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 $0$. 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 $u_{\mathbb{K},F}\log(S)\le \dim_{\mathbb{K}}(\Phi_{\alpha, S }) \le v_F S$ for any $S$, for some constants $u_{\mathbb{K},F}>0$ and $v_F>0$. This appears to be the first Diophantine result for values of $G$-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [{\em Linear independence of values of G-functions}, 46 pages, J. Europ. Math. Soc., to appear], where $\alpha\in \overline{\mathbb{Q}}^*$ was assumed to be in the disk of convergence. Its proof relies on an explicit construction of a Pad\'e 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 Andr\'e, Chudnovsky and Katz on $G$-operators, of a linear independence criterion \`a la Siegel over number fields, and of a far reaching generalization of Shidlovsky's lemma built upon the approach of Bertrand-Beukers and Bertrand.