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Number theory IV: Transcendental numbers. Edited by A. N. Parshin, I. R. Shafarevich and R. V. Gamkrelidze. Transl. from the Russian by Neal Koblitz. (English) Zbl 0885.11004

Encyclopaedia of Mathematical Sciences 44. Berlin: Springer (ISBN 3-540-61467-2/hbk). 345 p. (1998).
This fourth volume of Number Theory of Springer’s Encyclopaedia of Mathematical Sciences deals with transcendental numbers. The authors, who are two leading experts of the subject, provide an introduction not only to some of the most important results in this field, but also to the main methods. The basic ideas of the proofs are clearly explained. Few complete proofs are provided, so that technical details are omitted, but the main steps are sketched. The methods are introduced with enlightening motivations.
This book shows how important the activity is in this branch of number theory. In fact the subject, transcendental number theory, is so vast that some parts had to be omitted. For instance Mahler’s method is only briefly quoted, but a recent overview of this topic has just appeared [K. Nishioka, Mahler functions and transcendence, Lect. Notes Math. 1631. Springer-Verlag, Berlin (1996; Zbl 0876.11034)]. Another missing area is transcendence in finite characteristic, especially Jing Yu’s work on Drinfeld’s modules (see for instance the recent survey by W. D. Brownawell [Transcendence in positive characteristics, in Number Theory (Tiruchirapalli, India, 1996), Contemp. Math. 210, 317-332 (1998)]. The authors have chosen to address a public not necessarily acquainted with algebraic groups, so they concentrate on the properties of the usual exponential function and elliptic functions. A few results concerning higher dimensional abelian varieties are only briefly quoted. In the same way, they consider almost only functions of a single variable.
As the authors themselves put it, “on several occasions we had to shorten the length of this volume. We ended up deleting Chapter 7, which was devoted to the important question of estimates for the number of zeros of a function. We ask our colleagues not to judge us too harshly for our omission of some topics from the book.”
The book starts with an introduction which includes connections with other branches of number theory. Chapter 1 deals with approximation of algebraic numbers: of course the authors study the classical results and methods of Thue, Siegel, Roth and Schmidt. But they also go more deeply than other books on this subject in their thorough discussion of both old and new effective results, including works of Thue, Baker and Bombieri. This chapter also provides information on the number of solutions of diophantine equations.
The second chapter “Effective constructions in transcendental number theory” is arguably the most important of this book, let alone for its size. The goal is to produce a unified treatment for the proof of many results in transcendental number theory. In order to prove that some complex number \(\theta\) is transcendental, the principle is to produce a sequence of sufficiently good diophantine approximations to \(\theta\). If \(\theta\) were algebraic, then elimination using the minimal polynomial would produce a contradiction. Sometimes constructions of the desired sequence of good approximations are explicit; sometimes only their existence can be deduced from Dirichlet’s box principle. The authors give many examples to illustrate this strategy. The methods of Hermite, Lindemann, and Weierstraß are presented in this setup. Functional approximation including continued fractions and Padé approximation provide a valuable tool. Irrationality and transcendence measures are also discussed, including bounds for the rational approximation of numbers like \(\zeta(3)\) and values of generalized hypergeometric functions or polylogarithms. Galochkin’s so-called unimprovable estimates deserve to be mentioned.
Hilbert’s problem is the heart of Chapter 3. After the methods of Gel’fond and Schneider, Laurent’s interpolation determinants are introduced. The proof of Schneider’s theorems on Weierstraß elliptic functions, which provide an elliptic analog to the theorems of Hermite, Lindemann, Gel’fond and Schneider, are outlined. This section includes the statement of the very recent solution of the Mahler-Manin problem on the transcendence of the values of the modular function \(J\). The so-called criterion of Schneider-Lang is given, but only in the version which includes derivatives (the criteria of Lang and Ramachandra, which rest on Schneider’s method, are not included). Abelian functions are briefly quoted. Bounds for linear forms in two logarithms, mainly due to Gel’fond, but also including the first explicit estimates by A. Schinzel, are stated and applied to several problems in number theory. One of these applications deals with the Gauß class number problem for imaginary quadratic fields. Finally transcendence measures for numbers connected with exponential and elliptic functions are given.
