On the distribution of Jacobi sums
| Autor: | Lu, Qing, Zheng, Weizhe, Zheng, Zhiyong |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | J. Reine Angew. Math. 2018, no. 741, pp. 67-86 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1515/crelle-2015-0087 |
| Popis: | Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let $J(\chi_1,\dots, \chi_m)$ denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for $m=2$, the normalized Jacobi sum $q^{-1/2}J(\chi_1,\chi_2)$ ($\chi_1\chi_2$ nontrivial) is asymptotically equidistributed on the unit circle as $q\to \infty$, when $\chi_1$ and $\chi_2$ run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$. In this paper, we show a similar property for $m\ge 2$. More generally, we show that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$ ($\chi_1\dotsm \chi_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $\chi_1,\dots, \chi_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$ with two of the sets being sufficiently large. The case $m=2$ answers a question of Shparlinski. Comment: 18 pages. v3: fixed some typos; v2: improved some bounds |
| Databáze: | arXiv |
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