The exotic inverted Kloosterman sum

Let B be a product of finitely many finite fields containing Fq, ψ: Fq→ Q̄_^* a nontrivial additive character, and χ: B^*→ Q̄_^* a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum {eqnarray*} EIK( F_q, a):=∑_{{x∈ B^* \\ TrB/ F_q(x) =0\\ NB/ F_q(x)=a}} χ(x)ψ({1}{TrB/ F_q(x)}), \ \ a∈ F_q^*. {eqnarray*} We estimate this sum using -adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most n, where n+1 = FqB. Up to a trivial main term, this gives the expected square root cancellation.