Abstract

Let F be a number field and f in F[x₁,…,x_n] ∖ 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^K(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^K(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(P_Cⁿ⁻¹ ∖{f = 0})≠0 we prove the holomorphy conjecture. Together with some results in the case when χ(P_Cⁿ⁻¹ ∖{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 in F[x₁,x₂,x₃] with χ(P_C² ∖{f = 0})≠0.

Authors