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Basic structures of function field arithmetic. (English) Zbl 0874.11004
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 35. Berlin: Springer. xiii, 422 p. (1996).
The analogy between numbers fields and functions fields (especially fields of rational functions over a compact Riemann surface) has been observed since the 19th century (Dedekind, Kronecker, Riemann, Weber), but the modern story began with E. Artin’s “Dissertation” (1923), where, for the first time, tools of algebraic number theory have been introduced in the range of function fields having finite constant fields, making clear the parallel with the classical case (in the sequel, all function fields are supposed having finite constants fields of characteristic \(p\)).
It would have stopped here without the works of Carlitz in the thirties, followed by two main papers of D. Hayes about the explicit Abelian class field theory [Trans. Am. Math. Soc. 189, 77-91 (1974; Zbl 0292.12018); Adv. Math., Suppl. Stud. 6, 173-217 (1979; Zbl 0476.12010)], and also without the fundamental article of V. G. Drinfeld [Funct. Anal. Appl. 10, 107-115 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 29-40 (1976; Zbl 0346.14010)]; Carlitz and Hayes gave the foundation of the contemporary “algebraic number theory” for function fields, and Drinfeld has given the foundation of automorphic and Langlands theories for those fields.
One can divide the modern treatment of the arithmetic of function fields in the following topics: 1) Non-Abelian class field theory and reciprocity laws. 2) Modular theory of Drinfeld modular schemes (which are the “Shimura varieties” of positive characteristic), especially for the dimension 2, with applications to diophantine problems over function fields. 3) Transcendency questions about rigid analytic functions and some of their values (as the \(j\)-, Weierstrass-, zeta- and \(L\)-functions in positive characteristic…); algorithmic aspects. 4) Algebraic arithmetic (“algebraic number theory”) in positive characteristic (class numbers, extensions generated by torsion points, Galois structures of integers…). 5) Inner problems to the theory of Drinfeld modules and generalizations to dimension greater than 1 (as the T-motives of G. Anderson). 6) Arithmetical properties of special functions (gamma-, zeta- and \(L\)-functions…).
The present book is concerned with the fourth through sixth topics (for the first, see the books of G. Laumon [Cohomology of Drinfeld modular varieties. Part I, Cambridge Stud. Adv. Math. 41 (1996; Zbl 0837.14018) and Part II, Cambridge Stud. Adv. Math. 56 (1997)]; for the second, see E.-U. Gekeler [Drinfeld modular curves, Lect. Notes Math. 1231 (1986; Zbl 0607.14020)], and E.-U. Gekeler and M. Reversat [J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)]; although there does not exist yet a survey for the third topic, fundamental results can be found in papers of Allouche, Brownawell, Denis, Yu…).
Chapters 1 to 7 of the book contain all the basic material and basic results of the contemporary algebraic arithmetic of function fields. Ore (additive) polynomials, the Carlitz module, Drinfeld modules, T-motives, stukas, and (in chapter 7) constructions of Abelian extensions, cyclic and “cyclotomic” extensions; Abelian class field theory (as described by Hayes) is explained here. Chapter 8 is devoted to the author’s zeta- and \(L\)-functions. These are functions with values in a (complete and algebraically closed) field of the same characteristic \(p\) as the base function field. Today, one does not know all the expected arithmetic properties for such functions. For instance, there does not exist a functional equation for the zeta-function, but, one knows properties close to classical ones of the Riemann zeta function (as values at integers) and it seems that there exists a reasonable theory of zeros. Two notions of gamma functions (with, as before, values in characteristic \(p\)) are described in chapter 9. The first, called the “arithmetic gamma-function”, is from Thakur. It possesses a functional equation. Analogues of Gauss sums and Jacobi sums are also given and studied. The second, the “geometric gamma-function”, is from the author; it possesses some functional relations. The last chapter contains a quick description of some other (related) topics.
In addition to theories of zeta, \(L\) and gamma-functions with values in positive characteristic, the book gives, for the first time, an exhaustive survey of the basic tools and properties of the modern algebraic arithmetic of function fields, which is very good for young researchers and beginners in that theory. The book is also replete with illuminating remarks. Many problems are suggested.

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11T55 Arithmetic theory of polynomial rings over finite fields
11T24 Other character sums and Gauss sums
11R58 Arithmetic theory of algebraic function fields