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Number theory in function fields. (English) Zbl 1043.11079
Graduate Texts in Mathematics 210. New York, NY: Springer (ISBN 0-387-95335-3/hbk). xii, 358 p. (2002).
Since the times of Hasse, Weil, Deuring and Chevalley, wonderful work has been done by numerous mathematicians providing the fundamentals of the theory of “function fields in one variable” – at least over finite fields – in equivalence to the established theory of numbers.
The last fruitful result was the proof of the Langlands conjectures over function fields by L. Lafforgue [Invent. Math. 147, 1–241 (2002; Zbl 1038.11075)]. So, all “important” conjectures in number theory [Riemann, Brumer-Stark, etc.] are now solved for algebraic function fields. Everybody who wants to get in touch with these results should have a look into this book.
For example, Chapter VII (“Extension of function fields”) gives a beautiful application of the Riemann-Hurwitz theorem to the proof of the ABC-conjecture, or Chapter X explains a proof of Artin’s primitive root conjecture.
The audience eagerly awaits a second part of this book containing the class field theory in the sense of G. Gras [Class field theory. From theory to practice. Berlin: Springer (2003; Zbl 1019.11032)] including an introduction to Lafforgue’s proof (loc. cit.) – with an invitation to Drinfeld modules, the fundamentals of which are given in this volume. Then the ideas of H. Hasse [Zahlentheorie. Berlin: Akademie Verlag (1949; Zbl 0035.02002)] – this author is not mentioned in the bibliography!? – finally should be fulfilled.

11R58 Arithmetic theory of algebraic function fields
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
11T55 Arithmetic theory of polynomial rings over finite fields