A gap theorem for abelian varieties over differential fields
type: Item
instance of: scholarly article
title: A gap theorem for abelian varieties over differential fields
zbMATH Open document ID: 0924.14021
author: Alexandru Buium
author: Anand Pillay
published in: Mathematical Research Letters
publication date: 1997-10-06T00:00:00Z
review text: Let \(\mathcal F\) be a differential field of characteristic zero with derivation \(\delta\) and field of constants \(\mathcal C\). Assume that \(\mathcal F\) is differentially closed. Let \(\mathcal F\{y_1, \dots, y_N\}\) denote the ring of differential polynomials over \(\mathcal F\) (ordinary polynomials in indeterminates \(\delta^iy_j\) which are differentiated as the notation implies). A subset \(\Sigma\) of the affine space \(\mathcal F^N\) is closed in the \(\delta\) topology if it is the set of common zeros of a subset of \(\mathcal F\{y_1, \dots, y_N\}\). If \(\Sigma\) is irreducible and if \(I(\Sigma)\) is the ideal of differential polynomials vanishing on it, the absolute dimension \(a(\Sigma)\) is the transcendence degree of the quotient field of \(\mathcal F\{y_1, \dots, y_N\}/I(\Sigma)\). The notions of \(\delta\)-closed subset and absolute dimension generalize in an obvious way to arbitrary algebraic varieties over \(\mathcal F\). The main result of this paper is the following theorem: Let \(A\) be an abelian variety over \(\mathcal F\) of dimension \(g\) which contains no abelian subvariety that descends to \(\mathcal C\). Let \(\Gamma \subset A\) be a \(\delta\)-closed subgroup of finite absolute dimension. Then any \(\delta\)-closed subset \(\Sigma\) of \(\Gamma\) Zariski dense in \(A\) has absolute dimension at least \(g+1\). Let \(A^\#\) denote the \(\delta\)-closure of the torsion points of \(A\). As a consequence of the above theorem, and previous work of the first author, the authors obtain the following results of \textit{E. Hrushovski} and \textit{Z. Sokolovic} [``Minimal subsets of differential closed fields'', Trans. Am. Math. Soc. (to appear)]. Suppose \(A\) is simple and does not descend to \(\mathcal C\). Then: Any proper \(\delta\)-closed subset of \(A^\#\) is finite; any \(\delta\)-closed subset of \(A^\# \times A^\#\) is a finite union of translates of \(\delta\)-closed subgroups; if \(B\) is another simple abelian variety over \(\ma…
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 1053504
zbMATH Keywords: simple abelian variety over differential field
zbMATH Keywords: gap theorem
zbMATH Keywords: absolute dimension
zbMATH Keywords: universal schemes
reviewed by: Andy R. Magid
MaRDI profile type: Publication
DOI: 10.4310/MRL.1997.V4.N2.A4
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description: scientific article; zbMATH DE number 1053504