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

被引:1
作者
Xi, Ping [1 ]
机构
[1] Xi An Jiao Tong Univ, Sch Math & Stat, Xian 710049, Peoples R China
基金
中国国家自然科学基金;
关键词
Bilinear forms; l-adic sheaves; Riemann Hypothesis over finite fields; Sato-Tate distribution; Kloosterman sums; elliptic curves; EXPONENTIAL-SUMS; KLOOSTERMAN SUMS; BOUNDS; FAMILIES;
D O I
10.1007/s11425-022-2184-9
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 F-p of suitable sizes. This essentially recovers the Polya-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 Polya-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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页码:2819 / 2834
页数:16
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