The effective Shafarevich conjecture for abelian varieties of -type

In this article we establish the effective Shafarevich conjecture for abelian varieties over ${\mathbb Q}$ of ${\text {GL}_2}$ -type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

Représentations linéaires des groupes kählériens : factorisations et conjecture de Shafarevich linéaire

We extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach, based on an interversion lemma for fibrations with tori versus general type manifolds as fibers, gives a refinement of the classical work of Zuo. We extend to the Kähler case some general results on holomorphic convexity of coverings such as the linear Shafarevich conjecture.