Bounds on the size of isogeny classes (over number fields)

Question

I am reading B. Mazur's seminal paper "Rational isogenies of prime degree" (Invent. Math. 44 (1978), 129-162), and Theorem 5 of this paper caught my attention; it states that there exists an absolute constant $C$ such that any elliptic curve $E/\mathbb{Q}$ is $\mathbb{Q}$-isogenous to at most $C$ pairwise non-isomorphic elliptic curves $E^\prime/\mathbb{Q}$.

Has this statement been proved for any other genus? That is, take abelian varieties $A/\mathbb{Q}$, with $\dim A = g \geq 2$. Does there exist a number $N_g$ depending only on $g$ such that the number of pairwise non-isomorphic abelian varieties $B$ of dimension $g$ isogenous to $A$ is at most $N_g$? What about the analogous statement for number fields?

Answer 1

I imagine this is very much open. Even special cases of this question seem hard.

Consider weight two modular forms of level $\Gamma_0(N)$ with real quadratic coefficient field $K$. Galois orbits of these are in bijective correspondence with abelian surfaces $A$ over $\mathbb{Q}$ with real multiplication by $K$. The class number of $K$ gives a lower bound on your constant $N_g$, by considering isogenies where you mod out by the $I$-torsion subgroup $A[I]$ for some ideal $I$ in $\mathrm{End}(A)$.

It's a conjecture of Coleman that only finitely many $K$ should arise this way, but the conjecture is very open. And if we can't control the number of $K$, it's hard to imagine we can say their class numbers are bounded. You can approach this question in terms of rational points on Hilbert modular surfaces, and conjecturally there should be few rational points once the discriminant is large, but again very little is known. Anyway, "few" doesn't mean 0.

Maybe you'd like to focus on those $A$ with $\mathrm{End}(A) = \mathbb{Z}$. Then you're looking at rational points on Siegel modular 3-folds (somehow avoiding all the rational points coming from lower dimensional Shimura varieties). But Mazur's strategy seems to totally fall apart here since the 3-fold doesn't embed in to an abelian variety in an obvious way.