Holomorphy of local zeta functions for curves (Q1318029)

Let \(F\) be a number field and fix \(f\in F[x,y]\). For any (non- archimedean) completion of \(F\) with valuation ring \(R\), let \(\kappa: R^ \times\to \mathbb{C}^ \times\) be a character of the group of units of \(R\). To \(f\) and \(\kappa\) is associated Igusa's local zeta function \[ Z_ \kappa (s)=\int_{R^ 2} \kappa(\text{ac } f(x,y)) | f(x,y)|^ s | dx\wedge dy|, \] which is known to be meromorphic on \(\mathbb{C}\). We prove (for all except a finite number of completions) that if the order of \(\kappa\) does not divide the order of any eigenvalue of the local monodromy of \(f\) at any complex point of \(f^{-1}\{0\}\), then \(Z_ \kappa(s)\) is holomorphic on \(\mathbb{C}\). This settles the curve-case of a general conjecture of \textit{J. Denef} [Astérisque 201-203, 359-386 (1991; Zbl 0749.11054)].