##
**An introduction to \(G\)-functions.**
*(English)*
Zbl 0830.12004

Annals of Mathematics Studies. 133. Princeton, NJ: Princeton University Press. xxi, 323 p. $59.50; £50.00 / hbk; $ 29.95; £25.00 / pbk (1994).

\(G\)-functions were introduced by Siegel in his fundamental paper of 1929, where he showed how the Thue-Siegel’s lemma could be applied both to transcendence theory and to diophantine equations. His main result on the former aspect concerned the values of confluent hypergeometric functions (such as the Bessel function), or more generally of certain “\(E\)- functions”, but he also noticed that his method could yield irrationality results for the Laplace transforms of \(E\)-functions, which he called \(G\)-functions.

More precisely, a \(G\)-function is a formal power series \(f(z)\) in one variable, with coefficients in a number field \(K\), which satisfies a linear differential equation \(Ly = 0\) with coefficients in the field \(\mathbb{C}(z)\) of rational functions, with the further requirements that \(f\) has a positive radius of convergence at each archimedean place of \(K\), and that the common denominator of its \(n\) initial coefficients grows at most geometrically with \(n\). As typical examples of \(G\)-functions, Siegel pointed out algebraic functions (Eisenstein’s theorem) and their primitives (i.e. abelian integrals), (poly)logarithms and Gauss’ hypergeometric functions with rational parameters.

Siegel gave no hint of proof of his irrationality results for values of such \(G\)-functions. Actual proofs were first obtained by A. I. Galochkin [Mat. Sb., Nov. Ser. 95(137), 396-417 (1974; Zbl 0311.10035), and Mat. Zametki 18, 541-552 (1975; Zbl 0319.10039)]. However, rather than requiring that the derivatives \(d^nf/dz^n (0)\) of \(f\) at \(0\) behave “geometrically”, Galochkin insisted that the sequence of differential operators \((d/dz)^n\), when read modulo \(L\), should satisfy a similar condition. All the results of Siegel (including an announcement on values of abelian integrals) were finally justified, and sharpened, by E. Bombieri [Recent progress in analytic number theory (Proc. Conf. Durham, 1979), Vol. 2, 1-67 (1981; Zbl 0461.10031)], thanks to a new type of hypothesis on \(L\), involving the radii of convergence, at all places of \(K\), of the solutions of \(L\) at a generic point. In this way, Bombieri brought \(p\)-adic differential equations (in particular the work of B. Dwork and P. Robba in Trans. Am. Math. Soc. 259, 559-577 (1980; Zbl 0439.12016) into the study of \(G\)- functions, and started a new era for \(G\)-functions.

In the meantime, another approach to Siegel’s results was initiated in the case of algebraic functions by P. Bundschuh [Semin. Delange- Pisot-Poitou, Paris, 1977-78, Fasc. 2, Exp. No. 42 (1978; Zbl 0399.10032)] after previous work of Th. Schneider, and this was generalized by P. Dèbes [Acta Arith. 47, 371-402 (1986; Zbl 0605.12013)], who showed that under a condition on \(L\) which is essentially equivalent to Galochkin’s, Gel’fond’s classical transcendence method, rather than Siegel’s, could provide results of a nature similar to Bombieri’s.

Finally, a fundamental advance was made by G. and D. Chudnovsky, following the work of G. Chudnovsky [Lect. Notes Math. 751, 45-69 (1979; Zbl 0418.10031)] on Padé approximants and values of \(G\)- functions of Galochkin’s type, and the study by various authors of specific examples. They showed [Lect. Notes Math. 1135, 9-51 (1985; Zbl 0561.10016)] that if \(L\) is a generator of the annihilator in the ring \(D\) of differential operators over \(K(z)\) of a \(G\)-function \(f\), then \(L\) must verify Galochkin’s condition. Notice that going into the reverse direction, namely showing that a formal solution \(f\) of an operator \(L\) of Galochkin’s type will be a \(G\)-function, is an elementary exercise when \(O\) is an ordinary point of \(L\).

