$\ell$-adic GKZ hypergeometric sheaf and exponential sums

TL;DR

This paper develops an $\, ext{l}$-adic sheaf analogue of GKZ hypergeometric systems, analyzing its properties and applications to exponential sums over finite fields, extending classical hypergeometric function theory.

Contribution

It introduces and studies the $\, ext{l}$-adic GKZ hypergeometric sheaf, proving its perversity, irreducibility, and analyzing its weight filtration and lisse locus.

Findings

The sheaf is perverse and irreducible under non-resonance conditions.

Calculated the rank of the $\, ext{l}$-adic GKZ sheaf.

Applied results to understand weights of twisted exponential sums.

Abstract

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $\ell$-adic counterpart of the GKZ hypergeometric system, which we call the $\ell$-adic GKZ hypergeometric sheaf. It is an object in the derived category of $\ell$-adic sheaves on the affine space over a finite field. Traces of Frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by Gelfand and Graev. We prove that the $\ell$-adic GKZ hypergeometric sheaf is perverse, calculate its rank, and prove that it is irreducible under the non-resonance condition. We also study the weight filtration of the GKZ hypergeometric sheaf, determine its lisse locus, and apply our result to the study of weights of twisted exponential sums.