It is well known that if chi1, chi2, and chi1*chi2 are
nontrivial characters mod p then the Jacobi sum
JacobiSum(chi1,chi2) := sum(x=2,p-1,chi1(x)*chi2(1-x))
is equal to GaussSum(chi1) * GaussSum(chi2) / GaussSum(chi1*chi2),
and thus in particular that
|JacobiSum(chi1,chi2)|^2 = p.
Daniel Kane recently noticed that the formula for |JacobiSum(chi1,chi2)|
can be obtained directly, without going through the Gauss-sum trick:
the square is the sum of
chi1(x/y) * chi2((1-x)/(1-y))
over x and y in (1,p); the terms with x=y sum to p-2, and the remaining
ones can be evaluted by writing y=cx with c=2,3,...,p-1 and summing
over x for each c. (Likewise for Jacobi sums associated to characters
of the unit group of any finite field.) Is there a standard reference
for this alternative proof?
--Noam D. Elkies
