*(English)*Zbl 1145.11004

During the last 40 years, substantial progress has been made in transcendental number theory and its applications. New transcendence results have been obtained, a number of open problems have been solved, many applications have been developed, in particular in arithmetic algebraic geometry. One of the main source of the revival of the theory was the solution in 1966 by the first author of a problem raised by A. O. Gel’fond on proving linear independence measures for logarithms of algebraic numbers. This pioneering contribution of A. Baker to the theory has been extended in several directions by a number of specialists. Among them is the second author, and the book contributes to highlighting the fundamental works of each of the two authors. The main emphasis is therefore on the study of linear independence of logarithms of algebraic points on commutative algebraic groups, including linear independence measures as well as applications.

The applications which are discussed in this book are essentially those related with Diophantine Geometry, where effective results are obtained thanks to linear independence estimates arising from transcendental number theory.

The selection of topics by the authors does not cover all applications from Diophantine approximation and transcendence methods to Diophantine geometry. Further topics are discussed in the recent book [*E. Bombieri* and *W. Gubler*, Heights in Diophantine geometry. New Mathematical Monographs 4. Cambridge: Cambridge University Press (2006; Zbl 1115.11034)].

Also the authors do not consider all results from transcendental number theory related with linear forms in logarithms of algebraic numbers – in particular no mention of the fundamental progress by D. Roy is given; see for instance [*D. Roy*, “Matrices whose coefficients are linear forms in logarithms”, J. Number Theory 41, No. 1, 22–47 (1992; Zbl 0763.11030)].

However the work of each of the two authors dealing with applications to Diophantine Geometry of what they call logarithmic forms is fully covered, with a comprehensive list of references.

The first study of elliptic curves and abelian varieties in connection with transcendental numbers goes back to C.L. Siegel and Th. Schneider, in the first half of the XX-th century. The introduction of group varieties is due to S. Lang in the 1960’s, following a suggestion of P. Cartier. The classical results on the transcendence of values of the exponential function, due to Hermite, Lindemann, Weierstrass, Gel’fond, Schneider, Baker (and also the six exponentials Theorem of Siegel, Lang and Ramachandra, which is not quoted in the book under review) follow from results on commutative algebraic groups by taking a product of copies of the additive and multiplicative groups ${\mathbb{G}}_{a}$ and ${\mathbb{G}}_{m}$.

One of the main tools is the so-called “multipliticy estimate”, which is an extension of previous “zero estimates”. D. W. Masser was one of the first to investigate thoroughly this question. He introduced a variety of tools, analytic, algebraic and geometric ones. Elimination methods turned out to be most efficient. His early contribution with D. Brownawell is quoted in the book [*W. D. Brownawell* and *D. W. Masser*, “Multiplicity estimates for analytic functions. I.”, J. Reine Angew. Math. 314, 200–216 (1980; Zbl 0417.10027), “Multiplicity estimates for analytic functions. II.”, Duke Math. J. 47, 273–295 (1980; Zbl 0461.10027)] but not his founding paper which can be viewed as the start of the theory of zero estimates on algebraic group [*D. W. Masser*, “On polynomials and exponential polynomials in several complex variables”, Invent. Math. 63, 81–95 (1981; Zbl 0454.10019)]. The main contributions to this theory are due to Masser, Nesterenko, Philippon and Wüstholz.

The first chapter is a concise survey of selected topics from transcendental number theory. It covers the contributions of Liouville, Thue, Siegel and Roth on rational approximation of real numbers (Schmidt’s Subspace Theorem is postponed to the last chapter), the Theorems due to Hermite, Lindemann and Weierstrass (this last name is omitted) on the transcendence of exponentials and logarithms of algebraic numbers, the method of Siegel and Shidlovsky for $E$-functions, Mahler’s method for functions satisfying functional equations (the solution of a problem by Mahler and Manin on the modular function $J$ by a team from St. Etienne in 1995 is quoted in this section as well as Nesterenko’s work on modular functions and the algebraic independence of $\pi $, ${e}^{\pi}$ and ${\Gamma}(1/4)$), Riemann’s Hypothesis over finite fields and the related works of S. Stepanov, E. Bombieri and W. M. Schmidt.

