Horizontal sections of connections on curves and transcendence

Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a fibre bundle with connection having meromorphic poles on D. Let p1,...,ps ∈ X(K) and X := X \ {D,p1,...,ps} (the pj’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 α of the vector bundle E with respect to the points pj ; 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 α in an algebraic point different from the pj’s has maximal transcendence degree. Siegel Shidlowski theorem is a special case of the theorem proved. We give an application to isomonodromic connections.