A lower bound for classical Kloosterman sums and an application

Technical Summary: A Lower Bound for Classical Kloosterman Sums and an Application

Problem Statement
The paper addresses two interconnected problems in analytic number theory. First, it seeks to establish a non-trivial lower bound for the classical Kloosterman sum $S(a, b; c)$, defined as $\sum_{x \pmod c, (x,c)=1} e^{2\pi i (ax + bx^{-1})/c}$, under the conditions that $(ab, c) = 1$ and $c$ is an odd integer. While the non-vanishing of these sums has been utilized in recent literature to study eigenvalue distributions, no explicit lower bound for $|S(a, b; c)|$ had been previously established in the literature. Second, the authors aim to apply this new bound to derive an explicit lower bound for the discrepancy term in Petersson's trace formula, specifically for the space of cusp forms $S_k(N)$ where the weight $k$ and level $N$ are permitted to vary independently.

Methodology
The methodology proceeds in two primary stages:

1. Derivation of the Lower Bound for Kloosterman Sums:
   The authors decompose the modulus $c$ into $c = du$, where $u$ is squarefree and $d$ is a powerful number (with $(d, u)=1$). They treat the components separately:

   • Squarefree Case: For the squarefree part $u$, they utilize the algebraic properties of Kloosterman sums within cyclotomic fields. By considering the norm of the sum over the Galois group and applying Weil's bound, they derive a lower bound involving the number of divisors $\tau(u)$ and Euler's totient function $\phi(u)$.
   • Odd Prime Power Case: For the powerful part $d$ (composed of prime powers $p^\beta$), they employ explicit evaluation formulas for $S(a, b; p^\beta)$ found in standard texts (e.g., Iwaniec and Kowalski). By analyzing the real part of the resulting expression and utilizing trigonometric inequalities, they establish a lower bound of order $2/\sqrt{q}$ for prime powers, provided $ab$ is a quadratic residue modulo the prime.
   • General Case: Using the multiplicativity of Kloosterman sums, they combine the bounds for the squarefree and powerful components to obtain a global lower bound for $|S(a, b; c)|$.

2. Application to Petersson's Trace Formula:
   The authors apply the derived lower bound to the error term in Petersson's trace formula, denoted as $\Delta_{k,N}(m, n) - \delta(m, n)$. The formula expresses this difference as a sum involving Kloosterman sums and Bessel functions of the first kind, $J_{k-1}$.

   • They split the summation over the level $c$ into the term where $c=N$ and the tail where $c > N$.
   • The tail is bounded from above using integral tests and exponential decay estimates for the Bessel function.
   • The main term ($c=N$) is bounded from below using the newly established Kloosterman sum lower bound and explicit estimates for the Bessel function $J_{k-1}$ in the transition range (specifically when the argument is close to the order of the function).
   • The authors utilize specific lemmas regarding the behavior of $J_\nu(\nu)$ and $J_\nu(\nu + t\nu^{1/3})$ derived from Kraus's work to handle the asymptotic behavior of the Bessel functions.

Key Contributions and Results

• Theorem 1: The paper establishes the first explicit lower bound for $|S(a, b; c)|$ where $c$ is odd and $(ab, c)=1$. The bound depends on the decomposition of $c$ into squarefree and powerful parts, specifically involving $\omega(d)$, $\tau(u)$, and $\phi(u)$. The result also clarifies that the sum vanishes if $ab$ is a quadratic non-residue modulo any prime dividing the powerful part of $c$.
• Theorem 2: The authors derive an explicit lower bound for $|\Delta_{k,N}(m, n) - \delta(m, n)|$ for even weight $k$ and odd level $N$. Unlike previous asymptotic results (e.g., [JS20, Theorem 1.7]) that fix the level $N$ and let $k \to \infty$, this theorem allows $k$ and $N$ to vary independently, provided they satisfy specific inequalities involving a constant $D_0 \in (e^{H_0}, 1)$. The lower bound is of the order $J_{k-1}(k-1)e^{H_0(k-1)}$.
• Theorem 3: For a shorter range of the argument $4\pi\sqrt{mn}/N$ (specifically $[(k-1), (k-1) + (k-1)^{1/3}]$), the authors recover a lower bound of order $(k-1)^{-1/3}$, which is stronger than the exponential decay found in the wider range of Theorem 2.
• Corollaries: The paper provides specific corollaries (Corollary 1 and Corollary 3) that quantify these bounds for specific ranges of parameters, offering explicit constants for the lower bounds of the weighted trace of Hecke operators.

Significance and Claims
The authors claim that their work fills a gap in the literature by providing the first explicit lower bound for classical Kloosterman sums. This result is significant because it enables a more precise analysis of the non-vanishing of Petersson's trace formula terms.

The primary application highlighted is a "modified version" of [JS20, Theorem 1.7]. The modification allows the weight $k$ and level $N$ to vary independently, whereas previous results required $N$ to be fixed. The authors note that while their interval for the variable $4\pi\sqrt{mn}/N$ is larger (of size proportional to $k$) compared to the $k^{1/3}$ interval in [JS20], this comes at the cost of a weaker lower bound (exponential decay vs. polynomial decay). However, for a shorter interval, they successfully recover the polynomial order lower bound.

The paper concludes by providing a table of values for the derived constants $H(N)$, $k_0(N)$, and $k_1(N)$ for specific levels $N$, demonstrating the practical applicability of their explicit bounds for varying weights and levels. The work is presented as a foundational step for obtaining discrepancy results for the distribution of eigenvalues of Hecke operators, building upon the non-vanishing aspect of Kloosterman sums.