  We introduce the etale framework to study Igusa zeta functions in several
variables, generalizing the machinery of vanishing cycles in the univariate
case. We define the etale Alexander modules, associated to a morphism of
varieties F from X to affine r-space, a geometric point of X, and an object in
the derived category of constructible l-adic sheaves on the inverse image of
the r-torus under F. The Alexander modules are sheaves of modules on the scheme
of continuous characters of the tame fundamental group of the r-torus. We
formulate Loeser's Monodromy and Holomorphy Conjectures for multivariate p-adic
zeta functions, and prove them in the case where dim(X)=2, generalizing results
from the univariate case. Furthermore, we prove a comparison theorem with the
transcendent case, we study a formula of Denef's for the zeta function in terms
of a simultaneous embedded resolution, and we generalize a result concerning
the degree of the zeta function to our setting.
