  A conjecture of Coleman implies that only finitely many quaternion algebras
over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of
abelian surfaces over the complex numbers which can be defined over
$\mathbf{Q}$. One may think of this as a higher-dimensional version of the
Gauss Class Number problem. Before now, no one has ruled out quaternion
algebras over $\mathbf{Q}$ not already ruled out by the Albert classification.
We rule out infinitely many such quaternion algebras by showing that for
infinitely many $D$, the Atkin-Lehner quotient Shimura curve $X^D/w_D$ has no
$\mathbf{Q}$-rational points. Our principal method is to use the level
structure maps above $X^D$ to create torsors for use in the descent
obstruction. Numerous Diophantine and analytic results on Shimura curves are
also proved.
