Non-Abelian Amplification and Bilinear Forms with Kloosterman Sums
<jats:p> We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli <jats:inline-formula> <jats:alternatives> <jats:tex-math>$c$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , using Fourier analysis on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\mathrm{SL}_{2}(\mathbb{Z}/c\mathbb{Z})$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mi>c</mml:mi> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> and an amplification argument with non-abelian characters. For sums of length <jats:inline-formula> <jats:alternatives> <jats:tex-math>$\sqrt{c}$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>c</mml:mi> </mml:msqrt> </mml:math> </jats:alternatives> </jats:inline-formula> , our method produces a non-trivial bound for all moduli except near-primes, saving <jats:inline-formula> <jats:alternatives> <jats:tex-math>$c^{-1/12}$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -functions, and to large sieve inequalities for exceptional cusp forms with composite levels. </jats:p>