Motivation:

Let $Q_{\infty,p}$ be the field obtained by adjoining to $Q$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension $Q^{cycl}$ of $Q$. By Kronecker-Weber, $Q^{cycl}$ is also the maximal abelian extension $Q^{ab}$ of $Q$.

A well known conjecture due to Mazur (with known examples) asserts, for an elliptic curve $E$ with certain conditions, that $E(Q_{\infty,p})$ is finitely generated. This is the group of rational points of $E$ over $Q_{\infty,p}$ (not a number field!).

A theorem due to Ribet asserts the finiteness of the torsion subgroup $E(Q^{ab})$ for certain elliptic curves.

Questions:

(a) Can one expect to find elliptic curves (or abelian varieties) $A$ with $A(Q^{ab})$ finitely generated?

(c) Can one expect to find curves $C$ of genus $g >1$ with $C(Q^{ab})$ finite?

Thanks!