On the moments of Kloosterman sums and fibre products of Kloosterman curves

Let q=pr with p=3 and r⩾2. We give a recursion formula for the moments of a Kloosterman sum over the finite field Fq, which utilizes known weight formulae for the ternary Melas code M of length q−1. The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As an application for the formulae, and for their analogues obtained earlier in case p=2, we get the exact number of rational points on fibre products of certain Kloosterman curves. As a corollary we obtain identities between Ramanujan's tau-function, Kronecker class numbers, and Dickson polynomials.