Endomorphism Algebras of abelian varieties over a number field

Question

Let $K$ be a number field and let $g$ be an integer. Let $\mathcal{A}(K,g)$ be the set of absolutely simple $g$-dimensional abelian varieties over $K$. Is the set $\{\mathrm{End}^0(A_{\mathbb{\overline{Q}}}):A\in \mathcal{A}(K,g)\}$ of division algebras a finite set?

Answer 1

It is a folk conjecture that for each fixed positive integer $g$ and $d$, the set of isomorphism classes of rings arising as $\operatorname{End}(A) := \operatorname{End}(A_{\overline{F}})$ for a $g$-dimensional abelian variety defined over any degree $d$ number field $F$ is finite.

(For a time when I was in grad school I was of the opinion that this conjecture arose in a conversation I had with my thesis advisor, Barry Mazur. Later it turned out that this conjecture, or some close variant of it, was made earlier by Robert Coleman, and I now think it is likely that the idea passed from Coleman to Mazur to me.)

Some comments:

$\bullet$ The conjecture holds when $g = 1$ -- i.e. for elliptic curves. Here the content is that for an imaginary quadratic order $\mathcal{O}$, $[\mathbb{Q}(j(\mathbb{C}/\mathcal{O})):\mathbb{Q}]$ tends to infinity with the discriminant of $\mathcal{O}$, a theorem of Heilbronn from the 1930's.

$\bullet$ I am a bit embarrassed not to know off the top my of head the status of this conjecture restricted to abelian varieties with complex multiplication (CM) for arbitrary $g$ and $d$. I am pretty confident that the case of CM and $(g,d) = (2,1)$ is known, even with an explicit list of the possible rings.

$\bullet$ The case $(g,d) = (2,1)$ is already wide open. It can be very roughly translated as follows: as we range over all possible Hilbert modular surfaces and all quaternionic Shimura curves, only finitely many have "nontrivial $\mathbb{Q}$-rational points". (At any rate, this should also be true, and it would at least morally imply the case $(g,d) = (2,1)$. The equivalence runs afoul of some tricky field of moduli vs. field of definition issues.)

For all we know there could be infinitely many real quadratic *fields( arising as endomorphism algebras of abelian surfaces $A_{/\mathbb{Q}}$ and also infinitely many indefinite rational quaternion algebras so arising. Actually even to show that a particular real quadratic field or indefinite rational quaternion algebra cannot arise in this way is very difficult. The latter case is addressed by past work of Victor Rotger and work in progress of Jim Stankewicz.