View source for G-function

The concept of $G$-functions, not to be confused with the [[Meijer-G-functions|Meijer $G$-functions]], was introduced by C.L. Siegel around 1929 [[#References|[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 function]]s, and the ordinary [[hypergeometric function]]s ${}_2F_1$ with rational parameters. Siegel introduced them along with the related [[E-function|$E$-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 [[#References|[a2]]] and G.V. Chudnovsky [[#References|[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 [[#References|[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 [[#References|[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 [[#References|[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 [[#References|[a7]]]. See also [[#References|[a8]]], [[#References|[a9]]] for actual computations both in the Archimedean and $p$-adic domain.

See also [[E-function|$E$-function]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> C.L. Siegel, "Über einige Anwendungen diophantischer Approximationen" , ''Ges. Abhandlungen'' , '''I''' , Springer (1966) {{MR|}} {{ZBL|56.0180.05}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Bombieri, "On $G$-functions" , ''Recent Progress in Analytic Number Theory'' , '''2''' , Acad. Press (1981) pp. 1–67 {{MR|0185108}} {{ZBL|0136.06503}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Chudnovski, G. Chudnovski, "Applications of Padé-approximations to diophantine inequalities in values of $G$-functions" , ''Lecture Notes in Mathematics'' , '''1052''' , Springer (1982) pp. 1–51</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> B. Dwork, G. Gerotto, F.J. Sullivan, "An introduction to $G$-functions" , ''Ann. Math. Studies'' , '''133''' , Princeton Univ. Press (1994) {{MR|1274045}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top"> N.M. Katz, "On differential equations satisfied by period matrices" ''IHES Publ. Math.'' , '''35''' (1968) {{MR|}} {{ZBL|0159.22502}} </TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top"> Y. André, "G-functions and geometry" , ''Aspects of Mathematics'' , Vieweg (1989) {{MR|0990016}} {{ZBL|0688.10032}} </TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Wolfart, "Werte hypergeometrischer Funktionen" ''Invent. Math.'' , '''92''' (1988) pp. 187–216 {{MR|0931211}} {{ZBL|0649.10022}} </TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top"> F. Beukers, J. Wolfart, "Algebraic values of hypergeometric functions" , ''New Advances in Transcendence Theory'' , Cambridge Univ. Press (1988) {{MR|0971994}} {{ZBL|0656.10030}} </TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top"> F. Beukers, "Algebraic values of $G$-functions" ''J. Reine Angew. Math.'' , '''434''' (1993) pp. 45–65 {{MR|1195690}} {{ZBL|0753.11024}} </TD></TR>
</table>

{{TEX|done}}