#### Vol. 13, No. 3, 2019

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Algebraic independence for values of integral curves

### Tiago J. Fonseca

Vol. 13 (2019), No. 3, 643–694
##### Abstract

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

##### Keywords
algebraic independence, transcendence, integral curves, modular forms, Eisenstein series, differential equations, Nevanlinna theory, integrality, zero lemma
##### Mathematical Subject Classification 2010
Primary: 11J81
Secondary: 14G40, 37F75, 32A22
##### Milestones
Received: 17 May 2018
Accepted: 25 December 2018
Published: 23 March 2019
##### Authors
 Tiago J. Fonseca Mathematical Institute University of Oxford Oxford United Kingdom