When Kloosterman sums meet Hecke eigenvalues.

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from 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 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. [ABSTRACT FROM AUTHOR]