Resonance between automorphic forms and exponential functions

Abstract

Let f be a holomorphic cusp form of weight k for SL 2(ℤ) and λ f (n) its n-th Fourier coefficient. In this paper, the exponential sum ΣX < n ⩽ 2Xλ f (n)e(αn B ) twisted by Fourier coefficients λ f (n) is proved to have a main term of size |λ f (q)|X 3/4 when β = 1/2 and α is close to $$ \pm 2\sqrt q ,q \in \mathbb{Z} $$ , and is smaller otherwise for β < 3/4. This is a manifestation of the resonance spectrum of automorphic forms for SL 2(ℤ).