An arithmetic zeta function respecting multiplicities

2020/03/12 by Lukas Prader, Prader, Lukas
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2003.06057

openalex publication_date 2020/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the arithmetic zeta function \mathscrZ_X(s) = ∏_p ∏_\substackx ∈ X_p
closed ( \frac11-|κ(x)|^-s )^\mathfrakm_p(x) associated to a scheme X of finite type over ℤ, where κ(x) denotes the residue field and \mathfrakm_p(x) the multiplicity of x in X_p. If X is defined over a finite field, then \mathscrZ_X appears naturally in the context of point counting with multiplicities. We prove that \mathscrZ_X admits a meromorphic continuation to \s ∈ ℂ \colon Re(s) > dim(X)-1/2\ and determine the order of its pole at s = dim(X). Finally, we relate \mathscrZ_X to a zeta function ζ_f encoding the residual factorization patterns of a polynomial f.

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