On Finiteness Conjectures for Modular Quaternion Algebras

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL2-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.