Kloosterman sums and Hecke polynomials in characteristics 2 and 3

TL;DR

This paper provides a modular and p-adic interpretation of Kloosterman sums and their symmetric power L-functions in characteristics 2 and 3, including explicit Newton polygon calculations for specific cases.

Contribution

It extends previous results to include characteristics 2 and 3, offering explicit p-adic Newton polygons and modular interpretations for Kloosterman sums and L-functions.

Findings

Explicit 2-adic Newton polygons for odd k in characteristic 2.

Modular interpretation of symmetric power L-functions in characteristics 2 and 3.

Extension of previous prime restrictions to all p ≥ 2.

Abstract

In this paper we give a modular interpretation of the $k$-th symmetric power $L$-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of $p=2$ and $k$ odd give the precise 2-adic Newton polygon. We also give a $p$-adic modular interpretation of Dwork's unit root $L$-function of the Kloosterman family, and give the precise 2-adic Newton polygon when $k$ is odd. In a previous paper, we gave an estimate for the $q$-adic Newton polygon of the symmetric power $L$-function of the Kloosterman family when $p \geq 5$. We discuss how this restriction on primes was not needed, and so the results of that paper hold for all $p \geq 2$.