A supercharacter theory on (ℤ/pℤ)3 and 2-dimensional Kloosterman sums

In this article, we construct a supercharacter theory on [Formula: see text] induced by the action of a subgroup of GL[Formula: see text] for which the 2-dimensional Kloosterman sums appear as supercharacter values. Consequently, we deduce a general formula for mixed moments of 2-dimensional Kloosterman sums, and express the absolute fourth power moment of 2-dimensional Kloosterman sums in terms of traces of Frobenius endomorphism of certain families of elliptic curves. As a result, we obtain a non-trivial bound [Formula: see text] for the 2-dimensional Kloosterman sums.