On the $\ell$-adic valuation of certain Jacobi sums

TL;DR

This paper derives a new $ ext{ell}$-adic congruence for Jacobi sums associated with certain algebraic curves over finite fields, enhancing understanding of their valuations and Frobenius eigenvalues.

Contribution

It introduces a novel $ ext{ell}$-adic congruence for Jacobi sums linked to specific algebraic curves over finite fields.

Findings

Established a new $ ext{ell}$-adic congruence for Jacobi sums.

Connected Jacobi sums to Frobenius eigenvalues of algebraic curves.

Provided insights into the $ ext{ell}$-adic valuation of these sums.

Abstract

Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $\chi : \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(\chi^{\ell}, \chi^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.