On some exponential sums involving Maass forms over arithmetic progressions.

Let g ( z ) be a Maass cusp form for SL ( 2 , Z ) , and let λ g ( n ) be the n -th Fourier coefficient of g ( z ) . In this paper we investigate the nonlinear exponential sum ∑ n ∼ X n ≡ l mod q λ g ( n ) e ( α n β ) twisted by Fourier coefficient over the arithmetic progression. We prove an asymptotic formula when β = 1 / 2 and α is close to ± 2 k q , k ∈ Z + . [ABSTRACT FROM AUTHOR]