Which quaternion algebras act on a modular abelian variety?
  Let A be a modular abelian variety over \Q of arbitrary even dimension. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A over \bar\Q. We accomplish this by analyzing the representation of Gal(\bar\Q /\Q) on the points of N-torsion of A at primes N which ramify in B. 
Victor Rotger