On the non-commutative endomorphism rings of abelian surfaces

Abstract

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.