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Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?

Do there exist infinitely many real quadratic fields $F$ that are the coefficient field of a weight $2$ classical holomorphic eigenforms over $\mathbb Q$?

Of course the analogue of the first question for imaginary quadratic fields and elliptic curves the answer is yes. By studying the Hilbert surfaces, maybe one can prove conditionally on Bombieri-Lang that there are finitely many $F$ for which there are infinitely many isomorphism classes of such surfaces, but this of course is a long way off.

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  • $\begingroup$ @reuns Yes, I do. $\endgroup$
    – Will Sawin
    Commented Dec 13, 2018 at 3:31
  • $\begingroup$ @reuns What is a weight 2 Maass form? In any case my interest is with abelian varieties, and no Maass forms are expected to correspond to abelian varieties. $\endgroup$
    – Will Sawin
    Commented Dec 13, 2018 at 4:24
  • $\begingroup$ There is a conjecture of Coleman asserting that there are only finitely many such real quadratic fields, see Which quaternion algebras act on a modular abelian variety by Victor Rotger. Also it is known that abelian surfaces with real multiplication are modular, so unless I'm missing something the two questions are equivalent. $\endgroup$
    – François Brunault
    Commented Dec 13, 2018 at 8:22
  • $\begingroup$ @FrançoisBrunault I would be willing to accept such an answer explaining that it's a known open problem. $\endgroup$
    – Will Sawin
    Commented Dec 13, 2018 at 12:52
  • $\begingroup$ This question was earlier asked here: mathoverflow.net/q/207935/6518 Should we close one of these? $\endgroup$
    – Kimball
    Commented Jul 14, 2020 at 0:36

1 Answer 1

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A conjecture of Coleman asserts that only finitely many rings arise as the endomorphism ring of an abelian variety of given dimension defined over a number field of given degree. See [1] for an account of this conjecture. In your case, the relevant conjecture is denoted there by $C(1,2)$. To my knowledege, the only results on Coleman's conjecture in dimension >1 concern CM abelian varieties.

Regarding your second question, Ribet has shown that Serre's conjecture on representations of the absolute Galois group of $\mathbf{Q}$ [2, Conjecture (3.2.4)] implies that every abelian variety of $\mathrm{GL}_2$-type over $\mathbf{Q}$ is a quotient of $J_1(N)$ for some $N \geq 1$, see [3, Theorem 4.4]. Serre's conjecture is now a theorem of Khare-Wintenberger [4, Theorem 5.1] so that your second question is equivalent to the first.

[1] Bruin, Flynn, Gonzalez, Rotger, On finiteness conjectures for endomorphism algebras of abelian surfaces. Math. Proc. Camb. Philos. Soc. 141 (2006), No. 3, 383-408.

[2] Serre, Sur les représentations modulaires de degré 2 de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. Duke Math. J. 54 (1987), 179-230.

[3] Ribet, Abelian varieties over $\mathbf{Q}$ and modular forms. Modular curves and abelian varieties, 241–261, Progr. Math., 224, Birkhäuser, 2004.

[4] Khare, Serre's conjecture and its consequences, Japan. J. Math. 5 (2010), 103–125.

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