G-functions and geometry.

*(English)*Zbl 0688.10032
Max-Planck-Institut für Mathematik, Bonn. Aspects of Mathematics, 13. Wiesbaden etc.: Friedr. Vieweg & Sohn. xii, 229 p. DM 48.00 (1989).

As confirmed by the title, this is a book about G-functions. Moreover, as far as I know, it is the first book on this subject. So it is worth while to give an elaborate discussion. Before we consider its contents further, let us give a brief description of G-functions.

Consider a Taylor series of the form \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\), where the numbers \(a_ n\) belong to the same algebraic number field K \(([K:{\mathbb{Q}}]<\infty)\). Suppose it satisfies the following conditions, (i) f satisfies a linear differential equation with polynomial coefficients. (ii) \(| a_ n| =O(c^ n_ 1)\) for all \(n>0\) and a fixed \(c_ 1>0\). (iii) (common denominator of \(a_ 0,...,a_ n)=O(c^ n_ 2)\) for all \(n>0\) and a fixed \(c_ 2>0.\)

Roughly speaking, they can be considered as (very interesting) variations on the geometric series. They are not the same as the Meyer G-functions. The most common examples are -log(1-z), arctg(z) and the ordinary hypergeometric functions with rational parameters. They were defined by C. L. Siegel in 1929, along with their relatives, the E-functions, which can be considered as variations on \(e^ z\). Although Siegel states some irrationality results for values of G-functions at algebraic points, he never published the details of his computations. This turned out to be understandable when subsequent work of Galochkin and others showed that there are many more obstacles to get arithmetic results for values of G- functions then for E-functions. Significant progress was achieved in the 1980’s, notably by E. Bombieri and G. V. Chudnovsky. Much of this progress was related to the properties of G-functions themselves and to the question of what G-functions really are.

If one takes any linear differential equation with coefficients in \({\mathbb{Q}}(z)\), its power series solutions will usually not be G-functions. Having a G-function solution poses very large constraints on the arithmetic of a linear differential equation. There exist several conjectures in this direction, the most important being the Bombieri- Dwork conjecture that the differential equation should come from algebraic geometry in a suitable sense. All known G-functions actually arise in this way. The converse statement is known to be true. So statements on the arithmetic nature of values of G-functions can also have consequences for problems in algebraic geometry. This explains the ‘Geometry’ part of the title of the book. Despite all progress the amount of arithmetic results on values of G-functions at algebraic points is still very meager indeed.

The book under review is in the first place an account of recent developments in connection with G-function, which are otherwise scattered over the literature. Secondly, it is a very interesting attempt to point out directions in which it might be possible to have some ‘mature’ applications of G-function theory, notably to algebraic geometry. In addition there are a number of new results by the author. Let us try to give a summary of the contents. The first part deals with definitions and the introduction of two heights, \(\rho\) (f) and \(\sigma\) (f), of a G- function f. Then an important example (conjecturally the only), namely the case of geometric differential equations is presented. The second part deals with Fuchsian differential systems \(\Lambda\), their formal and arithmetic aspects. Again two heights are introduced, \(\rho\) (\(\Lambda)\) and \(\sigma\) (\(\Lambda)\). A corrected proof of Chudnosky’s remarkable theorem: ‘y cyclic and \(\sigma (y)<\infty\) implies \(\sigma (\Lambda)'\) is presented and finally the main results of this part are assembled on page 125. Part three deals with the arithmetic of values of G-functions. Here the author gives an unusual but original approach to produce linear independence results which is inspired by Gel’fond’s method. We also find Bombieri’s important idea of global relations in this part. It is on this principle that the author’s hope for future applications is based, although up till now I have not seen this hope vindicated by any example. Finally, in part four we find two applications, found by the author, of the previous results to algebraic geometry. One concerns Grothendieck’s conjecture on algebraic relations between periods of algebraic varieties. The other gives a bound for the heights of certain abelian varieties with a large endomorphism ring. Although both results apply to very limited situations it is very much worth while to keep these potential application areas for G-functions in mind.

In an appendix we find a new proof of the transcendence of \(\pi\) as a bonus. Unfortunately it contains an error. On line 6 on page 128 it is stated that \(\tau_ v\) belongs to the fundamental domain of SL(2,\({\mathbb{Z}})\) and on line 12 we find that \(\tau_ v\in \{ni,\frac{i+m}{n}| -[\frac{n}{2}]\leq m\leq [\frac{n-1}{2}]\}\). This is a contradiction. Moreover, I am afraid that this error is beyond repair.

To conclude, the book is written on a high level and not easy to read. Sometimes the author has a tendency to impress the reader unnecessarily. The proofs are written in concise, but usually intelligible way. They are not always reliable as is shown by the incorrect transcendence proof of \(\pi\). So the book should be handled with care in this respect. In particular someone ought to check the main theorem of chapter X very carefully. Despite these misgivings I enjoyed studying the book. Because of its originality and the stimulus it gives to provoke new research in the field of G-functions.

