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

TL;DR

This paper extends the understanding of infinite symmetric power L-functions for hyper-Kloosterman families, providing new p-adic estimates, a cohomological description, and a uniform lower bound for their Newton polygons.

Contribution

It generalizes previous results from the 1-dimensional Kloosterman family to the hyper-Kloosterman family, establishing a cohomological framework and p-adic bounds.

Findings

Established a uniform lower bound for the Newton polygon.

Provided a cohomological description of the L-functions.

Extended p-adic estimates to hyper-Kloosterman families.

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.