Algebraic independence for values of integral curves

We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over Q that are integral curves of some algebraic vector field (defined over Q). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields. This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series E2, E4, E6. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of Value Distribution Theory.