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Function field arithmetic. (English) Zbl 1061.11001
River Edge, NJ: World Scientific (ISBN 981-238-839-7/hbk). xv, 388 p. (2004).
The starting point of the theory presented in the book under review has its roots in the work of L. Carlitz in the 1930’s when he found an exponential function $$e_ C (s)$$ associated to the congruence rational function field $$k= {\mathbb F}_ q(T)$$. In his important paper [Duke Math. J. 1, 137–168 (1935; Zbl 0012.04904)] Carlitz attached $$e_ C(s)$$ to a rational curve which has many of the same properties of the classical exponential function $$e ^ z$$. The function $$e _ C(s)$$ is an analytical function defined on spaces of characteristic $$p>0$$ and with values in characteristic $$p$$. Furthermore, $$e_ C(s)$$ satisfies, for each $$M\in {\mathbb F}_ q[T]$$ the functional equation $$e_ C (Ms)= C_ M ( e_ C (s))$$ for some $$C_ M(u) \in {\mathbb F}_ q(T)[u]$$. The mapping $$M\mapsto C_ M(u)$$ is called the Carlitz module. In 1974, D. Hayes [Trans. Am. Math. Soc. 189, 77–91 (1974; Zbl 0292.12018)] established the analogue of the Kronecker-Weber theorem and found the maximal abelian extension of $$k$$.
In the cyclotomic theory, the ring $$A={\mathbb F} _ q[T]$$, the field $$k$$, the field $$k_ \infty$$, the completion of $$k$$ at the infinite prime and the field $$C_ \infty$$, the completion of an algebraic closure of $$k_ \infty$$, play roles analogous to those played by the ring of rational integers, the field of rational numbers, the field of real numbers and the field of complex numbers, respectively, for the classical exponential function.
In 1974, in his fundamental paper [Math. USSR, Sb. 23, 561–592 (1974); translation from Mat. Sb., N. Ser. 94(136), 594–627 (1974; Zbl 0321.14014)], V. G. Drinfeld began the theory of general exponential functions of rank $$r$$. In particular $$e_ C(s)$$ has rank $$1$$. He showed how these exponential functions provide some algebraic objects, the “elliptic” modules, which are now called Drinfeld modules. Based on Drinfeld’s work, D. Hayes [Studies in Algebra and Number Theory, Adv. Math., Suppl. Stud. 6, 173–217 (1979; Zbl 0476.12010)] developed an explicit class field theory for general function fields. Both approaches, by Drinfeld and by Hayes give class fields, the latter being analogous to the cyclotomic number fields and so we have several analogies with number fields. The goal of Hayes’ approach is to study arithmetic in positive characteristic.
Thakur’s book gives a comprehensive exposition of function field arithmetic related to recent developments on Drinfeld modules, arithmetic of special values of transcendental functions and Diophantine approximation. This is not an introductory book. It assumes a solid background in algebraic number theory, cyclotomic classic theory and algebraic geometry. With the publication of this book, together with the books of D. Goss [“Basic structures of function field arithmetic”, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 35. Berlin: Springer (1996; Zbl 0874.11004)] and E.-U. Gekeler [“Drinfeld modular curves”, Lecture Notes in Mathematics, 1231. Berlin etc.: Springer-Verlag (1986; Zbl 0607.14020)] we now have a very comprehensive treatise on function field arithmetic. Thakur’s book is a valuable contribution to the theory of arithmetic function fields. It covers many topics treated in the book of Goss and complements that book with the inclusion of recent developments. There is also a treatment of new topics such as transcendence, Diophantine approximation, hypergeometric functions, automata, and modular forms. The author has included several interesting examples and discusses many open problems.
The content of the book is the following:
Chapter 1 is a compilation of some analogies and differences between number fields and function fields. Along the book, function fields means congruence function fields. In this chapter, the author also presents the basic facts on function fields such as the Riemann-Roch theorem, the Riemann-Hurwitz formula, the Castelnuovo and Riemann inequalities, Weierstrass points, Zeta functions, Riemann hypothesis, class field theory by means of divisor groups, idèles and generalized Jacobians.
Chapter 2 deals with the basic theory of Drinfeld modules. Starting with a very pleasant introduction to the Carlitz exponential function and the Carlitz module, the author gives the basic general facts on Drinfeld modules including nonarchimedean analysis, Dwork’s trace formula and adjoints and duality.
Chapter 3 treats explicit class field theory. By the Kronecker-Weber theorem, the classic cyclotomic theory shows that the theory of extensions by adjoining roots of unity, that is, the torsion of the natural action of $${\mathbb Z}$$ on $${\mathbb C}^ \ast$$, is the same as the theory of abelian extensions over $${\mathbb Q}$$. In this way we obtain class fields explicitly. In the function field case the usual cyclotomic extensions give only the constant extensions. D. Hayes noticed the similarity between Lubin and Tate’s approach to explicit local class field theory and that of Carlitz global theory and then constructed explicitly the maximal abelian extension of $${\mathbb F}_ q(T)$$. Shortly after that, he constructed the maximal abelian extension of any congruence function field using the theory of Drinfeld modules. In this chapter, the author presents this theory of rank one Drinfeld modules to give explicit construction of class fields.
In Chapters 4 and 5 the author presents the study of zeta functions, Gauss sums and gamma functions. There are two kinds of gamma functions: the first one, the arithmetic gamma function, is closely linked with constant field extensions and the second one, the geometric gamma function, is linked with the Drinfeld cyclotomic case. In the function field case the study of special zeta values in finite characteristics is still in the initial stages.
Chapters 6 and 7 study the theory of objects of rank larger than $$1$$. The topics include elliptic modules, modular forms, DeRham cohomology and cohomology realizations.
Some applications of Gauss sums and of gamma and zeta values are given in Chapter 8.
Chapter 9 treats Diophantine approximation which is one of the few areas where the number field case is much better understood than the function field situation.
Transcendence results is the topic covered in Chapter 10 where the transcendence results are applied to zeta and gamma function values. In this chapter, most of the results are just stated without proofs.
Chapter 11 deals with automata and algebraicity and its applications to transcendence of gamma values, periods, monomials and modular functions.

MSC:
 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R58 Arithmetic theory of algebraic function fields 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) 11G20 Curves over finite and local fields 11J81 Transcendence (general theory) 11M38 Zeta and $$L$$-functions in characteristic $$p$$ 11R37 Class field theory 11T24 Other character sums and Gauss sums 11T55 Arithmetic theory of polynomial rings over finite fields 14D10 Arithmetic ground fields (finite, local, global) and families or fibrations 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14H25 Arithmetic ground fields for curves