Independence of Values of G-Functions

Zusammenfassung

In his paper of 1929 [56], C.L. Siegel, after defining G-functions and giving some examples, announced some results which one could obtain by the techniques he found (and described in the same paper) for studying the diophantine approximation properties of values of what he called E-functions. However no proof had appeared, and the first attempt in the direction of Siegel’s statements was in M.S. Numagomedov’s work, more than fourty years later. The successive work of A.I. Galockin [30], Y. Flicker, E. Bombieri [7] and D.V. & G.V. Chudnovsky [18], finally completed the proof of a G-function theorem that Siegel could have envisioned; roughly speaking, this is a quantitative result on the non-existence of too many algebraic relations among the values at some algebraic point ξ of certain G-functions, when ξ is “arithmetically” small enough (depending on the degree of the relations). Therefore, the diophantine theory of values of G-functions belongs to irrationality rather than transcendence theory. Nevertheless, its typical feature (and strength), discovered by Bombieri, is the possibility of a local-to-global setting.