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

Authors:
Marko Moisio
Affiliations:
Department of Mathematics and Statistics, Faculty of Technology, University of Vaasa, PO Box 700, FIN-65101, Finland
Venue:
Finite Fields and Their Applications
Year:
2008

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Abstract

Let q=p^r with p=3 and r=2. We give a recursion formula for the moments of a Kloosterman sum over the finite field F"q, 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.