On Modular Forms of Half Integral Weight

The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, but the forms of half integral weight have attracted only casual attention, in spite of their importance and ancientness. Indeed, the connection of such forms with zeta functions was never clarified. When Hecke developed his theory of Euler product forthe forms of integral weight, he pointed out the impossibility of a similar theory for the forms of half integral weight, and that only partial information could be obtained for the Fourier coefficients of such forms (Werke, p. 639). He explained this in more detail in his last paper 3, which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases. A treatment of a more general type of modular form was given by Wohlfahrt 12. In fact, he defined Hecke operators whose degree is the square of a prime, and showed a certain multiplicative relation, as predicted by Hecke, for the Fourier coefficients, but discussed neither Euler product, nor connection with zeta functions. In the present paper, we try to reveal a more affirmative aspect of the subject. To be specific, put, for each positive integer N,