Algebraic independence for values of integral curves

Abstract

We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasiprojective variety over $\overline{\mathbb{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbb{Q}}$). 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 $E_2$, $E_4$, $E_6$. 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.