Back to Search
Start Over
On the distribution of Jacobi sums.
- Source :
- Journal für die Reine und Angewandte Mathematik; 2018, Vol. 2018 Issue 741, p67-86, 20p
- Publication Year :
- 2018
-
Abstract
- 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]
Details
- Language :
- English
- ISSN :
- 00754102
- Volume :
- 2018
- Issue :
- 741
- Database :
- Complementary Index
- Journal :
- Journal für die Reine und Angewandte Mathematik
- Publication Type :
- Academic Journal
- Accession number :
- 131059866
- Full Text :
- https://doi.org/10.1515/crelle-2015-0087