Horizontal sections of connections on curves and transcendence

Let $K$ be a number field, $\text{UX}$ be a smooth projective curve over it and $D$ be a reduced divisor on $\text{UX}$. Let $(E,\mathrm{\nabla })$ be a fibre bundle with connection having meromorphic poles on $D$. Let $p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_s\in \text{UX}(K)$ and $X:=\text{UX}\setminus \{D,p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_s\}$ (the $p_j$’s may be in the support of $D$). Using tools from Nevanlinna theory and formal geometry, we give the definition of $E$–section of type $\alpha$ of the vector bundle $E$ with respect to the points $p_j$; this is the natural generalization of the notion of $E$ function defined in Siegel Shidlowski theory. We prove that the value of a $E$–section of type $\alpha$ in an algebraic point different from the $p_j$’s has maximal transcendence degree. Siegel Shidlowski theorem is a special case of the theorem proved. We give an application to isomonodromic connections.