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

Kojima的结果（1）在重数为2的假设下，我们确定了属于半整权Kohnen空间的具有任意本原特征的任意奇数阶模形式f的Fourier系数的平方与模形式F的zeta函数（f在Shimura对应下的象）的特殊值之间的显式关系。(2)我们构造了虚二次域上半整数权的Maass波型f到整数权的Maass波型g的Shimura对应S。我们将根据f的傅里叶系数明确地确定g的傅里叶系数。在Hecke算子重1定理的某些假设下，我们导出了虚二次域上半整权模形式f的Fourier系数的平方与S（f）的zeta函数的临界值之间的显式关系.此外，我们将这些结果推广到任意数域上半整数权的Maass波型f的情形.这产生一个推广的志村公式有关的傅立叶系数的希尔伯特模形式f的半整数重量在虚二次域。(3)在关于Hecke算子的重数1定理的关于f的假设下，我们导出了f的Fourier系数的平方与Shimura对应象的zeta函数的临界值之间的显式关系，其给出了结果（1）（4）的进一步简明改进本文明确地确定了有限域上某个线性群在SU（2，1）上的半整权模形式、Jacobi形式和自守形式空间中的表示迹。在某些情况下，我们可以确定表示的多重性。

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.