue of the local monodromy of f at some point of the complex
zero locus of f close to the origin.
  Essentially the entire paper is dedicated to proving that, for f as above,
certain candidate poles of Igusa's p-adic zeta function of f, arising from
so-called B_1-facets of the Newton polyhedron of f, are actually not poles.
This turns out to be much harder than in the topological setting. The
combinatorial proof is preceded by a study of the integral points in
three-dimensional fundamental parallelepipeds. Together with the work of
Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture
for the p-adic and motivic zeta function of a non-degenerated surface
singularity.
