Hecke theory over arbitrary number fields
Let K be an algebraic number field and h its class number. An adelic approach to Hecke theory over arbitrary K was outlined by \textit{A. Weil} [Dirichlet series and automorphic forms (Lect. Notes Math. 189) (1971; Zbl 0218.10046)]. The author develops an explicit Hecke theory along classical lines, reconsidering and extending ideas of \textit{O. Herrmann} [Math. Ann. 127, 357-400 (1954; Zbl 0055.312)] who treated the case of totally real K. The main features of this approach are the introduction of a set Z of ideal numbers for K in Hecke's sense, of h-length vectors of modular forms (one component for each ideal class), and of h-length vectors of matrix operators. Since addition is not in general possible within Z, care has to be taken to consider only matrices which have a determinant in Z. The components of the vector modular forms are functions on the half space \(H=H_ 1^{r_ 1}\times H_ 2^{r_ 2}.\) Here, \(r_ 1\) and \(r_ 2\) are the numbers of real, respectively complex embeddings of K, \(H_ 1=\{x+yk |\) \(x\in {\mathbb{R}}\), \(y>0\}\), \(H_ 2=\{x+yk |\) \(x\in {\mathbb{C}}\), \(y>0\}\), \({\mathbb{C}}=\{a+bi |\) \(a,b\in {\mathbb{R}}\}\), and i,k are the usual quaternion elements satisfying \(i^ 2=k^ 2=-1\), \(ik=-ki\). This choice of half-spaces allows an operation of GL(2,\({\mathbb{R}})\) on \(H_ 1\), and of GL(2,\({\mathbb{C}})\) on \(H_ 2\) such that, for example, \(\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} 0\\ - 1\end{matrix} \right)\in GL(2,{\mathbb{R}})\) acts as \(z \mapsto -\bar z.\) The definition of vector modular forms, their automorphy factors and Fourier expansions, and the definition of arithmetical groups associated with K and operating on H, are quite cumbersome and will not be reproduced here. But it is worth noting that since \(H_ 2\) has no complex structure, differentiable rather than holomorphic modular forms have to be considered. The definition of Hecke operators resembles Hecke's original formulas. (No double cosets show up.) Then the author treats the expected topics: effect of Hecke operators on Fourier expansions, Dirichlet series with Euler product and functional equation, Petersson inner product, self-adjointness of Hecke operators. Finally, his main theorem states that the principal component (corresponding to the principal ideal class) of an eigenform determines the entire form.
effect of Hecke operators on Fourier expansions
Dirichlet series with Euler product and functional equation
scientific article; zbMATH DE number 4130466