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In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for thelocal topological zeta function of a non-degenerated surface singularity. Westart from their work and obtain the same result for Igusa's p-adic and themotivic zeta function. In the p-adic case, this is, for a polynomial f inZ[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newtonpolyhedron, we show that every pole of the local p-adic zeta function of finduces an eigenvalue of the local monodromy of f at some point of the complexzero 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 fromso-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. Thecombinatorial proof is preceded by a study of the integral points inthree-dimensional fundamental parallelepipeds. Together with the work ofLemahieu and Van Proeyen, this main result leads to the Monodromy Conjecturefor the p-adic and motivic zeta function of a non-degenerated surfacesingularity.

Igusa's p-adic local zeta function and the Monodromy Conjecture for non-degenerated surface singularities