Families of generalized Kloosterman sums

TL;DR

This paper develops p-adic cohomology for families of generalized Kloosterman sums, enabling precise Frobenius action estimates and advancing understanding of associated L-functions.

Contribution

It introduces a new cohomological framework for generalized Kloosterman sums, extending classical methods and allowing sharper analysis of Frobenius actions.

Findings

Cohomology is acyclic outside top dimension under certain conditions

Sharp estimates for Frobenius action on cohomology

Applications to properties of L-functions from these sums

Abstract

We construct p-adic relative cohomology for a family of toric exponential sums which generalize the classical Kloosterman sums. Under natural hypotheses such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except in the top dimension. Our construction enables sufficiently sharp estimates for the action of Frobenius on cohomology so that our earlier work may be applied to the L-functions coming from linear algebra operations on these families to deduce a number of basic properties.