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).
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Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday
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Ren, X., Ye, Y. Resonance between automorphic forms and exponential functions. Sci. China Math. 53, 2463–2472 (2010). https://doi.org/10.1007/s11425-010-3150-4
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DOI: https://doi.org/10.1007/s11425-010-3150-4