Slopes and abelian subvariety theorem
type: Item
instance of: scholarly article
title: Slopes and abelian subvariety theorem
zbMATH Open document ID: 1080.11048
author: Evelina Viada
published in: Journal of Number Theory
publication date: 2005-05-04T00:00:00Z
review text: 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 p…
reviewed by: Vasyl I. Andriychuk
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 2164819
zbMATH Keywords: Abelian varieties
zbMATH Keywords: Slopes theory
zbMATH Keywords: transcendence theory
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.07.015
OpenAlex ID: W2057065509
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DOI: 10.1016/J.JNT.2004.07.015
description: scientific article; zbMATH DE number 2164819