Poles of the complex zeta function of a plane curve

TL;DR

This paper investigates the poles and residues of the complex zeta function associated with plane curves, providing conditions under which certain divisors contribute to poles and confirming Yano's conjecture for generic cases.

Contribution

It identifies which divisors contribute to poles of the zeta function and proves Yano's conjecture for plane branches with distinct monodromy eigenvalues.

Findings

Most non-rupture divisors do not contribute to poles or roots of the Bernstein-Sato polynomial.

For generic plane branches, all candidate poles are actual poles.

Yano's conjecture holds for any number of characteristic exponents when eigenvalues are distinct.

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.