Poles of the complex zeta function of a plane curve

Abstract

We study the poles and residues of the complex zeta function $ f^s $ of a plane curve. We prove that most non-rupture divisors do not contribute to poles of $ f^s $ or roots of the Bernstein-Sato polynomial $ b_f(s) $ of $ f $. For plane branches we give an optimal set of candidates for the poles of $ f^s $ from the rupture divisors and the characteristic sequence of $ f $. We prove that for generic plane branches $ f_{gen} $ all the candidates are poles of $ f_{gen}^s $. As a consequence, we prove Yano's conjecture for any number of characteristic exponents if the eigenvalues of the monodromy of $ f $ are different.