Partial zeta functions of algebraic varieties over finite fields

2000/06/14 by Daqing Wan, Wan, Daqing
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.math/0006236

arxiv created 2000/06/14 · openalex publication_date 2000/06/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well known rationality theorem. In general, the partial zeta function is probably not rational. But a theorem of Faltings says that the partial zeta function is always nearly rational.

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• Rationality of partial zeta functions
• Partial zeta functions, partial exponential sums, and p-adic estimates
• Computing zeta functions over finite fields
• Pure L-functions from algebraic geometry over finite fields
• Zeta functions over finite fields- An exposition of Dwork's methods