On the distribution of Jacobi sums.

Let 𝐅 q \mathbf{F}_{q} be a finite field of q elements. For multiplicative characters χ 1 , … , χ m \chi_{1},\ldots,\chi_{m} of 𝐅 q × \mathbf{F}_{q}^{\times} , we let J ⁢ ( χ 1 , … , χ m ) J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m = 2 m=2 , the normalized Jacobi sum q - 1 / 2 ⁢ J ⁢ ( χ 1 , χ 2 ) q^{-1/2}J(\chi_{1},\chi_{2}) ( χ 1 ⁢ χ 2 \chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q → ∞ q\to\infty , when χ 1 \chi_{1} and χ 2 \chi_{2} run through all nontrivial multiplicative characters of 𝐅 q × \mathbf{F}_{q}^{\times}. In this paper, we show a similar property for m ≥ 2 m\geq 2. More generally, we show that the normalized Jacobi sum q - ( m - 1 ) / 2 ⁢ J ⁢ ( χ 1 , … , χ m ) q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) ( χ 1 ⁢ ⋯ ⁢ χ m \chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when χ 1 , … , χ m \chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of 𝐅 q × \mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m = 2 m=2 answers a question of Shparlinski. [ABSTRACT FROM AUTHOR]