The Kloosterman circle method and weighted representation numbers of quadratic forms

TL;DR

This paper introduces a refined Kloosterman circle method with explicit constants, enabling precise asymptotics for weighted representation numbers of quadratic forms, and discusses potential applications to local-global principles in Kleinian sphere packings.

Contribution

It develops a new version of the Kloosterman circle method with explicit error constants tailored for quadratic forms.

Findings

Provides asymptotic formulas with explicit error bounds

Utilizes Gauss, Kloosterman, and Salié sums in the method

Suggests applications to local-global principles in sphere packings

Abstract

We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of nonsingular integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Sali\'{e} sums, and a principle of nonstationary phase. We briefly discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local-global principle for certain Kleinian sphere packings.