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On the distribution of Jacobi sums

Qing Lu, Weizhe Zheng and Zhiyong Zheng

Abstract

Let 𝐅q be a finite field of q elements. For multiplicative characters χ1,,χm of 𝐅q×, we let J(χ1,,χm) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m=2, the normalized Jacobi sum q-1/2J(χ1,χ2) (χ1χ2 nontrivial) is asymptotically equidistributed on the unit circle as q, when χ1 and χ2 run through all nontrivial multiplicative characters of 𝐅q×. In this paper, we show a similar property for m2. More generally, we show that the normalized Jacobi sum q-(m-1)/2J(χ1,,χm) (χ1χm nontrivial) is asymptotically equidistributed on the unit circle, when χ1,,χm run through arbitrary sets of nontrivial multiplicative characters of 𝐅q× with two of the sets being sufficiently large. The case m=2 answers a question of Shparlinski.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11371043

Award Identifier / Grant number: 11321101

Funding statement: The first-named author was partially supported by the National Natural Science Foundation of China under grant 11371043. The second-named author was partially supported by China’s Recruitment Program of Global Experts, the National Natural Science Foundation of China under grant 11321101, the Hua Loo-Keng Key Laboratory of Mathematics (Chinese Academy of Sciences) and the National Center for Mathematics and Interdisciplinary Sciences (Chinese Academy of Sciences). The third-named author was partially supported by Program 863 grant 2013AA013702 and Program 973 grant 2013CB834205.

Acknowledgements

The authors wish to thank Ming Fang, Ofer Gabber, and Nicholas Katz for useful discussions, as well as the referee for helpful comments. The first- and second-named author gratefully acknowledge the hospitality and support of l’Institut des Hautes Études Scientifiques, where part of this work was done during their visits.

References

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Received: 2013-05-17
Revised: 2015-09-21
Published Online: 2016-01-21
Published in Print: 2018-08-01

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