Slopes and abelian subvariety theorem
In a series of papers published during 1990--1995 Masser and Wüstholz studied the period relations for abelian varieties and obtained an alternative proof of the Tate Conjecture proved by Faltings in 1983. They have obtained a bound for the degree of a minimal abelian subvariety \(B\) of an abelian variety \(A\), the tangent space of which at the origin contains a given period of the lattice associated to \(A\). The approach of Masser and Wüstholz was later developed by Bost who gave an intrinsic geometric interpretation of original argument and introduced several new tools. In the paper under review the author improves the bounds in Bost's Abelian Subvariety Theorem and extends this theorem to subspaces instead of hyperplanes. The main result of the paper asserts that if \(A\) is an abelian variety over a number field \(k\), and \(\gamma\) is a non-trivial period in a proper subspace \(W\) of the stalk \(t_{A_K}\) at zero of the tangent bundle of \(A_K\) (\(K\) is a finite extension of \(k\)) then there exists a proper abelian subvariety \(B\) of \(A_{\overline{\mathbb{Q}}}\) the degree of which is bounded in terms of the height of \(W\), the norm of \(\gamma\), the degree of \(k\) and the degree and dimension of \(A\). The main ingredients of the proof are Hermitian vector bundles, Moret-Bailly models, non-reduced subschemes, Faltings heights, Slope Inequality and Phragmén-Lindelöf Theorem. Let \(\Sigma_K\) be an \(N\)-torsion subscheme of \(A\), and \(L\) be a symmetric ample line bundle on \(A\). The author considers a Moret-Baily model \((\mathcal{A}, \mathcal{L}, \Sigma)\) of \((A,L,\Sigma_K)\) and the restriction morphism \(\phi:E \rightarrow F\) of Hermitian \(\mathcal{O}_K\)-modules, where \(E\) is given by global section \(\mathcal{L}^D\) with \(D\) a positive integer, and \(F\) is given by the restriction of these sections at a highly non-reduced subscheme with support in \(E\). If \(\phi\) is injective it satisfies the so called slopes inequality which provides the upper bound for the normalized Arakelov degree \(\widehat{\text{deg}}_n\overline{E}\) in terms of some quantities related to \(\varphi\), filtration on \(F\), maximal slopes and \(K\). The author considers an arithmetic filtration related to \(\phi\), defines the evaluation operators, and establishes the analytic inequalities for the norms of these operators which allow him to choose the parameters for which the slopes inequality fails. Thus \(\phi\) can not be injective and there exists a section \(s\in E\) in its kernel. Then a version of Multiplicity Estimate Theorem guarantees the existence of an abelian subvariety \(B\) of \(A_{\overline{\mathbb{Q}}}\) with bounded degree. The author concludes with an appendix that discusses the basic properties of Hermitian vector bundles and of the Arakelov degree.
scientific article; zbMATH DE number 2164819
scientific article