Exponential sums over finite fields and the large sieve

Abstract

By using a variant of the large sieve for Frobenius in compatible systems developed in [Kow06a] and [Kow08], we obtain zerodensity estimates for arguments of ℓ-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.