L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates)

TL;DR

This paper develops a p-adic cohomology framework to estimate the p-adic absolute values of roots of L-functions associated with symmetric powers of Kloosterman sums, advancing understanding of their p-adic properties.

Contribution

It introduces a novel Dwork-type p-adic cohomology theory for symmetric power L-functions of Kloosterman sums and derives p-adic estimates of Frobenius eigenvalues.

Findings

Established bounds for p-adic absolute values of roots

Developed a new p-adic cohomology approach

Provided a continuity-based method for p-adic estimates

Abstract

The L-function of symmetric powers of classical Kloosterman sums is a polynomial whose degree is now known, as well as the complex absolute values of the roots. In this paper, we provide estimates for the p-adic absolute values of these roots. Our method is indirect. We first develop a Dwork-type p-adic cohomology theory for the two-variable infinite symmetric power L-function associated to the Kloosterman family, and then study p-adic estimates of the eigenvalues of Frobenius. A continuity argument then provides the desired p-adic estimates.