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Fourier coefficients and modular forms of half-integral weight. (English) Zbl 0542.10018
In two important papers [J. Math. Pures Appl., IX. Sér. 59, 1–32 (1980; Zbl 0412.10019); ibid. 60, 375–484 (1981; Zbl 0431.10015)] J.-L. Waldspurger showed that under the Shimura correspondence between Hecke eigenforms of weight $$k+1/2$$ and weight $$2k$$ the square of the $$m$$th Fourier coefficient ($$m$$ squarefree) of a form of half-integral weight is essentially proportional to the value at $$s=k$$ (center of the critical strip) of the $$L$$-series of the corresponding form of integral weight twisted with the quadratic character of $$\mathbb Q(\sqrt{(-1)^ km})$$.
The main purpose of the present paper is to give a formula for the product $$c(m)\overline{c(n)}$$ of two arbitrary Fourier coefficients of a Hecke eigenform $$g$$ of half-integral weight and of level $$4N$$ with $$N$$ odd and squarefree in terms of certain cycle integrals of the corresponding form $$f$$ of integral weight. This includes as a special case $$(m=n)$$ Waldspurger’s result for odd squarefree level and also generalizes [(*) the author and D. Zagier, Invent. Math. 64, 175-198 (1981; Zbl 0468.10015)], where for level 1 the constant of proportionality between the values of the twists and the squares of the Fourier coefficients in Waldspurger’s theorem was given explicitly.
As corollaries we obtain results already proved for level 1 in (*), e.g. the nonnegativity of the values of the twists at $$s=k$$ or the fact that the square of the Petersson norm of $$g$$ divided by one of the periods of $$f$$ is algebraic (the latter result was also obtained by G. Shimura [J. Math. Soc. Japan 33, 649–672 (1981; Zbl 0494.10018)]). As another corollary we also deduce that $$c(m)=O(m^{k/2+\varepsilon})$$ for every $$\varepsilon>0$$. This has been previously proved by different methods, namely by combining Waldspurger’s theorem with estimates for $$L$$-series on the critical line à la Phragmén-Lindelöf [cf. the remark in D. Goldfeld, J. Hoffstein and S. J. Patterson, Progr. Math. 26, 153–193 (1982; Zbl 0499.10029), Introduction].

##### MSC:
 11F30 Fourier coefficients of automorphic forms 11F37 Forms of half-integer weight; nonholomorphic modular forms
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##### References:
 [1] Abramowitz, M., Stegun, I.: Handbook of mathematical functions. New York: Dover 1965 · Zbl 0171.38503 [2] Goldfeld, D., Hoffstein, J., Patterson, S.J.: On automorphic functions of half-integral weight with applications to elliptic curves. In: Number theory related to Fermat’s last theorern, Proc. of a Conf. sponsored by the Vaughn Foundation, ed. N. Koblitz. Progress in Maths. Vol. 26. Boston: Birkhäuser 1982 · Zbl 0499.10029 [3] Gross, D., Zagier, D.: Points de Heegner et dérivées des fonctionsL. Preprint 1983 [4] Kohnen, W.: Beziehungen zwischen Modulformen halbganzen Gewichts und Modulformen ganzen Gewichts. Dissertation. Bonn. Math. Schr.131 (1981) · Zbl 0451.10016 [5] Kohnen, W.: Newforms of half-integral weight. J. reine angew. Math.333, 32-72 (1982) · Zbl 0475.10025 · doi:10.1515/crll.1982.333.32 [6] Kohnen, W., Zagier, D.: Values ofL-series of modular forms at the center of the critical strip. Invent. Math.64, 175-198 (1981) · Zbl 0468.10015 · doi:10.1007/BF01389166 [7] Kohnen, W., Zagier, D.: Modular forms with rational periods. In: Modular forms, ed. R.A. Rankin, Ellis Horwood Limited Publishers, Chichester, 1984 · Zbl 0618.10019 [8] Manin, Y.: Periods of parabolic forms andp-adic Hecke series. Math. USSR Sb.,21, 371-393 (1973) · Zbl 0293.14008 · doi:10.1070/SM1973v021n03ABEH002022 [9] Niwa, S.: Modular forms of half-integral weight and the integral of certain theta-functions. Nagoya Math. J.56, 147-161 (1974) · Zbl 0303.10027 [10] Niwa, S.: On certain thera functions and modular forms of half-integral weight. Preprint 1983 [11] Rankin, R.A.: The scalar product of modular forms. Proc. Lond. Math. Soc.2, 198-217 (1972) · Zbl 0049.33904 · doi:10.1112/plms/s3-2.1.198 [12] Sarnak, P.: Class numbers of indefinite binary quadratic forms. J. Number Theory15, 229-247 (1982) · Zbl 0499.10021 · doi:10.1016/0022-314X(82)90028-2 [13] Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure appl. Math.29, 783-804 (1976) · Zbl 0348.10015 · doi:10.1002/cpa.3160290618 [14] Shimura, G.: On modular forms of half-integral weight. Ann. Math.97, 440-481 (1973) · Zbl 0266.10022 · doi:10.2307/1970831 [15] Shimura, G.: The critical values of certain zeta functions associated with modular forms of half-integral weight. J. Math. Soc. Japan33, 649-672 (1981) · Zbl 0494.10018 · doi:10.2969/jmsj/03340649 [16] Shintani, T.: On construction of holomorphic cusp forms of half-integral weight. Nagoya Math. J.58, 83-126 (1975) · Zbl 0316.10016 [17] Siegel, C.L.: Über die Klassenzahl quadratischer Zahlkörper. Acta Arithm.1, 83-86 (1935) · JFM 61.0170.02 [18] Waldspurger, J.L.: Correspondances de Shimura et Shintani. J. Math. Pures Appl.59, 1-133 (1980) · Zbl 0412.10019 [19] 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 [20] Zagier, D.: Modular forms associated to real quadratic fields. Invent. Math.30, 1-46 (1975) · Zbl 0308.10014 · doi:10.1007/BF01389846 [21] Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In: Modular forms of one variable VI. Lect. Notes Math., vol. 627, pp. 105-169. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0372.10017
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