Arithmetic study of Fourier coefficients of automorphic forms and modular forms and zeta functions

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Kojima's results(1) Under the assumption of the multiplicity 2 theorem, we determined an explicit relation between the square of Fourier coefficients of modular forms f belonging to the Kohnen's space of half integral weight and of arbitrary oddlevel with arbitrary primitive character and special values of the zeta function of the modular form F which is the image of f under the Shimura correspondence.(2) We constructed the Shimura correspondence S from Maass wave forms f of half integral weight over imaginary quadratic fields to those g of integral weight. We shall determine explicitly the Fourier coefficients of g in terms of these of f. Under some assumptions about the multiplicity one theorem with respect to Hecke operators, we deduced an explicit connection between the square of Fourier coefficients of modular forms f of half integral weight over imaginary quadratic fields and the critical value of the zeta function associated with S(f). Moreover, we generalized those results in the case of Maass wave forms f of half integral weight over arbitrary number fields. This yield a generalization of Shimura's formula concerning Fourier coefficients of Hilbert modular forms f of half integral weight over imaginary quadratic fields.(3) Under the assumptions about f concerning the multiplicity one theorem with respect to Hecke operators, we deduced an explicit connection between the square of Fourier coefficients of f and the critical value of the zeta function associated with the image of Shimura correspondence, which gives a further concise improvement of the results (1)(4) We determined explicitly the trace of representations of certain linear group over finite field into the spaces of modular forms of half integral weight, Jacobi forms and automorphic forms on SU(2,1). In the some case, we can determine the multipicity of representation.