Unit roots of the unit root L-functions of Kloosterman family

Abstract

As a consequence of Wan's theorem about Dwork's conjecture, the unit root L-functions of the n-dimensional Kloosterman family are p-adic meromorphic. By studying the symmetric power L-functions associated to the Kloosterman family, we prove that for each 0≤j≤n, the unit root L-function coming from slope j has a unique unit root which equals 1.