On finiteness conjectures for endomorphism algebras of abelian surfaces

It is conjectured that there exist finitely many isomorphism classes of
simple endomorphism algebras of abelian varieties of GL_2-type over \Q of
bounded dimension. We explore this conjecture when particularized to quaternion
endomorphism algebras of abelian surfaces by giving a moduli interpretation
which translates the question into the diophantine arithmetic of Shimura curves
embedded in Hilbert surfaces. We address the resulting problems on these curves
by local and global methods, including Chabauty techniques on explicit
equations of Shimura curves.