G-function

The concept of $G$-functions, not to be confused with the Meijer $G$-functions, was introduced by C.L. Siegel around 1929 [a1], p. 239, in connection with transcendence questions. Consider a Taylor series of the form $$ f(z) = \sum_{n=0}^\infty a_n z^n $$ where the numbers $a_n$ all belong to a fixed algebraic number field (cf. also Algebraic number; Field) $K$ ($[K:\mathbf{Q}] < \infty$) Suppose it satisfies the following conditions:

i) $f$ satisfies a linear differential equation (cf. also Linear differential operator) with polynomial coefficients;

ii) there exists a $c>0$ such that $H(a_0,a_1,\ldots,a_n) = O(c^n)$. Then $f$ is called a $G$-function. Here, the notation $H$ stands for the so-called projective height, given by $$ H(x_0,\ldots,x_n) = \prod_\nu \max(|x_0|_\nu,\ldots,|x_n|_\nu) $$ for any $(n+1)$-tuple $(x_0,\ldots,x_n) \in K^{n+1}$. The product is taken over all valuations $|\cdot|_\nu$ of $K$ (cf. also Norm on a field). When the $x_i$ are rational numbers, $H(x_0,\ldots,x_n)$ is simply the maximum of the absolute values of the $x_i$ times their common denominator. Roughly speaking, Siegel's $G$-functions can be considered as (very interesting) variations on the geometric series, hence the name.

The most common examples are $-\log(1-z)$, $\arctan z$, algebraic functions, and the ordinary hypergeometric functions ${}_2F_1$ with rational parameters. Siegel introduced them along with the related $E$-functions, which can be considered as variations on $\exp z$.

Although Siegel states some irrationality results for values of $G$-functions at algebraic points, he never published the details of his computations. Subsequent work of A.I. Galoschkin and others showed that there are many more obstacles to get arithmetic results for values of $G$-functions than for $E$-functions. Significant progress was achieved in the 1980s, notably by E. Bombieri [a2] and G.V. Chudnovsky [a3]. 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 $\mathbf{Q}(z)$, its power series solutions will usually not be $G$-functions. An introduction to these arithmetical problems is given in [a4]. Having a $G$-function solution poses considerable 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, see [a5]. So, statements on the arithmetic nature of values of $G$-functions can also have consequences for problems in algebraic geometry. This point of view was notably adopted by Y. André [a6].

Presently (1996), one can say that Siegel's method and its improvements have not yet given a transcendence proof for a single value of a $G$-function. The fact that such transcendence proofs are a subtle matter was once more confirmed by J. Wolfart's discovery of transcendental $G$-functions having algebraic values at algebraic points [a7]. See also [a8], [a9] for actual computations both in the Archimedean and $p$-adic domain.

See also $E$-function.

References

[a1]	C.L. Siegel, "Über einige Anwendungen diophantischer Approximationen" , Ges. Abhandlungen , I , Springer (1966) Zbl 56.0180.05
[a2]	E. Bombieri, "On $G$-functions" , Recent Progress in Analytic Number Theory , 2 , Acad. Press (1981) pp. 1–67 MR0185108 Zbl 0136.06503
[a3]	D.V. Chudnovsky, G.V. Chudnovsky, "Applications of Padé-approximations to diophantine inequalities in values of $G$-functions" , Lecture Notes in Mathematics , 1052 , Springer (1982) pp. 1–51
[a4]	B. Dwork, G. Gerotto, F.J. Sullivan, "An introduction to $G$-functions" , Ann. Math. Studies , 133 , Princeton Univ. Press (1994) MR1274045 Zbl 0830.12004
[a5]	N.M. Katz, "On differential equations satisfied by period matrices" IHES Publ. Math. , 35 (1968) Zbl 0159.22502
[a6]	Y. André, "G-functions and geometry" , Aspects of Mathematics , Vieweg (1989) MR0990016 Zbl 0688.10032
[a7]	J. Wolfart, "Werte hypergeometrischer Funktionen" Invent. Math. , 92 (1988) pp. 187–216 MR0931211 Zbl 0649.10022
[a8]	F. Beukers, J. Wolfart, "Algebraic values of hypergeometric functions" , New Advances in Transcendence Theory , Cambridge Univ. Press (1988) MR0971994 Zbl 0656.10030
[a9]	F. Beukers, "Algebraic values of $G$-functions" J. Reine Angew. Math. , 434 (1993) pp. 45–65 MR1195690 Zbl 0753.11024