Slopes and Abelian Subvariety Theorem

In this article, we show how to modify the proof of the Abelian Subvariety Theorem by Bost (Périodes et isogénies des variétés abeliennes sur les corps de nombres, Séminaire Bourbaki, 1994-95, Theorem 5.1) in order to improve the bounds in a quantitative respect and to extend the theorem to subspaces instead of hyperplanes. Given an abelian variety A defined over a number field κ and a non-trivial period γ in a proper subspace W of tAK with K a finite extension of κ, the Abelian Subvariety Theorem shows the existence of a proper abelian subvariety B of AQ¯, whose degree is bounded in terms of the height of W, the norm of γ, the degree of κ and the degree and dimension of A. If A is principally polarized then the theorem is explicit.