Arithmetic study of Fourier coefficients of modular forms of half integral weight and Siegel modular forms

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(1) We shall constract the Shimura correspondence S from Hilbert-Maass wave forms f of half integral weight over algebraic number field to Hilbert-Maass wave forms S(f) of integral weight over algebraic number fields. Moreover, we establish an explicit connection between the square of Fourier coefficients of f and the central value of quadratic twisted L-series associated with the image S(f) off.(2) W.Kohnen reformulated the Ikeda lifting as a linear mapping and he formulated a Maass space of Siegel modular forms of degree 2n. Moreover, he purposed a conjecture that the image of Ikeda lifting is equal to the Maass space of degree 2n. In a joint work with W.Kohnen, we proved that this conjecture is ture in the case where 4|n and 4|n-1.