In 1988, Y. André made a synthesis of the results of Bombieri, Dèbes, Chudnovsky, and of his own work, in his book “\(G\)-functions and geometry” [Vieweg, Asp. Maths., E 13 (1989; Zbl 0688.10032)]. Although geared towards diophantine problems, this was the first work where \(G\)- functions were treated for their own sake. As tentative answers to his questions “What are \(G\)-functions, and what should they conjecturally be?”, André formalized the definitions of the numerical invariants whose finiteness express the various conditions introduced above (size and global radius of a power series \(f\), resp. of a \(D\)-module \(D/DL\)), studied their variations under the classical constructions of linear algebra, sharpened Bombieri’s analysis of their comparison, thereby checking the equivalence of Bombieri’s and Galochkin’s conditions, and extended to singular points the “converse” to Chudnovsky’s theorem, by a repeated use of the methods of Dwork and of G. Christol [Astérisque 119-120, 151-168 (1984; Zbl 0553.12014)] for weak Frobenius structures. But André’s book is densely written (half of it is in fact concerned with global diophantine results in the style of Bombieri’s), and we still lacked an introduction, in the everyday’s sense of the word, to the theory of \(G\)-functions.

True to its title, the book under review provides such an introduction, and can in fact be viewed as a general introduction to \(p\)-adic analysis, with its main methods clearly displayed and sharp estimates proved in full details. After a first chapter on \(p\)-adic fields, we are treated in Chapter II with a self-contained proof of Dwork’s theorem on the rationality of the zeta function of a hypersurface over a finite field. As explained in the introduction, \(D\)-modules are hidden in this proof, but they arrive openly in Chapter III with Honda’s theory of differential equations in finite characteristics and its applications to Katz’ theorem on global nilpotence. From then onwards, the book’s theme is close to the first part of André’s. The study of \(p\)-adic differential equations starts in Chapter IV, where the Dwork-Robba theorem and first examples of Dwork’s transfer principles are discussed. Chapters V and VI provide the main ingredients in the reverse direction to Chudnovsky’s theorem, based again on an iteration process: Dwork-Robba type estimates are established for the growth of the solutions at the boundary of a singular disk, following a theorem of G. Christol and G. Dwork [Duke Math. J. 62, 689-720 (1991; Zbl 0762.12004)] in case of nilpotent monodromy, while problems connected with the diophantine nature of the exponents are studied in Chapter VI. Finally, the last two chapters deal with global problems: equivalence of Bombieri’s and Galochkin’s conditions, Chudnovsky’s theorem and its converse. The Leitfaden appearing on p. 262 gives an enlightening picture of the whole story.

In their introduction, the authors express the hope that their book will make Bombieri’s and André’s work more accessible (and they could have added Chudnovsky’s theorem to this list). Their carefully written monograph certainly completes this programme, even if one may regret that precise cross-references to these works have not been inserted in the text. But their book does more than that: within the ‘elementary’ limits they have set to themselves, they are here presenting the reader with an enticing invitation to the whole theory of \(p\)-adic differential equations.

More precisely, a \(G\)-function is a formal power series \(f(z)\) in one variable, with coefficients in a number field \(K\), which satisfies a linear differential equation \(Ly = 0\) with coefficients in the field \(\mathbb{C}(z)\) of rational functions, with the further requirements that \(f\) has a positive radius of convergence at each archimedean place of \(K\), and that the common denominator of its \(n\) initial coefficients grows at most geometrically with \(n\). As typical examples of \(G\)-functions, Siegel pointed out algebraic functions (Eisenstein’s theorem) and their primitives (i.e. abelian integrals), (poly)logarithms and Gauss’ hypergeometric functions with rational parameters.

Siegel gave no hint of proof of his irrationality results for values of such \(G\)-functions. Actual proofs were first obtained by A. I. Galochkin [Mat. Sb., Nov. Ser. 95(137), 396-417 (1974; Zbl 0311.10035), and Mat. Zametki 18, 541-552 (1975; Zbl 0319.10039)]. However, rather than requiring that the derivatives \(d^nf/dz^n (0)\) of \(f\) at \(0\) behave “geometrically”, Galochkin insisted that the sequence of differential operators \((d/dz)^n\), when read modulo \(L\), should satisfy a similar condition. All the results of Siegel (including an announcement on values of abelian integrals) were finally justified, and sharpened, by E. Bombieri [Recent progress in analytic number theory (Proc. Conf. Durham, 1979), Vol. 2, 1-67 (1981; Zbl 0461.10031)], thanks to a new type of hypothesis on \(L\), involving the radii of convergence, at all places of \(K\), of the solutions of \(L\) at a generic point. In this way, Bombieri brought \(p\)-adic differential equations (in particular the work of B. Dwork and P. Robba in Trans. Am. Math. Soc. 259, 559-577 (1980; Zbl 0439.12016) into the study of \(G\)- functions, and started a new era for \(G\)-functions.

