Kloosterman sums and Hecke polynomials in characteristics 2 and 3

Abstract

In this paper we give a modular interpretation of the k-th symmetric power L-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of p=2 and k odd give the precise 2-adic Newton polygon. We also give a p-adic modular interpretation of Dwork's unit root L-function of the Kloosterman family, and give the precise 2-adic Newton polygon when k is odd.In a previous paper, we gave an estimate for the q-adic Newton polygon of the symmetric power L-function of the Kloosterman family when p≥5. We discuss how this restriction on primes was not needed, and also provide an improved estimate.