Zeros of the Goss zeta function

2023/12/03 by Joe Kramer-Miller, Kramer-Miller, Joe, James Upton +1
Arts and Humanities · Mathematics · #11M38 #11R59 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.01264

openalex publication_date 2023/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth proper curve over a finite field and let ∞ ∈ X be a closed point. Let A be the ring of functions on X - ∞. The Goss zeta function ζ_A of A is an equicharacteristic analogue of the Riemann zeta function. In this article we study the zeros of ζ_A under the generic condition that X is ordinary. We prove an analogue of the Riemann hypothesis, which verifies a corrected version of a conjecture of Goss. We also show that the zeros of ζ_A at negative even integers are `simple' and that ζ_A is nonzero at negative odd integers. This answers questions posed by Goss and Thakur. Both of these results were previously only known under the restrictive hypothesis that A has class number one. Finally, we prove versions of these results for v-adic interpolations of the Goss zeta function.

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