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.


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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[1] Bateman, H.: Higher transcendental functions II. New York-Toronto-London: McGraw-Hill 1953 · Zbl 0051.34703
[2] Burgess, D.: On character sums and primitive roots. Proc. Lond. Math. Soc. (3)12, 179-192 (1962) · Zbl 0106.04003
[3] Deligne, P.: La conjécture de Weil I. Publ. Math. Inst. Hautes Etud. Sci.43, 273-307 (1974) · Zbl 0287.14001
[4] Flicker, Y.: Automorphic forms on covering groups ofGL(2). Invent. math.57, 119-182 (1980) · Zbl 0431.10014
[5] Gelbart, S.: Weil’s representation and the spectrum of the metaplectic group. Lecture Notes in Math., vol. 530. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0365.22017
[6] Goldfeld, D., Hoffstein, J.: Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series. Invent. math.80, 185-208 (1985) · Zbl 0564.10043
[7] Hooley, C.: On the greatest prime factor of a cubic polynomial. J. Reine Angew. Math.303/304, 21-50 (1978) · Zbl 0391.10028
[8] Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann.271, 237-268 (1985) · Zbl 0553.10020
[9] Mordell, L.J.: The sign of the Gaussian sum. Ill. J. Math.6, 177-180 (1962) · Zbl 0101.28001
[10] Niwa, S.: Modular forms of half-integral weight and the integral of certain theta-functions. Nagoya Math. J.56, 147-161 (1975) · Zbl 0303.10027
[11] Petersson, H.: Über die Entwicklungskoeffizienten der automorphen Formen. Acta Math.58, 169-215 (1932) · Zbl 0003.35002
[12] Rankin, R.: Modular forms and functions. Cambridge-London-New York: Cambridge University Press 1977 · Zbl 0376.10020
[13] Salié, H.: Über die Kloostermanschen SummenS(u, v; q). Math. Z.34, 91-109 (1931) · JFM 57.0211.01
[14] Serre, J-P., Stark, H.M.: Modular forms of weight 1/2. Springer Lecture Notes in Math., vol. 627, pp. 27-67. Berlin-Heidelberg-New York: Springer 1977
[15] Shintani, T.: On construction of holomorphic cusp forms of half-integral weight. Nagoya Math. J.58, 83-126 (1975) · Zbl 0316.10016
[16] Shimura, G.: On modular forms of half-integral weight. Ann. Math.97, 440-481 (1973) · Zbl 0266.10022
[17] Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl.60, 375-484 (1981) · Zbl 0431.10015
[18] Williams, K.S.: Note on Salié’s sum. Proc. Am. Math. Soc.30(2), 393-394 (1971) · Zbl 0224.10039
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