On The L-Adic Valuation Of Certain Jacobi Sums

Fix distinct primes l and f, a finite field F-q such that q equivalent to 1 (mod lf), multiplicative characters chi(l), chi(f) : F-9(x) -> (Q) over bar (x) of orders l and f, and let J(chi(l), chi(f)) be the associated Jacobi sum. We prove new a l-adic congruence for J(chi(l), chi(f)). More specifically, we give a necessary and sufficient condition for J(chi(l), chi(f)) equivalent to -1 (mod (1 - zeta(l))(i)) when i <= l in terms of certain cyclotomic units of F-q(x) being lth powers.