Chapter 4 deals with lower bounds for linear forms in logarithms of algebraic numbers and their applications. The early ineffective estimates of Gel’fond are followed by some of the main results dealing with Baker’s method together with sketches of proofs. Some of the most important results in this section, which are not given a reference, are due to the first author; this is the case for instance with the effective strengthening of Liouville’s theorem. Many applications are given: effective upper bounds for the solutions of Thue and Thue-Mahler equations, also for solutions in \(S\)-integers (including work of Shorey and Tijdeman), Tijdeman’s result on Catalan’s equation, elliptic and hyperelliptic diophantine equations, the \(abc\) conjecture, class number problems, application to algebraic number theory, recurrence sequences, primitive divisors of successive terms in some sequences. Elliptic results of Baker, Coates, Masser and Wüstholz are also briefly quoted.
The title of Chapter 5 is “Values of analytic functions that satisfy linear differential equations”. This subject started with Siegel’s work in 1929, and then in 1949, on so-called \(E\)- and \(G\)-functions. Shidlovskij’s fundamental results from the 50’s are described and the method clearly explained. Consequences of Shidlovskij’s results concern the values of \(E\)-functions which satisfy a differential equation, especially hypergeometric \(E\)-functions, Bessel’s functions and Kummer’s functions. A lot of such results have been produced by Shidlovskij and his students. These results of transcendence and algebraic independence are effective: lower bounds for linear forms and polynomials in the values of \(E\)-functions are extensively studied. The second part of this chapter is devoted to \(G\)-functions and their values. Important contributions are due to Galochkin, Bombieri, Chudnovsky and André.
The subject of the last chapter is Gel’fond’s method for algebraic independence and its recent developments. Two main conjectures are
(1) Gel’fond’s problem on algebraic independence of numbers \(\alpha^{\beta_1},\ldots,\alpha^{\beta_n}\) for algebraic \(\alpha\) and \(\beta_1,\ldots,\beta_n\) and
(2) the conjecture of algebraic independence of logarithms of algebraic numbers.
This Chapter 6 deals mostly with the first issue as well as related problems. Gel’fond’s method, going back to 1949, is described with some details. Next the inductive procedure proposed by Lang with the notion of “transcendence type” is explained. The first results on large transcendence degree occur in Chudnovsky’s work. Section 3 is especially important: under the title “Application of general elimination theory”, it includes the bases of the second author’s new approach involving Chow forms. In this section one finds Philippon’s criterion for algebraic independence together with Diaz’ results on the values of the exponential functions, including the lower bound \([(d+1)/2]\) for the transcendence degree of the field \(\mathbb{Q}(\alpha^{\beta},\ldots,\alpha^{\beta^d})\) (when \(\beta\) is algebraic of degree \(d\)), which remains the best known estimate on Gel’fond’s conjecture (1). Also Brownawell’s results on an effective version of Hilbert’s Nullstellensatz as well as their applications to algebraic independence are explained. In the next sections the authors are interested in an elliptic analog of the previous results of algebraic independence, including the elliptic Lindemann Weierstraß Theorem (works of Chudnovsky, Philippon and Wüstholz) and the statement of the second author’s result of algebraic independence on values of modular functions, which yields the algebraic independence of \(\pi\), \(e^\pi\) and \(\Gamma(1/4)\). Finally quantitative results and measures of algebraic independence are discussed.
The last 34 pages of this volume are devoted to a long list of references (not all of which are quoted elsewhere in the book).
This book is the right reference for those who wish to know the state of the art on transcendental numbers. The compilation of results is associated with clear explanations of the basic ideas; it is an excellent introduction to the subject.

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11J81 Transcendence (general theory)
11J82 Measures of irrationality and of transcendence
11J86 Linear forms in logarithms; Baker’s method
11J25 Diophantine inequalities
11J85 Algebraic independence; Gel’fond’s method
11J89 Transcendence theory of elliptic and abelian functions
11J68 Approximation to algebraic numbers
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