Hecke theory over arbitrary number fields

Abstract

In 1954, Hermann [6] developed an explicit Hecke theory over totally real fields by using ideal numbers. For a field of class number h, he introduced modular forms which were actually vectors of h different functions. Recently, persons have studied modular forms over totally complex fields (Goldfeld et al. [2]) or over arbitrary number fields (Stark [8]) with h = 1. When h > 1, however, complications arise. This paper will develop the tools of Hecke theory for study of modular forms over arbitrary number fields when h> 1. Section 2 introduces ideal numbers following Hecke [S]. Section 3 defines the appropriate upper half space and the action of matrix operators. In Section 4, we define the notion of a vector modular form, with the crucial point being the introduction of h-length vectors, with h-length vectors of matrix operators. These ideas follow Hermann [6]. Section 5 describes Hecke operators, and their effect on the Fourier expansion. Section 6 outlines the theory of Dirichlet series, their Euler products, and functional equations. In Section 7, we describe the Petersson inner product and prove the self adjointness of the Hecke operators. Section 8 contains the main theorem. Intuitively, one might suspect that ideal numbers are unnecessary-the principal elements should suffice. In Section 8, we show the principal component of a vector modular form and its action under “principal” Hecke operators do determine the entire form. Throughout the paper, we illustrate the general theory with examples of Eisenstein series over Q(a). These concrete examples demonstrate the value of this approach to obtain results analogous to the classical Eisenstein series. It is well known that an adelic version of Hecke theory exists over any number field. For instance, Weil outlines an adelic approach, saying “technical difficulties arose when the number of ideal classes is greater than