Non-Abelian Amplification and Bilinear Forms with Kloosterman Sums

Abstract We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$ c , using Fourier analysis on $\mathrm{SL}_{2}(\mathbb{Z}/c\mathbb{Z})$ SL 2 ( Z / c Z ) and an amplification argument with non-abelian characters. For sums of length $\sqrt{c}$ c , our method produces a non-trivial bound for all moduli except near-primes, saving $c^{-1/12}$ 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 $L$ L -functions, and to large sieve inequalities for exceptional cusp forms with composite levels.