Consider a Taylor series of the form \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\), where the numbers \(a_ n\) belong to the same algebraic number field K \(([K:{\mathbb{Q}}]<\infty)\). Suppose it satisfies the following conditions, (i) f satisfies a linear differential equation with polynomial coefficients. (ii) \(| a_ n| =O(c^ n_ 1)\) for all \(n>0\) and a fixed \(c_ 1>0\). (iii) (common denominator of \(a_ 0,...,a_ n)=O(c^ n_ 2)\) for all \(n>0\) and a fixed \(c_ 2>0.\)

Roughly speaking, they can be considered as (very interesting) variations on the geometric series. They are not the same as the Meyer G-functions. The most common examples are -log(1-z), arctg(z) and the ordinary hypergeometric functions with rational parameters. They were defined by C. L. Siegel in 1929, along with their relatives, the E-functions, which can be considered as variations on \(e^ z\). Although Siegel states some irrationality results for values of G-functions at algebraic points, he never published the details of his computations. This turned out to be understandable when subsequent work of Galochkin and others showed that there are many more obstacles to get arithmetic results for values of G- functions then for E-functions. Significant progress was achieved in the 1980’s, notably by E. Bombieri and G. V. Chudnovsky. Much of this progress was related to the properties of G-functions themselves and to the question of what G-functions really are.

If one takes any linear differential equation with coefficients in \({\mathbb{Q}}(z)\), its power series solutions will usually not be G-functions. Having a G-function solution poses very large constraints on the arithmetic of a linear differential equation. There exist several conjectures in this direction, the most important being the Bombieri- Dwork conjecture that the differential equation should come from algebraic geometry in a suitable sense. All known G-functions actually arise in this way. The converse statement is known to be true. So statements on the arithmetic nature of values of G-functions can also have consequences for problems in algebraic geometry. This explains the ‘Geometry’ part of the title of the book. Despite all progress the amount of arithmetic results on values of G-functions at algebraic points is still very meager indeed.

The book under review is in the first place an account of recent developments in connection with G-function, which are otherwise scattered over the literature. Secondly, it is a very interesting attempt to point out directions in which it might be possible to have some ‘mature’ applications of G-function theory, notably to algebraic geometry. In addition there are a number of new results by the author. Let us try to give a summary of the contents. The first part deals with definitions and the introduction of two heights, \(\rho\) (f) and \(\sigma\) (f), of a G- function f. Then an important example (conjecturally the only), namely the case of geometric differential equations is presented. The second part deals with Fuchsian differential systems \(\Lambda\), their formal and arithmetic aspects. Again two heights are introduced, \(\rho\) (\(\Lambda)\) and \(\sigma\) (\(\Lambda)\). A corrected proof of Chudnosky’s remarkable theorem: ‘y cyclic and \(\sigma (y)<\infty\) implies \(\sigma (\Lambda)'\) is presented and finally the main results of this part are assembled on page 125. Part three deals with the arithmetic of values of G-functions. Here the author gives an unusual but original approach to produce linear independence results which is inspired by Gel’fond’s method. We also find Bombieri’s important idea of global relations in this part. It is on this principle that the author’s hope for future applications is based, although up till now I have not seen this hope vindicated by any example. Finally, in part four we find two applications, found by the author, of the previous results to algebraic geometry. One concerns Grothendieck’s conjecture on algebraic relations between periods of algebraic varieties. The other gives a bound for the heights of certain abelian varieties with a large endomorphism ring. Although both results apply to very limited situations it is very much worth while to keep these potential application areas for G-functions in mind.

In an appendix we find a new proof of the transcendence of \(\pi\) as a bonus. Unfortunately it contains an error. On line 6 on page 128 it is stated that \(\tau_ v\) belongs to the fundamental domain of SL(2,\({\mathbb{Z}})\) and on line 12 we find that \(\tau_ v\in \{ni,\frac{i+m}{n}| -[\frac{n}{2}]\leq m\leq [\frac{n-1}{2}]\}\). This is a contradiction. Moreover, I am afraid that this error is beyond repair.

To conclude, the book is written on a high level and not easy to read. Sometimes the author has a tendency to impress the reader unnecessarily. The proofs are written in concise, but usually intelligible way. They are not always reliable as is shown by the incorrect transcendence proof of \(\pi\). So the book should be handled with care in this respect. In particular someone ought to check the main theorem of chapter X very carefully. Despite these misgivings I enjoyed studying the book. Because of its originality and the stimulus it gives to provoke new research in the field of G-functions.

Reviewer: F.Beukers

##### MSC:

11J81 | Transcendence (general theory) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

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

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |

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

34M99 | Ordinary differential equations in the complex domain |