By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from $$\ell $$ ℓ -adic cohomology, as well as a Bombieri–Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke–Maass cusp form of trivial nebentypus, the eigenvalue of the n -th Hecke operator does not coincide with the Kloosterman sum $$\mathrm {Kl}(1,n)$$ Kl ( 1 , n ) for infinitely many squarefree n with at most 100 prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterman sums.

中文翻译：

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当 Kloosterman 和遇到 Hecke 特征值时

通过从 $$\ell $$ ℓ -adic 上同调和 Bombieri-Vinogradov 类型的等差数列中的 Kloosterman 和的中值定理阐述了具有渐近性和 Kloosterman 和等分布的二维塞尔伯格筛，证明对于任何给定的原始 Hecke-Maass 尖点形式的平凡 nebentypus，第 n 个 Hecke 算子的特征值与 Kloosterman 和 $$\mathrm {Kl}(1,n)$$ Kl ( 1 , n ) 不重合许多无平方的 n 最多有 100 个质因数。这对 Katz 关于 Kloosterman 和的模结构的问题提供了部分否定的答案。