Motivation:
Let $\mathbb{Q}_{\infty,p}$ be the field obtained by adjoining to $\mathbb{Q}$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal cyclotomic extension $\mathbb{Q}^\text{cycl}$ of $\mathbb{Q}$. By Kronecker–Weber, $\mathbb{Q}^\text{cycl}$ is also the maximal abelian extension $\mathbb{Q}^\text{ab}$ of $\mathbb{Q}$.
A well known conjecture due to Mazur (with known examples) asserts, for an elliptic curve $E$ with certain conditions, that $E(\mathbb{Q}_{\infty,p})$ is finitely generated. This is the group of rational points of $E$ over $\mathbb{Q}_{\infty,p}$ (not a number field!).
A theorem due to Ribet asserts the finiteness of the torsion subgroup $E(\mathbb{Q}^\text{ab})$ for certain elliptic curves.
Questions:
(a) Can one expect to find elliptic curves (or abelian varieties) $A$ with $A(\mathbb{Q}^\text{ab})$ finitely generated?
(c) Can one expect to find curves $C$ of genus $g >1$ with $C(\mathbb{Q}^\text{ab})$ finite?
Best Answer
Actually Ken Ribet proved that if $K$ is a number field and $K(\mu_{\infty})$ is its infinite cyclotomic extension generated by all roots of unity then for every abelian variety $A$ over $K$ the torsion subgroup of $A(K(\mu_{\infty}))$ is finite: http://math.berkeley.edu/~ribet/Articles/kl.pdf .
On the other hand, Alosha Parshin conjectured that if $K_{p}$ is the extension of $K$ generated by all $p$-power roots of unity (for a given prime $p$) then the set $C(K_{p})$ is finite for every $K$-curve $C$ of genus $>1$: http://arxiv.org/abs/0912.4325 (see also http://arxiv.org/abs/1001.3424 ).