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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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