On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions
In 1981 \textit{J.-L. Waldspurger} [J. Math. Pures Appl. (9) 60, 375--484 (1981; Zbl 0431.10015)] proved important new results concerning proportionality between Fourier coefficients of modular forms of half integral weight and special values of twisted \(L\)-series defined by Shimura correspondences. Soon afterwards \textit{W. Kohnen} and \textit{D. Zagier} [Invent. Math. 64, No. 2, 175--198 (1981; Zbl 0468.10015)] and \textit{W. Kohnen} [J. Reine Angew. Math. 333, 32--72 (1982; Zbl 0475.10025) and Math. Ann. 271, No. 2, 237--268 (1985; Zbl 0542.10018)] found the explicit value of the constant of proportionality in certain special cases. (In particular, for newforms of square-free level belonging to Kohnen's spaces with quadratic characters.) In 1999 the first author [J. Math. Soc. Japan 51, No. 3, 715--730 (1999; Zbl 1012.11035)] generalized results of Kohnen and Zagier to situations of Kohnen's spaces of arbitrary odd level and of arbitrary characters under the assumption of multiplicity two theorem of Hecke operators. In the present paper the authors extend this work. They generalize Kohnen's results and under the assumption of multiplicity one theorem they prove that the square of Fourier coefficients of modular forms of half integral weight, belonging to Kohnen's spaces of general odd level and of odd character, coincides with the critical values of zeta functions twisted by quadratic characters.
scientific article; zbMATH DE number 2158077
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