TITLE: Index of the endomorphism ring of an abelian surface
QUESTION [8 upvotes]: For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]$ of the ring inside the maximal order $O_K$?

REPLY [2 votes]: There is a conjecture of Coleman asserting that up to isomorphism, there are only finitely many possible rings $\operatorname{End}_{\mathbf{Q}}(A)$ where $A$ varies among the abelian surfaces defined over $\mathbf{Q}$ (see Bruin, Victor Flynn, Gonzalez, Rotger, On finiteness conjectures for endomorphism algebras of abelian surfaces). This would imply that the answer to your question is yes. On the other hand, this is a very difficult conjecture, so there might be an easier way to tackle your question.