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.

arXiv comment36 pages. Minor changes. References added. Results unchanged