Algebraic independence for values of integral curves

Fonseca, Tiago J.

doi:10.2140/ant.2019.13.643

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Algebraic independence for values of integral curves

Tiago J. Fonseca

Vol. 13 (2019), No. 3, 643–694

DOI: 10.2140/ant.2019.13.643

Abstract

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

Vol. 13, No. 3, 2019