Factorization estimates for Abelian varieties
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 not appear directly in this paper, but is used indirectly via the endomorphism estimates quoted above. The argument goes essentially as follows. First one describes various ``class indices'' for division algebras in order to quantify the classical Jordan-Zassenhaus theorem. These class indices are bounded in terms of appropriate discriminants. Then all these discriminants are related to the Rosati discriminants defined via the Rosati involution on the endomorphism ring of a polarised abelian variety. A use of Zarhin's trick then enables one to get down to a principally polarised abelian variety on which the endomorphism estimates give the required bounds for the Rosati discriminants in terms of the Faltings height.
scientific article; zbMATH DE number 847746
scientific article