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Estimates for the shifted convolution sum involving Fourier coefficients of cusp forms of half-integral weight. (English) Zbl 1526.11020

The estimates for the shifted convolution sums involving the Fourier coefficients of automorphic forms have received attention continuously. In the paper under review, the authors consider the same problem for the half-integral weight modular forms. Let \(f\) and \(g\) be the cusp forms of half-integral weight with level \(4N\) and consider the following sum with fixed positive integer \(b\) and smooth function \[ S(f,g,b)= \sum_{n \geq 1} a_f(n+b)a_g(n)G(n), \] where \(a_f\) and \(a_g\) are Fourier coefficients of cusp forms \(f\) and \(g\), respectively. There are two estimates for \(S(f,g,b)\) in the paper under review, one of them is unconditional whereas for the second one the authors assume the truth of the Ramanujan-Petersson conjecture for the half-integral weight modular forms. The proofs are based on nice calculations via the Poincaré series since they generate the space of the half-integral weight cusp forms.

MSC:

11F30 Fourier coefficients of automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
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References:

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