Holomorphy of Igusa’s and Topological Zeta Functions for Homogeneous Polynomials

Let F be a number field and f ∈ F [x1, . . . , xn] \ F . To any completion K of F and any character κ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z(κ, f, s). The holomorphy conjecture states that for all except a finite number of completions K of F we have that if the order of κ does not divide the order of any eigenvalue of the local monodromy of f at any complex point of f−1{0}, then Z(κ, f, s) is holomorphic on C. The second author already showed that this conjecture is true for curves, i.e., for n = 2. Here we look at the case of an homogeneous polynomial f , so we can consider {f = 0} ⊆ Pn−1. Under the condition that χ(Pn−1 C \ {f = 0}) = 0 we prove the holomorphy conjecture. Together with some results in the case when χ(Pn−1 C \ {f = 0}) = 0, we can conclude that the holomorphy conjecture is true for an arbitrary homogeneous polynomial in three variables. We also prove the so-called monodromy conjecture for a homogeneous polynomial f ∈ F [x1, x2, x3] with χ(PC \ {f = 0}) = 0.