In the meantime, another approach to Siegel’s results was initiated in the case of algebraic functions by P. Bundschuh [Semin. Delange- Pisot-Poitou, Paris, 1977-78, Fasc. 2, Exp. No. 42 (1978; Zbl 0399.10032)] after previous work of Th. Schneider, and this was generalized by P. Dèbes [Acta Arith. 47, 371-402 (1986; Zbl 0605.12013)], who showed that under a condition on \(L\) which is essentially equivalent to Galochkin’s, Gel’fond’s classical transcendence method, rather than Siegel’s, could provide results of a nature similar to Bombieri’s.

Finally, a fundamental advance was made by G. and D. Chudnovsky, following the work of G. Chudnovsky [Lect. Notes Math. 751, 45-69 (1979; Zbl 0418.10031)] on Padé approximants and values of \(G\)- functions of Galochkin’s type, and the study by various authors of specific examples. They showed [Lect. Notes Math. 1135, 9-51 (1985; Zbl 0561.10016)] that if \(L\) is a generator of the annihilator in the ring \(D\) of differential operators over \(K(z)\) of a \(G\)-function \(f\), then \(L\) must verify Galochkin’s condition. Notice that going into the reverse direction, namely showing that a formal solution \(f\) of an operator \(L\) of Galochkin’s type will be a \(G\)-function, is an elementary exercise when \(O\) is an ordinary point of \(L\).

In 1988, Y. André made a synthesis of the results of Bombieri, Dèbes, Chudnovsky, and of his own work, in his book “\(G\)-functions and geometry” [Vieweg, Asp. Maths., E 13 (1989; Zbl 0688.10032)]. Although geared towards diophantine problems, this was the first work where \(G\)- functions were treated for their own sake. As tentative answers to his questions “What are \(G\)-functions, and what should they conjecturally be?”, André formalized the definitions of the numerical invariants whose finiteness express the various conditions introduced above (size and global radius of a power series \(f\), resp. of a \(D\)-module \(D/DL\)), studied their variations under the classical constructions of linear algebra, sharpened Bombieri’s analysis of their comparison, thereby checking the equivalence of Bombieri’s and Galochkin’s conditions, and extended to singular points the “converse” to Chudnovsky’s theorem, by a repeated use of the methods of Dwork and of G. Christol [Astérisque 119-120, 151-168 (1984; Zbl 0553.12014)] for weak Frobenius structures. But André’s book is densely written (half of it is in fact concerned with global diophantine results in the style of Bombieri’s), and we still lacked an introduction, in the everyday’s sense of the word, to the theory of \(G\)-functions.

True to its title, the book under review provides such an introduction, and can in fact be viewed as a general introduction to \(p\)-adic analysis, with its main methods clearly displayed and sharp estimates proved in full details. After a first chapter on \(p\)-adic fields, we are treated in Chapter II with a self-contained proof of Dwork’s theorem on the rationality of the zeta function of a hypersurface over a finite field. As explained in the introduction, \(D\)-modules are hidden in this proof, but they arrive openly in Chapter III with Honda’s theory of differential equations in finite characteristics and its applications to Katz’ theorem on global nilpotence. From then onwards, the book’s theme is close to the first part of André’s. The study of \(p\)-adic differential equations starts in Chapter IV, where the Dwork-Robba theorem and first examples of Dwork’s transfer principles are discussed. Chapters V and VI provide the main ingredients in the reverse direction to Chudnovsky’s theorem, based again on an iteration process: Dwork-Robba type estimates are established for the growth of the solutions at the boundary of a singular disk, following a theorem of G. Christol and G. Dwork [Duke Math. J. 62, 689-720 (1991; Zbl 0762.12004)] in case of nilpotent monodromy, while problems connected with the diophantine nature of the exponents are studied in Chapter VI. Finally, the last two chapters deal with global problems: equivalence of Bombieri’s and Galochkin’s conditions, Chudnovsky’s theorem and its converse. The Leitfaden appearing on p. 262 gives an enlightening picture of the whole story.

In their introduction, the authors express the hope that their book will make Bombieri’s and André’s work more accessible (and they could have added Chudnovsky’s theorem to this list). Their carefully written monograph certainly completes this programme, even if one may regret that precise cross-references to these works have not been inserted in the text. But their book does more than that: within the ‘elementary’ limits they have set to themselves, they are here presenting the reader with an enticing invitation to the whole theory of \(p\)-adic differential equations.

Reviewer: D.Bertrand (Paris)

### MSC:

12H25 | \(p\)-adic differential equations |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11J91 | Transcendence theory of other special functions |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14G20 | Local ground fields in algebraic geometry |

32S40 | Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) |

33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |

41A21 | Padé approximation |