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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:
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