## Algebraic number theory. (Algebraische Zahlentheorie.)(German)Zbl 0747.11001

Berlin etc.: Springer-Verlag. xiii, 595 S. (1992).
During the recent decades, algebraic number theory has undergone an enormous and far-reaching development. This process is essentially characterized by the fact that the geometric point of view has gained a central significance in the arithmetic theory of fields. The framework of modern algebraic geometry, its general concepts, methods, and results as well as its complex-analytic aspects have penetrated algebraic number theory in a natural way and to a growing extent. The interrelation between classical algebraic number theory and modern algebraic geometry has led to a particular branch of mathematics, which is commonly called “arithmetic algebraic geometry”. The recent epoch-making results concerning, for example, the Weil conjectures, the Mordell conjecture, the Birch-Swinnerton-Dyer conjecture, and other long-standing problems have been achieved by systematically using methods of (arithmetic) algebraic geometry, in the course of which the algebro-geometric viewpoint has crucially contributed to a deeper systematization, conceptual unification, and methodical clarity in algebraic number theory as a whole. Eventually, the recent methods and results of arithmetic algebraic geometry have also proved to be of increasing significance for constructing models of physical theories, in particular quantum field theories.
In regard to this development, it is unquestionable that algebraic number theory has reached a new stage of both theoretical compactness in itself and intertwining with diophantine geometry. Consequently, it became highly desirable that this new quality would be suitable reflected, in a fundamental and updated manner, in a comprehensive textbook on algebraic number fields itself. On the one hand, there are many excellent textbooks on algebraic number theory, and there are now also a few treatises on topics in arithmetic algebraic geometry. On the other hand, however, the established textbooks on algebraic number theory are rather aged, in the meantime, and mainly impart the classical material, nevertheless from different points of view and in varying degree of theoretical uniformity, whereas the available texts on arithmetic algebraic geometry are, throughout, fairly advanced and specific, already assuming that the reader is sufficiently familiar with both the fundamentals of classical algebraic number theory and the basic framework of modern algebraic geometry. The present book has as its aim to resolve this discrepancy in the relevant textbook literature and, apparently for the first time in such a throughgoing way, to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. In contrast to the existing textbooks, the present work focuses on elaborating thoroughly the higher theoretical completeness of the subject, with special emphasis on the general functorial and algebro- geometric methods. In this sense, the author has given preference, wherever possible, to the powerful general (and generalizable) geometric aspects, instead of the many common — though fascinating and ingenious — special artifices and ad-hoc methods in number theory.
Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner, who is required to only have the basic knowledge in university algebra, and it develops the modern abstract framework only as far as necessary for a profound understanding of the rich classical theory of algebraic number fields in its modern, unifying geometric interpretation. Chapter I presents the foundations of the global theory of algebraic number fields from the classical ideal- theoretic point of view. In addition, this chapter includes, as a first special feature, a geometric approach to algebraic orders in number fields. This already illustrates the geometric background, namely by the analogy between algebraic orders and the theory of singular algebraic curves; and it motivates, in the sequel, the introduction of one- dimensional algebraic schemes (curves) in the light of their arithmetic examples. Chapter II deals with the local aspects of number fields, that is, with the classical theory of valuations. This encompasses sections on $$p$$-adic numbers, $$p$$-adic valuations, general valuations and valuation rings, completions, local fields, Henselian fields, unramified and tamely ramified extensions, the Galois theory of valuations, and an introduction to higher ramification groups. The basic material developed so far is completed in the following Chapter III. This chapter represents, in particular, the afore-mentioned new character of the textbook under review, in that it turns towards the modern algebro-geometric approach to algebraic number fields. The author discusses the classical concepts and results in number field theory from the actual viewpoint of (one- dimensional) Arakelov theory which, for its part, essentially consists in transferring results from algebraic geometry over algebraically closed fields (i.e., from the function field case) to the number field case. This geometric viewpoint is not entirely new; it can be traced back to an earlier paper by A. Weil [Sur l’analogie entre les corps de nombres algébriques et les corps de fonctions algébriques (1939) in: Collected Papers. Vol. 1 (Springer 1979; Zbl 0424.01027)]. However, the decisive break-through in understanding this analogy was provided by S. Yu. Arakelov in 1974, who succeeded in constructing an intersection theory for divisors in arithmetic surfaces, which had precisely the crucial properties known from the theory of nonsingular projective complex surfaces in algebraic geometry [cf. Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1179-1192 (1974; Zbl 0355.14002)]. Arakelov’s basic idea of completing the set of points (and divisors) of an arithmetic surface by including those at infinity (i.e., the Archimedean places of the arithmetic ground field) and considering Hermitean structures on the extended sheaves, so that complex analysis got naturally involved, gave the possibility to transpose the fundamental geometric invariants (e.g., divisor class groups, Picard groups, Chow groups, Grothendieck groups, cohomology groups) adequately to arithmetic-geometric objects. As for the one-dimensional case of arithmetic curves, i.e., for spectra of rings of integers in number fields, the analogous concepts of Arakelov’s theory provide a new, geometrically disposed insight into classical algebraic number theory. This was sketchily pointed out, for the first time, by L. Szpiro in 1983 [Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell, Astérisque 127, 11-28 (1985; Zbl 0588.14028)] and, later on, generally worked out by E. Hübschke [Arakelovtheorie für Zahlkörper, Der Regensburger Trichter 20, Diss. (1987; Zbl 0743.14020)].
