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Bilinear forms with trace functions over arbitrary sets and applications to Sato-Tate

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Abstract

We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves, where the supports of two variables can be arbitrary subsets in Fp of suitable sizes. This essentially recovers the Pólya-Vinogradov range, and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves. In the case of hyper-Kloosterman sums, we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski, Michel and Sawin (2017) on sum-products of Kloosterman sheaves. Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12025106 and 11971370). I am very grateful to Bryce Kerr for letting me know the work of Shkredov [27] which yields an improved version of Theorem 1.2, and to Igor Shparlinski for pointing out the reference [1]. I also thank the referees for valuable comments and suggestions. It is my great honour to be invited to acknowledge the 50th anniversary of the detailed proof of Chen’s celebrated theorem on the Goldbach problem and the twin prime conjecture. I am very lucky that my academic career has been guided by Chen’s mathematics and spirits.

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  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Ping Xi

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  1. Ping Xi
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Professor Jingrun Chen in memoriam

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Xi, P. Bilinear forms with trace functions over arbitrary sets and applications to Sato-Tate. Sci. China Math. 66, 2819–2834 (2023). https://doi.org/10.1007/s11425-022-2184-9

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