Hecke theory over arbitrary number fields
type: Item
instance of: scholarly article
title: Hecke theory over arbitrary number fields
zbMATH Open document ID: 0691.10017
DOI: 10.1016/0022-314X(89)90001-2
author: Robert Styer
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: 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…
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11R99
zbMATH DE Number: 4130466
zbMATH Keywords: algebraic number field
zbMATH Keywords: Hecke theory
zbMATH Keywords: ideal numbers
zbMATH Keywords: h-length vectors of modular forms
zbMATH Keywords: h-length vectors of matrix operators
zbMATH Keywords: vector modular forms
zbMATH Keywords: automorphy factors
zbMATH Keywords: Fourier expansions
zbMATH Keywords: effect of Hecke operators on Fourier expansions
zbMATH Keywords: Dirichlet series with Euler product and functional equation
zbMATH Keywords: Petersson inner product
zbMATH Keywords: self-adjointness of Hecke operators
zbMATH Keywords: principal component
zbMATH Keywords: eigenform
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4130466