$L$-functions for families of generalized Kloosterman sums and $p$-adic differential equations

TL;DR

This paper investigates the $L$-functions associated with generalized Kloosterman sums over the torus, employing $p$-adic cohomology, GKZ systems, and Dwork's theory to analyze their properties and provide evidence for Dwork's conjecture.

Contribution

It introduces a new method to compute Newton polygons of $L$-functions using explicit cohomology bases and establishes the Frobenius structure, advancing understanding of $p$-adic properties of these sums.

Findings

Derived explicit bases for top cohomology spaces

Established Dwork's deformation equation for the family

Proved strong Frobenius structure supporting Dwork's conjecture

Abstract

In this paper, we focus on a family of generalized Kloosterman sums over the torus. With a few changes to Haessig and Sperber's construction, we derive some relative $p$-adic cohomologies corresponding to the $L$-functions. We present explicit forms of bases of top dimensional cohomology spaces, so to obtain a concrete method to compute lower bounds of Newton polygons of the $L$-functions. Using the theory of GKZ system, we derive the Dwork's deformation equation for our family. Furthermore, with the help of Dwork's dual theory and deformation theory, the strong Frobenius structure of this equation is established. Our work adds some new evidences for Dwork's conjecture.