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Fourier analysis on number fields. (English) Zbl 0916.11058

Graduate Texts in Mathematics. 186. New York, NY: Springer. xxi, 350 p. (1999).
This book imparts a close insight into that part of mathematics which is concerned with \(\zeta\)-functions and \(L\)-series of number fields, and is of valuable help when studying \(L\)-functions of Galois representations or \(p\)-adic \(L\)-functions, as for example in connection with the Chinburg conjectures and problems in Galois module structure theory, or representation theory of reductive groups. Anyone who is familiar with the basic concepts in algebra and topology can dig into the book right away. The reader will continuously be led towards the main chapter regarding Tate’s work on local and global zeta functions, in particular to his proof of the functional equation and analytic continuation of \(L\)-functions of idèle class group characters and the many consequences thereof. The road to Tate’s thesis starts off from a chapter on topological groups with the main ingredients Haar measure and profinite groups (including their topological characterization and some material on pro-\(p\)-groups).
The second chapter addresses the representation theory of locally compact groups and touches on the Gelfand theory for Banach spaces and the two spectral theorems; its main topic is the discussion of unitary representations of Hilbert spaces.
Then, in chapter 3, duality for locally compact abelian groups is presented. The following three chapters (The structure of arithmetic fields; Adeles, ideles, and the class groups; A quick tour of class field theory) are concerned with number theory and include the classification of locally compact fields, some ramification theory, completions of global fields \(K\), the approximation theorem, the geometry of \(\mathbb{A}_K/K\), the relationship between the idèle class group, and ray class groups, Chebotarev’s density theorem, the transfer, and Artin reciprocity. As far as class field theory is dealt with, proofs are omitted; however, as an application the Kronecker-Weber theorem is shown.
Finally, in chapter 7, the authors bring in the harvest by turning to Tate’s thesis. This chapter has also a section on the analogy between the Poisson summation formula for number fields and the Riemann-Roch theorem for curves over finite fields, as well as a section with the well-known theorem of Hecke, by which two idèle class characters of a global field, which coincide for a set of primes of positive density, only differ by some character of finite order.
The book finishes with two appendices, the first one on normed linear spaces and the second on Dedekind domains.
The individual chapters are accompanied with exercises which often lead deeper into the presented theory and may stimulate the reader to consult books specializing on the subject.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
43A70 Analysis on specific locally compact and other abelian groups
11-02 Research exposition (monographs, survey articles) pertaining to number theory
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
22D10 Unitary representations of locally compact groups
22B05 General properties and structure of LCA groups
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
11R56 Adèle rings and groups
11R37 Class field theory
11R45 Density theorems
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