Symmetric Power L-functions of the hyper-Kloosterman Family

Abstract

The symmetric power L-function of the hyper-Kloosterman family is a rational function over the integers. Its degree and complex absolute values of its zeros and poles are now known through the work of Fu and Wan. The purpose of this paper is to study the p-adic absolute value of these zeros and poles. In particular, we give a uniform lower bound, independent of the symmetric power, of the q-adic Newton polygon of this $L$-function under suitable conditions. We also give similar results for any other linear algebra operation of the hyper-Kloosterman family, such as tensor, exterior, symmetric powers, or combinations thereof.