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Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider the endomorphism algebra $End^0(A) = End(A) \otimes \mathbf{Q}$, which lies inside the semisimple algebra $End^0(A_{\overline F})$.

Now let $F = \mathbf{Q}$ and let $K$ be a fixed real quadratic field with maximal order $\mathbf{Z}_K$. Let $D$ be the discriminant of $\mathbf{Z}_K$. We can form the coarse moduli space $Y_K$ of abelian varieties $A$ of dimension 2 with an embedding of $\mathbf{Z}_K$ into $End(A)$ (and other conditions), forcing $End^0(A) = K$. Although something could potentially happen with nonmaximal orders, let's just say that $Y_K$ is the moduli space of abelian surfaces with real multiplication by $K$. Alternately we could call it the Hilbert modular surface associated to $K$ and let $Y_{-}(D)$ be its compactification. Elkies-Kumar have a method for computing equations for these surfaces ( http://arxiv.org/pdf/1209.3527v3.pdf ) and in several cases produce rational points.

My question is

Question 1: Are there infinitely many $Y_K$ with rational points?

Of course I'm wondering about this in regards to the following.

Question 2: Are there infinitely many real quadratic fields $K$ appearing as $End^0(A_{\overline{\mathbf{Q}} })$ for some abelian surface $A$ over the rational numbers?

One possible tack to take here (as in http://www.math.harvard.edu/~elkies/banff07.pdf ) is the Eichler-Shimura construction, whereby if we take a normalized weight 2 cuspidal newform $f = \sum a_n q^n$ of level $N$ with $\mathbf{Q}(\ldots, a_n,\ldots) = K$ then we can find an abelian surface over $\mathbf{Q}$ with endomorphism algebra $K$ ( and with conductor $N$). Each $S_2(N)$ is finite-dimensional, but lots of them have 2-dimensional Galois orbits of Hecke eigenspaces. If it were possible to get our hands on the ramification of these coefficient fields while fixing the degree that would do the job, but saying anything about these coefficient fields seems to be hard. The one result I have been told about is that of Dieulefait, Jimenez-Urroz, and Ribet saying that the degrees of these coefficient fields are unbounded, extending Mazur's work on prime levels.

Question 3: Are there infinitely many real quadratic fields $K$ appearing as the coefficient field $\mathbf{Q}(\ldots, a_n(f),\ldots)$ for a normalized cuspidal weight 2 newform $f$ of level $N$?

None of these questions have an immediately obvious obstruction so I'd like to think the answer to all 3 is yes, and I could imagine a slick proof using modular forms existing somewhere but I can't see it myself.

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I had forgotten that I had posted this question, but in the time since I was pointed by John Voight to the following paper of Bruin, Flynn, Gonzalez, and Rotger: https://www-ma2.upc.edu/vrotger/docs/BFGR.pdf

This paper begins with a Conjecture of Coleman on the Endomorphism rings of abelian varieties. In particular, a solution to Conjecture $\mathbf{C}(1,2)$ would imply an answer of no to my questions. That said, this is a very hard conjecture. Another consequence of the truth of this conjecture is Mazur's theorem on rational isogenies of prime degree.

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