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

2022/11/27 by Xi, Ping
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2211.14702

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 F_p 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 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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