ON THE POLES OF IGUSA'S LOCAL ZETA FUNCTION FOR ALGEBRAIC SETS

Abstract

Let $K$ be a $p$-adic field, let $Z_{\Phi }(s,f)$, $s\,{\in}\,\mathbb{C}$, with Re$(s)\,{>}\,0$, be the Igusa local zeta function associated to $f(x)\,{=}\,(f_{1}(x),\ldots,f_{l}(x))\,{\in}\,[ K( x_{1},\ldots,x_{n})]^{l}$, and let $\Phi $ be a Schwartz–Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of $Z_{\Phi }(s,f)$. Using resolution of singularities it is possible to express $Z_{\Phi }(s,f)$ as a finite sum of $p$-adic monomial integrals. These monomial integrals are computed explicitly by using techniques of toroidal geometry. In this way, an explicit list of the candidates for poles of $Z_{\Phi }(s,f)$ is obtained.