Effective bound for the geometric Lang conjecture
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)].
scientific article; zbMATH DE number 495429
scientific article