Factorization estimates for Abelian varieties
type: Item
instance of: scholarly article
title: Factorization estimates for Abelian varieties
zbMATH Open document ID: 0854.11030
DOI: 10.1007/BF02699374
author: Gisbert Wüstholz
author: D. Masser
published in: Publications Mathématiques
publication date: 1996-03-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104105
review text: It is well known (Poincaré's complete irreducibility theorem) that every abelian variety \(A\) is isogenous to a product of the form \(A_1^{e_1}\times \cdots \times A_k^{e_k}\), where the \(A_i\)'s are simple abelian varieties which are mutually non-isogenous. In this work, the authors provide a quantitative version of this fact for abelian varieties defined over a number field. More precisely, given a number field \(K\) of absolute degree \(d\), an integer \(g\), and an abelian variety \(A\) defined over \(K\) of dimension \(g\), let \(h(A)\) denote the logarithmic absolute (stable) Faltings height of \(A\). The authors then prove (in Theorem I) that such a decomposition exists, and moreover (and here is the extra information) that the induced isogeny can be chosen of degree at most \(C(g, d)\max \{1, h(A)\}^{\kappa (g)}\), where \(\kappa (g)\) is an explicitly computable function of the dimension \(g\) (in fact it is quite large) of \(A\), and the implied constant \(C(g, d)\) depends polynomially on the degree \(d\) (with the exponent roughly of the size of \(\kappa (g)\)) and the dependence on \(g\) is unclear. They also prove an ``isogeny'' estimate giving a bound for the degree of an isogeny linking two isogenous abelian varieties defined over a number field (Theorem II) of the same form as the one provided for in Theorem I. It should be recalled that such an estimate was already provided by the authors [Ann. Math., II. Ser. 137, 459-472 (1993; Zbl 0804.14019) and Math. Z. 215, 641-653 (1994; Zbl 0826.14025)] but with one crucial difference: the abelian varieties were assumed to be polarized, and more precisely, both were supposed to be equipped with a polarisation of degree say less than \(\delta\), and the constant \(C(g, d)\) there was of the form \(C(g, d, \delta)\). Although the main result needed for these kinds of estimates is the ``periods'' estimates obtained previously by the authors [Ann. Math., II. Ser. 137, 407-458 (1993; Zbl 0796.11023)], it does …
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14K02
zbMATH DE Number: 847746
zbMATH Keywords: factorization estimates
zbMATH Keywords: abelian varieties
zbMATH Keywords: Faltings height
zbMATH Keywords: isogeny
zbMATH Keywords: class indices
zbMATH Keywords: Rosati discriminants
zbMATH Keywords: endomorphism estimates
Wikidata QID: Q101086289
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 847746