Chapter III of the present textbook provides a detailed account of this modern viewpoint in the one-dimensional case and, in addition, a rigorous embedding of this arithmetic subject into the general (geometric) Grothendieck-Riemann-Roch theory for algebraic schemes. This careful, comprehensive introduction takes the reader to the forefront of current research in arithmetic geometry, in that it enables him to study the very recent developments in the higher-dimensional case [e.g., G. Faltings, Lectures on the arithmetic Riemann-Roch theorem, Ann. Math. Stud. 127, Princeton Univ. Press (1992; Zbl 0744.14016)] by already knowing the philosophy from the example of arithmetic curves.
The reamaining Chapters IV-VII, which form the second half of the book, are devoted to two other central areas of algebraic number theory: class field theory and the theory of algebraic zeta functions and $$L$$- series.
More precisely, Chapters IV-VI consecutively treat, in an introductory yet very thoroughgoing manner, the main contents of general class field theory, local class field theory, and global class field theory. The presentation given here differs from the one offered in the author’s earlier standard textbook “Class Field Theory” (Springer-Verlag 1986; Zbl 0587.12001), in that the material is not only structurally appended to the foregoing introductory chapters, but also arranged in another way which leads again to a higher level of coherency, theoretical compactness, methodical effectivity, and alluding actual relatedness. Moreover, the presentation is slightly more detailed than in other textbooks on class field theory, including the author’s afore-mentioned standard work, and particularly rich in illustrating complements, hints for further study, and concrete examples.
A peculiar feature is certainly given by the early introduction and use of profinite groups and infinite Galois theory (in Chapter IV), as well as by the thorough treatment of formal groups, the Lubin-Tate theory, and higher ramification theory. This is perhaps more than a textbook is expected to offer, but perfectly serves the author’s ambitious aim of exhibiting the actual width of modern algebraic number theory, together with its relations to classical problems.
The concluding Chapter VII on zeta functions and $$L$$-series is another outstanding advantage of the present textbook. This classical topic, which relates complex analysis and number theory in a fascinating and deep-going way, has gained new interest and central significance during the recent decades, too. Since Hecke’s pioneering work on $$L$$-series [cf. E. Hecke, Mathematische Werke, 2nd. ed. (Göttingen 1970; Zbl 0205.28902)], which is still barely accessible to the non-specialist, this subject had never been updatedly included in a general textbook on number theory. Certainly, there are some special treatises on $$L$$-series, for example J. Tate’s famous Ph. D. thesis (Princeton 1950) which provides an ingenious approach to Hecke’s $$L$$-series by methods of harmonic analysis [as for a published version, cf. J. W. S. Cassels and A. Fröhlich (ed.), Algebraic Number Theory, Proc. Int. Conf., London, New York, Acad. Press (1967; Zbl 0153.07403)], or the omnibus volume [A. Fröhlich (ed.), Algebraic Number Fields ($$L$$-functions and Galois properties). Academic Press (1977; Zbl 0339.00010)]; however, here the author has not only included this topic in a general textbook on algebraic number theory but, beyond that, he has also entered upon the very rewarding and creative task of elaborating the first modern version of Hecke’s original approach, followed by a likewise comprehensive account on the Artin theory of $$L$$-series and their functional equations. This updated representation of Hecke’s theory fills a longstanding gap in number theory. Apart from its didactic value, it is an essential contribution of the author to the further theoretical consistency of number theory, in particular, and of pure mathematics in general. Likewise, the detailed treatment of the Artin $$L$$-series, in this context, is new and unique in the textbook literature, also in regard to included hints to the recently discovered (infinite-dimensional) geometric generalization of $$L$$-series [cf., e.g., M. Rapoport, N. Schappacher, P. Schneider (ed.), Beilinson’s Conjectures on Special Values of $$L$$-Functions, Perspectives in Mathematics 4, Boston, Academic Press (1988; Zbl 0635.00005)].
Altogether, the present work is an outstanding textbook. It is, without any doubt, the most actual, systematic and theoretically comprehensive textbook on algebraic number field theory available. Every section goes with a set of selected exercises. Many of them are intended as an invitation to further studies, just to complete the material by related (or deeper-going) topics. This provides the reader with additional guidance to the current literature, besides the excellent preparation of the text itself, which is enhanced by numerous motivating explanations, didactic recalls, and indications to related recent developments. For all that, the book is widely self-contained and, despite the vast material, of great clarity and intelligibility. It is perfectly suited as a modern reference book, too, because of its nearly encyclopedic character, and it is likewise a distinguished source for lectures and seminars. In this preface, the author proposes several variants for teaching by using this book.
Undoubtedly, Professor Neukirch’s recent book will quickly become a standard text on contemporary algebraic number theory, likewise for students, teachers, researchers, and interested non-specialists in mathematics or theoretical physics.

### MSC:

 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 11-02 Research exposition (monographs, survey articles) pertaining to number theory 14Gxx Arithmetic problems in algebraic geometry; Diophantine geometry 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 11Gxx Arithmetic algebraic geometry (Diophantine geometry) 11Rxx Algebraic number theory: global fields 11Sxx Algebraic number theory: local fields 14Hxx Curves in algebraic geometry 11M35 Hurwitz and Lerch zeta functions