Infinite symmetric power L-functions of the hyper-Kloosterman family

Abstract

The infinity symmetric power $L$-functions play a fundamental role in Wan's groundbreaking work on Dwork's conjecture[16]. Building upon this foundation, Haessig[8] established the $p$-adic estimates for these $L$-functions in the case of the one-dimensional Kloosterman family. In this paper, we extend Haessig's results by deriving a uniform lower bound for the $q$-adic Newton polygon of the infinite symmetric power $L$-functions associated with the hyper-Kloosterman family. For the $1$-dimensional Kloosterman family, Haessig[8] showed that there is a $p$-adic cohomology theory for the infinity symmetric power $L$-function. In this paper, we prove there is also a cohomological description of the infinity symmetric power $L$-function for the hyper-Kloosterman family. By applying the Frobenius endomorphism to this cohomology, we derive a uniform lower bound for the corresponding $L$-function.