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 ¯ that are integral curves of some algebraic vector field (defined over ¯). 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.

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