Algebraic independence for values of integral curves

TL;DR

This paper establishes a transcendence theorem for values of holomorphic maps from a disk to a quasi-projective variety, specifically for integral curves of algebraic vector fields, generalizing Nesterenko's algebraic independence results.

Contribution

It introduces a new geometric growth condition replacing polynomial growth, broadening the scope of algebraic independence results for integral curves.

Findings

Proves algebraic independence of values of certain holomorphic maps.

Generalizes Nesterenko's theorem on Eisenstein series.

Uses a geometric notion of moderate growth via Value Distribution Theory.

Abstract

We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are integral curves of some algebraic vector field (defined over $\overline{\mathbf{Q}}$). 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.