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Fourier coefficients of modular forms of half-integral weight. (English) Zbl 0606.10017
The Fourier coefficients \(a_n\) of normalized cusp forms for \(\Gamma_0(4N)\) of half-integral weight \(k\geq 5/2\) are shown to satisfy for \(n\) square-free \[ a_ n=O(n^{k/2-2/7+\varepsilon}). \] This exponent is \(3/14\) more than the Ramanujan-Petersson conjecture would give. This result implies an estimate of values at the center of the critical strip of twisted \(L\)-series related to these cusp forms.
The proof uses Petersson’s formula for Fourier coefficients of Poincaré series. The resulting sums of Kloosterman sums are estimated by clever averaging. A striking feature is the use of averaging over \(N\). Interesting are also some lemmas on Kloosterman and Salié sums.

MSC:
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F30 Fourier coefficients of automorphic forms
11L05 Gauss and Kloosterman sums; generalizations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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