A characterization of Jacobi sums

2024/11/22 by Andrew Snowden, Snowden, Andrew
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2411.15011

openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbM be the group of multiplicative characters of a finite field \mathbbF, and let \mathbbJ(α, β) be the Jacobi sum, for α, β∈ \mathbbM. We observe that the function \mathbbJ \colon \mathbbM × \mathbbM → C satisfies three elementary properties. We show that these properties (very nearly) characterize Jacobi sums: if M is an arbitrary non-trivial finite abelian group and J \colon M × M → C is a function satisfying these properties then M is naturally the group of multiplicative characters of a finite field and J is the Jacobi sum.

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