Poles of the complex zeta function of a plane curve

2018/05/04 by Blanco, Guillem
#14E15 (Secondary) #14F10 (Primary) 32S40 #14H20 #32S05 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.01683

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.

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