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On the Fourier coefficients of Hilbert modular forms of half-integral weight. (English) Zbl 0802.11017

In [Ann. Math. (2) 97, 440–481 (1973; Zbl 0266.10022)] the author proved that there is a certain correspondence (now well-known under the name Shimura correspondence) between elliptic modular forms \(f\) of half-integral weight \(k+{1\over 2}\) and level \(4N\) and elliptic modular forms \(F\) of weight \(2k\) and level \(2N\). Later on J.-L. Waldspurger [J. Math. Pures Appl. (9) 60, 375–484 (1981; Zbl 0468.10014)] showed that the special value at the center of the critical strip \(s=k\) of the twist of the Hecke \(L\)-series \(L(F,s)\) by the quadratic character of conductor \(t\) is essentially proportional to the square of the \(t\)-th Fourier coefficient of \(f\). A more exact formula giving the constant of proportionality in an explicit form was proved in [W. Kohnen and D. Zagier, Invent. Math. 64, 175–198 (1981; Zbl 0468.10015); W. Kohnen, Math. Ann. 271, 237–268 (1985; Zbl 0553.10020)] in the case where \(N\) is squarefree and the character is trivial and also in [S. Niwa, Proc. Jap. Acad., Ser. A 58, 90–92 (1982; Zbl 0524.10018)] in the case where \(N\) is 4 and the character is \(({{} \over p})\) with \(p\) a prime.
In the present paper the author proves several formulas of the same exact nature for Hilbert modular forms of arbitrary level over an arbitrary totally real number field.
Reviewer: W.Kohnen (Bonn)

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F30 Fourier coefficients of automorphic forms
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References:

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