Fourth power mean of the general 4-dimensional Kloosterman sum mod p

Abstract

In this article, we prove an asymptotic formula for the fourth power mean of a general 4-dimensional Kloosterman sum. We use a result of P. Deligne, which counts the number of $$\mathbb {F}_p$$ -points on the surface $$\begin{aligned} (x-1)(y-1)(z-1)(1-xyz)-uxyz=0, ~ u\ne 0, \end{aligned}$$ and then take an average of the error terms over u to prove the asymptotic formula. We also find the number of solutions of certain congruence equations mod p which are used to prove our main result.