On Dirichlet Series and Abelian Varieties Attached to Automorphic Forms

Abstract

Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its generalization for other types of automorphic forms. In the case of the Hilbert modular group of level one, Herrmann [14] succeeded in this problem; he has shown the necessity of considering not only the product of the upper half-planes but also the domain r consisting of the points (zl, * * , Zr) of the r-dimensional complex vector space Cr such that Im(z,) # 0, * ., Im(zr) # 0, and h distinct discontinuous groups commensurable to each other, h being the class number of the totally real number field in the problem. On the other hand, the unit-group of an order in an indefinite quaternion algebra over the rational number field Q yields a fuchsian group. In this case, Eichler [6] defined Hecke's operators as representations of algebraic correspondences, called modular correspondences, and proved a formula for the trace of the operators. The trace-formula was proved also by Selberg [22] in a more general formulation. Recently, Godement [9] has given a theory of zeta-functions attached to division algebras; namely, he has shown the possibility of applying the adele-idele method of Iwasawa-Tate [15, 28] to automorphic functions and forms with respect to the unit-group of a division algebra over Q. The case of non-holomorphic automorphic functions of this type has been investigated by Tamagawa [27]. In [25] I have treated cusp-forms with respect to the unit-group of an indefinite quaternion algebra over Q. Now, the purpose of Part I of the present paper is to develop an analogous theory for automorphic forms on the domain ; mentioned above with respect to the unit-groups of an indefinite quaternion algebra. Let F be a totally real algebraic number field of degree t, and D be a quaternion algebra over F. Denote by r the number of infinite prime spots of F unramified in D, and suppose that r > 0. Let D,1 and D-2 denote the product of r copies of the total matric algebra M2(R) over the real number field R and the product of t r copies of the division ring K of real quaternions, respectively. Then, D( Q R is isomorphic to D,1 x D12. Let o be a maximal order in D and I' be the group of units in o. Let y, and 72 237