Trivial factors for L-functions of symmetric products of Kloosterman sheaves

In this paper, we determine the trivial factors of L-functions of both finite and infinite symmetric products of Kloosterman sheaves. Let Fq be a finite field of characteristic p with q elements, let l be a prime number distinct from p, and let ψ : Fq → Q ∗ l be a nontrivial additive character. Fix an algebraic closure F of Fq. For any integer k, let Fqk be the extension of Fq in F with degree k. Let n ≥ 2 be a positive integer. If λ lies in Fqk , we define the (n− 1)-variable Kloosterman sum by Kln(Fqk , λ) = ∑