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Uniform bounds for Kloosterman sums of half-integral weight with applications

Qihang Sun

TL;DR

This work extends uniform bounds for sums of generalized Kloosterman sums to half-integral weight multipliers on congruence subgroups, with uniform control in the parameters $x$, $m$, and $n$. The authors develop a Kuznetsov trace formula in the mixed-sign case, together with Maass-Poincaré series and harmonic Maass form techniques, to obtain a main-term from exceptional spectrum plus power-saving error terms for a broad class of multipliers satisfying level-lifting and average-Weil bounds. A key application is a Rademacher-type exact formula for a partition statistic tied to mod $3$ ranks, yielding convergent tail bounds via uniform Kloosterman-sum estimates. They also establish admissibility criteria for multipliers, provide lower bounds on the exceptional spectrum through level lifting, and connect half-integral weight phenomena to integral-weight spectral data, thereby broadening the range of arithmetic problems accessible via uniform spectral methods.

Abstract

Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to $x$ with implied constants depending on $m$ and $n$. Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in $x$, $m$ and $n$. The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of $x$ with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in $x$, $m$ and $n$. Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to $m$ and $n$, which resulted in a convergent error estimate on the Rademacher exact formula of the partition function $p(n)$. We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo $3$, which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.

Uniform bounds for Kloosterman sums of half-integral weight with applications

TL;DR

This work extends uniform bounds for sums of generalized Kloosterman sums to half-integral weight multipliers on congruence subgroups, with uniform control in the parameters , , and . The authors develop a Kuznetsov trace formula in the mixed-sign case, together with Maass-Poincaré series and harmonic Maass form techniques, to obtain a main-term from exceptional spectrum plus power-saving error terms for a broad class of multipliers satisfying level-lifting and average-Weil bounds. A key application is a Rademacher-type exact formula for a partition statistic tied to mod ranks, yielding convergent tail bounds via uniform Kloosterman-sum estimates. They also establish admissibility criteria for multipliers, provide lower bounds on the exceptional spectrum through level lifting, and connect half-integral weight phenomena to integral-weight spectral data, thereby broadening the range of arithmetic problems accessible via uniform spectral methods.

Abstract

Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to with implied constants depending on and . Recently, in 2009, Sarnak and Tsimerman obtained a bound uniformly in , and . The generalized Kloosterman sums are defined with multiplier systems and on congruence subgroups. Goldfeld and Sarnak bounded sums of them with main terms corresponding to exceptional eigenvalues of the hyperbolic Laplacian. Their error term is a power of with implied constants depending on all the other factors. In this paper, for a wide class of half-integral weight multiplier systems, we get the same bound with the error term uniformly in , and . Such uniform bounds have great applications. For the eta-multiplier, Ahlgren and Andersen obtained a uniform and power-saving bound with respect to and , which resulted in a convergent error estimate on the Rademacher exact formula of the partition function . We also establish a Rademacher-type exact formula for the difference of partitions of rank modulo , which allows us to apply our power-saving estimate to the tail of the formula for a convergent error bound.
Paper Structure (8 sections, 20 theorems, 141 equations)

This paper contains 8 sections, 20 theorems, 141 equations.

Table of Contents

  1. Introduction: sum of Kloosterman sums
  2. Introduction: rank of partitions and Maass-Poincaré series
  3. Background and notations
  4. Proof of Theorem \ref{['exactFmlThmMod3']}
  5. Examples of admissible multipliers and a lower bound of the spectrum
  6. Proof of Proposition \ref{['Section3Prop']}
  7. Lower bound for the exceptional spectrum
  8. Kuznetsov trace formula in the mixed-sign case

Key Result

Theorem 1.2

Suppose $\nu$ is an admissible weight $k=\pm\frac{1}{2}$ multiplier on $\Gamma_0(N)$, $\tilde{m}>0$ and $\tilde{n} <0$. Then when $B\tilde{\ell}$ is square-free or coprime to $M$ for $\ell\in\{m,n\}$, we have where the sum runs over exceptional eigenvalues $\lambda_j<\frac{1}{4}$ of the hyperbolic Laplacian $\Delta_k$: $\lambda_j=\frac{1}{4}+r_j^2$, $s_j=\frac{1}{2}+\mathop{\mathrm{Im}}\nolimits

Theorems & Definitions (38)

  • Definition 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Remark
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark
  • ...and 28 more