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?