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

TL;DR

This paper proves that for the Kloosterman family, the associated unit root $L$-functions are $p$-adic meromorphic and have a unique unit root equal to 1 for each slope, advancing understanding of their $p$-adic properties.

Contribution

It establishes the $p$-adic meromorphicity of unit root $L$-functions and identifies their unique unit roots for all slopes, extending previous results on Kloosterman families.

Findings

Unit root $L$-functions are $p$-adic meromorphic.

Each slope $j$ has a unique unit root equal to 1.

Results apply to all dimensions $n$ of the 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 $0\le j\le n$, the unit root $L$-function coming from slope $j$ has a unique unit root which equals 1.