When Kloosterman sums meet Hecke eigenvalues

TL;DR

This paper demonstrates that for any primitive Hecke--Maass cusp form, the eigenvalues of Hecke operators rarely match Kloosterman sums for infinitely many squarefree integers with limited prime factors, using advanced sieve and equidistribution techniques.

Contribution

It introduces a novel combination of a two-dimensional Selberg sieve and equidistribution results to address a problem related to Kloosterman sums and Hecke eigenvalues.

Findings

Eigenvalues of Hecke operators rarely match Kloosterman sums for infinitely many n.

The results apply to squarefree n with up to 100 prime factors.

Provides a partial negative answer to Katz's problem on modular structures of Kloosterman sums.

Abstract

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from $\ell$-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 $\mathrm{Kl}(1,n)$ 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.