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

作　　者：Ping Xi 

机构地区：[1]School of Mathematics and Statistics,Xi'an Jiaotong University,Xi'an 710049,China

出　　处：《Science China Mathematics》2023年第12期2819-2834,共16页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 12025106 and 11971370)。

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 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.

关 键 词：bilinear forms l-adic sheaves Riemann Hypothesis over finite fields Sato-Tate distribution Kloosterman sums elliptic curves 

分 类 号：O156[理学—数学]