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 $O_K$?
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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.
answered Nov 20, 2018 at 15:20
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$\begingroup$ Thanks Francois, I am aware of this conjecture of Coleman, which would of course imply the answer to my question. I guess what I'm asking is whether it is known if the quantity $[O_K : R]$ is bounded, even for a fixed $K$? $\endgroup$– Sun RaCommented Nov 20, 2018 at 16:12
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$\begingroup$ @SunRa The index of the Hecke algebra is linked with congruences between the two associated eigenforms, so maybe the theory of congruences helps here. $\endgroup$ Commented Nov 20, 2018 at 16:49
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$\begingroup$ Do you know how they are linked exactly? $\endgroup$– Sun RaCommented Nov 20, 2018 at 22:45
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$\begingroup$ @SunRa As an example, take $A=A_f$ with $f$ a newform of weight 2 with coefficients in the order $R=\mathbf{Z}[m\sqrt{2}]$, then the Galois conjugate newform $f'$ is congruent to $f$ modulo $2m\sqrt{2}$. $\endgroup$ Commented Nov 21, 2018 at 8:15
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$\begingroup$ Yes that could be helpful. When $A_f$ is associated to $f$, Is it that End$_{\mathbb{Q}}(A_f) = \mathbb{Z}[ \{a_i(f)\}_i ]$ or the weaker statement that End$_{\mathbb{Q}}(A_f) \otimes \mathbb{Q} = \mathbb{Q}( \{a_i(f)\}_i )$. I hoped it was the former but I cannot find a precise reference for this. Also Prop 2.5.4 of J. Wilson's thesis implies that associated to $f$ there should also be a surface with real multiplication by the maximal order (because there is an isogeny). $\endgroup$– Sun RaCommented Nov 21, 2018 at 13:46