Relations algébriques entre valeurs de E-fonctions ou de M-fonctions

We prove that all algebraic relations over $\bar{\mathbb{Q}}$ between values of Siegel’s $E$-functions at some non-zero algebraic point have a functional source, in that they can be obtained as degeneration of $\delta$-algebraic relations over $\bar{\mathbb{Q}}(z)$ between the functions at stake. We also obtain an analogous result when considering Mahler’s $M_q$-functions. In this case, the so-called ${\sigma }_q$-algebraic relations substitute for the $\delta$-algebraic relations. We also give several consequences of this result concerning some descent phenomena. The point of view we adopt here reveals some striking similarities between the theory of $E$-functions and the one of $M_q$-functions.