Hyperkloosterman sums revisited

TL;DR

This paper revisits hyperkloosterman sums using $p$-adic cohomology, improving previous results by removing characteristic restrictions and employing a better basis choice guided by recent work on $p$-integrality.

Contribution

It introduces a new basis for cohomology and utilizes Dwork's $ heta_ty$-splitting function to enhance earlier hyperkloosterman sum analyses, removing previous characteristic constraints.

Findings

Removed all characteristic restrictions from earlier results.

Improved congruence estimates for hyperkloosterman sums.

Utilized a better basis guided by recent $p$-integrality work.

Abstract

We return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's $\theta_\infty$-splitting function and a better choice of basis for cohomology. To a large extent, we are guided to this choice of basis by our recent work on the $p$-integrality of coefficients of $A$-hypergeometric series[3]. In the earlier work, congruence estimates were limited to $p>n+2$. We are here able to remove all characteristic restrictions from earlier results.