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The Quarterly Journal of Mathematics Advance Access published online on February 19, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap008
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ON CUSP FORM COEFFICIENTS IN NONLINEAR EXPONENTIAL SUMS

Qingfeng Sun{dagger}

Department of Mathematics, Shandong University, Jinan 250100, People's Republic of China

{dagger} E-mail: qfsun{at}mail.sdu.edu.cn

Received 2 December 2008; revised 14 January 2009
  Abstract

Let f be either a holomorphic Hecke eigenform of weight {kappa} for SL2(Z) with


Formula

or a Maass Hecke eigenform for SL2(Z) with Laplace eigenvalue 1/4 + {nu}2. In the latter case,


Formula

Here Ki{nu} is the modified Bessel function of the third kind and e(z) = e2{pi}iz. This paper studied the cancelation of the coefficients {lambda}(n) or {rho}(n) in nonlinear exponential sums with amplitude n{theta}, 0 < {theta}≤ 1/2.


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