This is not a complete survey of transcendental number theory, but in only 23 pages it states a substantial number of results, with few proofs. It provides a good reference for a mathematician who wishes to be acquainted with the subject.

With the second chapter entitled “Logarithmic Forms” starts the main theme of the book. The source of this subject is Hilbert’s seventh problem on the transcendence of ${\alpha}^{\beta}$ for algebraic numbers $\alpha $ and $\beta $ and its solution in 1934 by A. O. Gel’fond and Th. Schneider. A sketch of proof is given, following Gel’fond’s method which is central to the proofs of all transcendence results which are discussed in the book. The method of Schneider is not considered. This chapter includes the statement of the Schneider-Lang Theorem in one variable. There are only a few lines on the several variables extension, with the applications to Schneider’s Theorem on the transcendence of values of the Beta function at rational points, the work of E. Bombieri, S. Lang and G. V. Chudnovsky. Then comes Baker’s Theorem on linear independence of logarithms with some effective results; the sketch of proof they give starts with a description of Fel’dman’s Delta function. The authors state explicitly the estimates they proved in a joint paper giving effective measures of linear independence for logarithms of algebraic numbers; they give a few references for later stronger estimates, including those by E. Matveev – see [Nesterenko, Yu. V.(ed.); Philippon, Patrice (ed.), Introduction to algebraic independence theory. With contributions from F. Amoroso, D. Bertrand, W. D. Brownawell, G. Diaz, M. Laurent, Yu. V. Nesterenko, K. Nishioka, P. Philippon, G. Rémond, D. Roy, M. Waldschmidt. Lecture Notes in Mathematics. 1752 (2001; Zbl 0966.11032)]. Also $p$-adic estimates by Yu Kunrui are quoted – the latest results are published in [*K. Yu*, “$p$-adic logarithmic forms and group varieties. III.”, Forum Math. 19, No. 2, 187–280 (2007; Zbl 1132.11038)].

The third chapter deals with applications of transcendental number theory: to class numbers, to the unit equation of Siegel and Lang, to the Thue equation, to Diophantine curves including practical computations, to exponential Diophantine equations with the solution of Catalan’s Conjecture [see *Yu. F. Bilu*, “The Many Faces of the Subspace Theorem (after Adamczewski, Bugeaud, Corvaja, Zannier...)”, Séminaire Bourbaki, Exposé 967, 59ème année (2006-2007)], to the $abc$ Conjecture and its connection with linear independence measures of logarithms of algebraic numbers [see also *S. Lang*, Elliptic curves: Diophantine analysis. Grundlehren der Mathematischen Wissenschaften. 231. Berlin etc.: Springer-Verlag (1978; Zbl 0388.10001) and *P. Philippon*, “Quelques remarques sur des questions d’approximation diophantienne”, Bull. Aust. Math. Soc. 59, No. 2, 323–334 (1999; Zbl 0927.11040); Addendum, Bull. Aust. Math. Soc. 61, No. 1, 167–169 (2000; Zbl 1099.11508)]. At the end, further applications are briefly quoted.

According to the authors, the first three chapters may be regarded as a new rendering and updating of Chapters 1 to 5 of the first author’s main book [*A. Baker*, Transcendental number theory. (Repr. of 1975 with additional material). Cambridge University Press (1979; Zbl 0497.10023)]. By contrast, Chapters 4, 5 and 6 contain an original and enlightening presentation of the analytic subgroup estimate due to the second author, including new material and more proofs than in the three first chapters.

Chapter 4 is an introduction to commutative algebraic groups, starting with basic concepts: the additive and multiplicative group. The Lie algebra and subgroup varieties are studied. There is also a short section on the geometry of numbers.

