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Non-abelian amplification and bilinear forms with Kloosterman sums

Alexandru Pascadi

TL;DR

This work introduces a novel non-abelian amplification method to bound Type II bilinear sums of Kloosterman sums with composite moduli, leveraging Fourier analysis on SL2(Z/cZ) and a carefully chosen amplifier built from normal subgroups. The main technical engine is a bound for the non-abelian Fourier coefficients, which is reduced to counting solutions to PSL2-congruences and completed via precise counting lemmas; for square-root-length sums with c=dd'e and d~sqrt{c}, the method delivers a saving of roughly c^{-1/12}, and it extends to general moduli by combining with prime-modulus results. The authors derive general and near-prime moduli bounds that surpass the Pólya–Vinogradov barrier in broad regimes, and apply these to moments of twisted L-functions and to large-sieve inequalities for exceptional cusp forms on composite levels. The results yield asymptotics for averaged second moments across primitive characters modulo q and provide improved exceptional-spectrum large sieve bounds at composite levels, thereby broadening the applicability of spectral methods in analytic number theory. Overall, the paper advances the toolkit for bilinear forms with Kloosterman sums, opening avenues for further generalizations to other trace functions and GL2-structures.

Abstract

We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums of length $\sqrt{c}$, our method produces a non-trivial bound for all moduli except near-primes, saving $c^{-1/12}$ for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the Pólya-Vinogradov range for all moduli. We give applications to moments of twisted cuspidal $L$-functions, and to large sieve inequalities for exceptional cusp forms with composite levels.

Non-abelian amplification and bilinear forms with Kloosterman sums

TL;DR

This work introduces a novel non-abelian amplification method to bound Type II bilinear sums of Kloosterman sums with composite moduli, leveraging Fourier analysis on SL2(Z/cZ) and a carefully chosen amplifier built from normal subgroups. The main technical engine is a bound for the non-abelian Fourier coefficients, which is reduced to counting solutions to PSL2-congruences and completed via precise counting lemmas; for square-root-length sums with c=dd'e and d~sqrt{c}, the method delivers a saving of roughly c^{-1/12}, and it extends to general moduli by combining with prime-modulus results. The authors derive general and near-prime moduli bounds that surpass the Pólya–Vinogradov barrier in broad regimes, and apply these to moments of twisted L-functions and to large-sieve inequalities for exceptional cusp forms on composite levels. The results yield asymptotics for averaged second moments across primitive characters modulo q and provide improved exceptional-spectrum large sieve bounds at composite levels, thereby broadening the applicability of spectral methods in analytic number theory. Overall, the paper advances the toolkit for bilinear forms with Kloosterman sums, opening avenues for further generalizations to other trace functions and GL2-structures.

Abstract

We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli , using Fourier analysis on and an amplification argument with non-abelian characters. For sums of length , our method produces a non-trivial bound for all moduli except near-primes, saving for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the Pólya-Vinogradov range for all moduli. We give applications to moments of twisted cuspidal -functions, and to large sieve inequalities for exceptional cusp forms with composite levels.

Paper Structure

This paper contains 31 sections, 40 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Brief background
  3. Main results.
  4. Applications
  5. Acknowledgements
  6. Outline
  7. Structure of the paper
  8. First steps: Fourier analysis
  9. The key step: Amplification
  10. Final steps: Combinatorics
  11. Comments on prime moduli
  12. Preliminaries
  13. Analytic and arithmetic notation
  14. Bounds for Kloosterman sums
  15. Fourier analysis on finite groups
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $c, M, N \in \mathbb{Z}_+$ with $M, N \ll c^{1/2+o(1)}$. Then for any complex sequences $(\alpha_m)_{m \le M}$, $(\beta_n)_{n \le N}$ and $a \in (\mathbb{Z}/c\mathbb{Z})^\times$, one has Moreover, if $|\alpha_m| \le 1$ for all $m$ (so $\|\alpha\| \le \sqrt{M}$), then

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Corollary 1.4
  • Remark
  • Theorem 1.5
  • Remark
  • Corollary 1.6
  • Remark
  • Lemma 3.1
  • ...and 97 more