On the holomorphy conjecture for Igusa’s local zeta function

Abstract:

To a polynomial f over a p-adic field K and a character $\chi$ of the group of units of the valuation ring of K one associates Igusa’s local zeta function $Z(s,f,\chi )$, which is a meromorphic function on $\mathbb {C}$. Several theorems and conjectures relate the poles of $Z(s,f,\chi )$ to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of $\chi$ does not divide the order of any eigenvalue of monodromy of f, then $Z(s,f,\chi )$ is holomorphic on $\mathbb {C}$. We prove mainly that if the holomorphy conjecture is true for $f({x_1}, \ldots ,{x_{n - 1}})$, then it is true for $f({x_1}, \ldots ,{x_{n - 1}}) + x_n^k$ with $k \geq 3$, and we give some applications.