A lower bound for classical Kloosterman sums and an application

TL;DR

This paper establishes a new lower bound for classical Kloosterman sums and applies it to derive explicit bounds in Petersson's trace formula, enabling improved estimates for traces of Hecke operators on cusp forms.

Contribution

It introduces a novel lower bound for Kloosterman sums and applies it to refine bounds in Petersson's trace formula with independent weight and level parameters.

Findings

Derived a lower bound for classical Kloosterman sums.

Applied the bound to obtain explicit bounds in Petersson's trace formula.

Established a lower bound for weighted traces of Hecke operators.

Abstract

We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$.