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Gaussian distributions of Kloosterman sums: vertical and horizontal

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Abstract

To illustrate vertical and horizontal equidistributions of Kloosterman sums, we prove two relevant central limit theorems by some analytic and combinatorial properties of Chebyshev polynomials and independence of monodromy groups of Kloosterman sheaves. The first one supports a short interval version of Katz’s vertical Sato–Tate distribution theorem, and the second one coincides with the horizontal conjecture of Katz in the statistical sense.

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Acknowledgements

The present work was completed during my stay in EPF Lausanne and contained as part of my PhD thesis. I am grateful to Professors Philippe Michel and Yuan Yi for helpful discussions and the Mathematics Department in EPF Lausanne for the excellent working environment.

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  1. Department of Mathematics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Ping Xi

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  1. Ping Xi
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Correspondence to Ping Xi.

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Partially supported by China Scholarship Council, EPF Lausanne, CPSF (No. 2015M580825) and NSF (No. 11601413) of P.R. China.

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Xi, P. Gaussian distributions of Kloosterman sums: vertical and horizontal. Ramanujan J 43, 493–511 (2017). https://doi.org/10.1007/s11139-017-9905-6

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  • DOI: https://doi.org/10.1007/s11139-017-9905-6

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