The emphasis of Chapter 5 is on multiplicity estimates. It includes Hilbert functions in degree theory, differential length, algebraic degree theory, calculation of the Jacobi rank; there is a subsection with title “the Wüstholz Theory”; this chapter ends with the study of algebraic subgroups of the torus. Another introduction to the same topic has been given by D. Roy (in the book under review, the authors restrict themselves to the product of one copy of the additive group with multiplicative groups, while Roy considers the general situation of a product of arbitrary many copies of the additive and multiplicative groups which is required for a whole part of transcendental number theory which is not mentioned by Baker and Wüstholz) – see [Chap. 5 and 7 of *M. Waldschmidt*, Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables. Grundlehren der Mathematischen Wissenschaften. 326. Berlin: Springer (2000; Zbl 0944.11024)].

The analytic subgroup Theorem of the second author is the central theme of Chapter 6. This fundamental result is the main tool for proving results on the transcendence of “periods” as defined by *M. Kontsevich* and *D. Zagier* in [“Periods”, Engquist, Björn (ed.) et al., Mathematics unlimited - 2001 and beyond. Berlin: Springer. 771–808 (2001; Zbl 1039.11002)]. The authors quote [*V. I. Arnol’d*, Huygens and Barrow, Newton and Hooke. Pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals. Transl. from the Russian by Eric J. F. Primrose. Basel etc.: Birkhäuser Verlag (1990; Zbl 0721.01001)] for early references to such questions. This chapter includes also further results like the analog of the Lindemann-Weierstrass Theorem for elliptic curves, which was proved independently by P. Philippon and G. Wüstholz shortly before the analytic subgroup Theorem.

Quantitative theory is discussed in Chapter 7. The authors discuss in detail sharp estimates of their own on the usual logarithms, and they mention more briefly further results due to N. Hirata-Kohno, S. David and E. Gaudron (further references are [*S. David* and *N. Hirata-Kohno*, “Recent progress on linear forms in elliptic logarithms”, J. Reine Angew. Math., to appear, *É. Gaudron*, “Formes linéaires de logarithmes effectives sur les variétés abéliennes”, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 5, 699–773 (2006; Zbl 1111.11038),“ Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif”, Invent. Math. 162, No. 1, 137–188 (2005; Zbl 1120.11031),“ Study of the rational case of the theory of linear forms in logarithms”, J. Number Theory 127, No. 2, 220–261 (2007; Zbl 1197.11085)]). A missing item in this section is the method of J.-B. Bost involving slopes inequalities based on Arakelov theory – see [*J.-B. Bost*, “Périodes et isogénies des variétés abéliennes sur les corps de nombres [d’après D. Masser et G. Wuestholz]”, Séminaire Bourbaki. Volume 1994/95. Exposés 790–804. Astérisque 237, 115–161, Exp. No. 795 (1996; Zbl 0970.23932)] and [*A. Chambert-Loir*, “Théorèmes d’algébricité en géométrie diophantienne d’après J.-B. Bost, Y. André, D. et G. Chudnovsky”, Bourbaki seminar. Volume 2000/2001. Exposés 880–893. Astérisque 282, 175–209, Exp. No. 886 (2002; Zbl 1044.11055)]. There are a few lines on the transcendence theory in finite characteristic: the contributions of G. Anderson, W.D. Brownawell, L. Denis, M. Papanikolas, D. Thakur and Jing Yu produce remarkably deep statements which are very far ahead for Drinfeld motives compared to the classical theory in zero characteristic [*F. Pellarin*, “Aspects de l’indépendance algébrique en caractéristique non nulle, d’après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu ...”, 59, 2006/07, No. 973]. This chapter 7 includes a number of applications of the theory, due mainly to D. W. Masser and G. Wüstholz: the isogeny theorems, discriminant, polarisations and Galois groups, the conjecture of Mordell–Tate.

The last chapter deals with further aspects of Diophantine geometry, where Diophantine approximation methods yield to ineffective results (in contrast with the rest of the book where the transcendence arguments yield to effective statements): the Schmidt Subspace Theorem, Faltings’s product Theorem, the André-Oort conjecture, hypergeometric functions and the Manin-Mumford conjectures.

The quantity of recent results quoted in this book reveals the intense vitality of the subject.