Algebraic groups and small transcendence degree. I
type: Item
instance of: scholarly article
title: Algebraic groups and small transcendence degree. I
zbMATH Open document ID: 0608.10036
DOI: 10.1016/0022-314X(87)90033-3
author: Robert Tubbs
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: This paper contains many results on algebraic independence of transcendental numbers of the following form: in certain specified sets of numbers (connected with values of exponential or elliptic functions, for instance), at least two elements are algebraically independent. One of the most interesting features of this paper is that all these results are deduced from only two general theorems, while in previous works on this subject, each single statement needed a specified proof [see for instance \textit{G. V. Chudnovsky}, Contributions to the theory of transcendental numbers (Math. Surv. Monogr. 19) (1984; Zbl 0594.10024)]. Here are the two main results. Let \(G={\mathbb{G}}_ a^{d_ 0}\times {\mathbb{G}}_ m^{d_ 1}\times G_ 2\) be a commutative algebraic group of dimension \(d=d_ 0+d_ 1+d_ 2\geq 1\), over a subfield K of \({\mathbb{C}}\). Let W be a subspace of \(T_ G({\mathbb{C}})\), of dimension \(n\geq 1\), and let V be a one-dimensional subspace of W. Assume that W is defined over \({\bar {\mathbb{Q}}}\). Further, let Y be a finitely generated subgroup of V, of rank \(\ell\) over \({\mathbb{Z}}\), such that \(\exp_ G Y\subset G(K)\). Assume furthermore that either (i) \(\ell (d-n)\geq \ell +2d_ 2+d_ 1\), with strict inequality in case \(d_ 0=n=1\), or (ii) \(Y\cap \ker \exp_ G\neq 0\) and \(2\ell (d-n)\geq \ell +2d_ 2+d_ 1\). Then the transcendence degree of K over \({\mathbb{Q}}\) is at least 2. Several new interesting consequences are derived. The proof involves the method of algebraic independence introduced by Gel'fond, an adaptation of Baker's method (for the case \(n\geq 2)\), and a refinement, due to Philippon, of a zero estimate of Masser-Wüstholz [\textit{P. Philippon}, Bull. Soc. Math. Fr. 114, 355-383 (1986)].
Mathematics Subject Classification ID: 11J85
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 3983283
zbMATH Keywords: Gel'fond method
zbMATH Keywords: exponential function
zbMATH Keywords: algebraic independence
zbMATH Keywords: transcendental numbers
zbMATH Keywords: elliptic functions
zbMATH Keywords: commutative algebraic group
zbMATH Keywords: Baker's method
zbMATH Keywords: zero estimate
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
cites work: The Size Function on Abelian Varieties
cites work: Q3208748
cites work: Q3724416
cites work: Algebraic independence of values of elliptic functions
cites work: Zero estimates on group varieties. I
cites work: Zero estimates on group varieties. II
cites work: Abelian varieties and algebraic independence. II: An abelian analogue of the Lindemann-Weierstrass theorem
cites work: Q5613281
cites work: Elliptic curves in two-dimensional abelian varieties and the algebraic independence of certain numbers
cites work: Q3718785
cites work: On the measure of algebraic independence of certain values of elliptic functions
cites work: Q3864596
cites work: Sous-groupes analytiques de groupes algébriques
cites work: Q3869420
cites work: Linear forms in elliptic logarithms
Recommended article: Algebraic groups and small transcendence degree. II
Recommended article: On the transcendence degree of group algebras of nilpotent groups
Recommended article: Small isomorphisms between group algebras
Recommended article: Small infinite dimensional modules for algebraic groups
Recommended article: Algebraization, transcendence, and \(D\)-group schemes
Recommended article: Q3835533
Recommended article: On the rationality of almost simple algebraic groups.
Recommended article: Q3881019
Recommended article: Algebraic structure of small countably compact Abelian groups
Recommended article: FINITE LINEAR GROUPS OF SMALL DEGREE. II
description: scientific article; zbMATH DE number 3983283