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

Abstract

Fix distinct primes $\ell$ and $f$, a finite field ${\mathbf{F}}_q$ such that $q\equiv 1(\mathrm{mod}\ell f)$, 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 $y^{\ell }=x^f+1$.

18 pages; improved notation, reorganized sections accordingly, clarified a small subtlety with ell=2 case