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The origin and development of the Igusa local zeta function, by Igusa and others, is presented. In particular, we discuss the various conjectures Igusa made and the notable results that have so far been obtained. We also explain how topological and motivic zeta functions arose from the Igusa local zeta function and present the current status of the analogous conjectures. Igusa's conjecture on exponential sums that are related to his zeta function is described, and along with the progress made towards its resolution.

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In this paper we consider a local zeta function in two complex variables associated to an affine algebraic variety defined over a finite algebraic extension of Qp" The zeta function combines the Well and Igusa zeta functions for this variety in a natural way. We investigate the properties of this function and in particular show that it has a meromorphic continuation to C 2.Let K 1 denote a fixed finite algebraic extension of Qp. Let The numbers N1, e are those that appear in the Weil zeta function. The coefficient of w e is Pc(z)= Y~ S~,eq-~ z e e>O which is the Igusa zeta function for the field K e which was shown to be rational in [6] and [7].