Effective bound for the geometric Lang conjecture
type: Item
instance of: scholarly article
title: Effective bound for the geometric Lang conjecture
zbMATH Open document ID: 0812.14029
DOI: 10.1215/S0012-7094-93-07120-7
published in: Duke Mathematical Journal
publication date: 1995-05-14T00:00:00Z
review text: Let \(A\) be an abelian variety of dimension \(g\) over the complex field \(\mathbb{C}\), let \(X \subset A\) be a smooth closed subvariety of dimension \(s\) and \(\Gamma \subset A\) a subgroup of finite rank \(r\). After Faltings' proof of a celebrated conjecture of S. Lang, there exist a finite number of abelian subvarieties \(B_ i \subset A\) and points \(a_ i \in A\) such that: \(X \cap \Gamma \subset \bigcup^ n_{i=1} (B_ i + a_ i) \subset X\). Denote by \(\lambda (X \cap \Gamma)\) the minimum number \(n\) of such translates of abelian subvarieties of \(A\) contained in \(X\) and whose union contains \(X \cap \Gamma\). In this paper, the author finds an effective bound for \(\lambda (X \cap \Gamma)\) in the ``geometric case'', when either \(A\) has no abelian subvariety which descends to \(\overline \mathbb{Q}\) or \(X\) is a curve not descending to \(\overline \mathbb{Q}\). Under this assumption he is able to prove that: \(\lambda (X \cap \Gamma) \leq N(!)^{3 N^ 2}\), where, if \(D\) is an ample divisor on \(A\) defining a polarisation of degree \(d^ 2\) and \(e = (X \cdot D^ s)\), one sets \(N = \max \{g,r,d,e,2s + 1\}\) and the notation \(N(!)^ m\) indicates iterated factorial taken \(m\) times. The methods rely heavily on the ``differential algebraic'' constructions of \textit{A. Buium} [``Intersections in jet spaces and a conjecture of S. Lang,'' Ann. Math., II. Ser. 136, No. 3, 557-567 (1992)].
Mathematics Subject Classification ID: 14K05
zbMATH DE Number: 495429
zbMATH Keywords: subvariety of abelian variety
Wikidata QID: Q123253110
author: Alexandru Buium
reviewed by: Enric Nart
MaRDI profile type: Publication
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full work available at URL: s0012-7094-93-07120-7
OpenAlex ID: W2063701439
description: scientific article; zbMATH DE number 495429