When Kloosterman sums meet Hecke eigenvalues

摘要

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from ?-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-thHecke 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.