Linear independence of values of G-functions, II: outside the disk of convergence

Given any non-polynomial G-function \(F(z)=\sum _{k=0}^\infty A_kz^k\) of radius of convergence R and in the kernel a G-operator \(L_F\), we consider the G-functions \(F_n^{[s]}(z)=\sum _{k=0}^\infty \frac{A_k}{(k+n)^s}z^k\) for every integers \(s\ge 0\) and \(n\ge 1\). These functions can be analytically continued to a domain \({\mathcal {D}}_F\) star-shaped at 0 and containing the disk \(\{\vert z\vert <R\}\). Fix any \(\alpha \in {\mathcal {D}}_F \cap \overline{{\mathbb {Q}}}^*\), not a singularity of \(L_F\), and any number field \({\mathbb {K}}\) containing \(\alpha \) and the \(A_k\)’s. Let \(\Phi _{\alpha , S}\) be the \({\mathbb {K}}\)-vector space spanned by the values \(F_n^{[s]}(\alpha )\), \(n\ge 1\) and \(0\le s\le S\). We prove that \(u_{{\mathbb {K}},F}\log (S)\le \dim _{\mathbb {K}}(\Phi _{\alpha , S })\le v_FS\) for any S, for some constants \(u_{{\mathbb {K}},F}>0\) and \(v_F>0\). This appears to be the first general Diophantine result for values of G-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [Linear independence of values of G-functions, J. Europ. Math. Soc. 22(5), 1531–1576 (2020)], where \(\alpha \in \overline{{\mathbb {Q}}}^*\) was assumed to be such that \(\vert \alpha \vert <R\). Its proof relies on an explicit construction of a Padé approximation problem adapted to certain non-holomorphic functions associated to F, and it is quite different of that in the above mentioned paper. It makes use of results of André, Chudnovsky and Katz on G-operators, of a linear independence criterion à la Siegel over number fields, and of a far reaching generalization of Shidlovsky’s lemma built upon the approach of Bertrand–Beukers and Bertrand.