Non-abelian amplification and bilinear forms with Kloosterman sums

TL;DR

This paper develops a new Fourier-analytic method using non-abelian characters to bound bilinear sums of Kloosterman sums with composite moduli, leading to improved bounds and applications in number theory.

Contribution

Introduces a novel non-abelian Fourier analysis technique with amplification for bounding bilinear Kloosterman sums over composite moduli, extending previous prime modulus results.

Findings

Achieves non-trivial bounds for sums of length √c for most moduli.

Provides savings beyond the Pólya-Vinogradov range for all moduli.

Applies results to moments of twisted L-functions and large sieve inequalities.

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 P\'olya-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.