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Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It is easy to prove that it is not finitely generated and the theorem of K.Ribet asserts that its torsion is finite. What else can we say about it? Does it look more like $\bigoplus \mathbb Z$ or $\mathbb C/\Gamma$? Is it divisible? (I don't think so). I apologize that the questions are not precise, I just need to know which results do we have in this direction.

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$E (\mathbb Q^{ab})/\operatorname{tors}$ is a sum of countably many copies of $\mathbb Z$.

To prove this, take a countable basis $x_1,x_2,\dots$ of $E (\mathbb Q^{ab})/\operatorname{tors}\otimes \mathbb Q$. Suppose we show that $\oplus_{i=1}^n x_i \mathbb Q \cap E (\mathbb Q^{ab})/\operatorname{tors}$ is finitely generated. Then it is necessarily isomorphic to $\mathbb Z^n$. Moreover, the only embeddings $\mathbb Z^n \to \mathbb Z^{n+1}$ where the elements of $\mathbb Z^n$ don’t get any more divisible are isomorphic to the standard embedding. So if $x_1,\dots,x_n$ form a $\mathbb Z$-basis of $\oplus_{i=1}^n x_i \mathbb Q \cap E (\mathbb Q^{ab})/\operatorname{tors}$, we can change $x_{n+1}$ so it generates the same $\mathbb Q$-subspace and also $x_1,\dots,x_{n+1}$ form a $\mathbb Z$-basis of $\oplus_{i=1}^{n+1} x_i \mathbb Q \cap E (\mathbb Q^{ab})/\operatorname{tors}$. Iterating, we form a $\mathbb Z$-basis of the whole group.

Now let’s check the claim. Let $k$ be a cyclotomic field containing $x_1,\dots,x_n$ and all the torsion points of $E (\mathbb Q^{ab})/\operatorname{tors}$. Let $N$ be the maximum order of a torsion point. Let $y$ be any point of $E (\mathbb Q^{ab})/\operatorname{tors}$ such that $m y \in E(k)$ for some natural number $m$. We will check that $N y \in E(k)$. It will follow that$ \oplus_{i=1}^n x_i \mathbb Q \cap E (\mathbb Q^{ab})/\operatorname{tors}$ is contained in $(1/N) E(k)$, which is finitely generated, so is finitely generated.

To check this, let $y’$ be any Galois conjugate of $y$ relative to $k$. Because $\mathbb Q^{ab}/\mathbb Q$ is Galois, $y’ \in E(\mathbb Q^{ab})$, so $y’-y \in E(\mathbb Q^{ab})$. Because the Galois conjugation is relative to $k$, $m y’=my $, so $m(y’-y) =0$ and $y’-y$ is torsion, hence it is $N$-torsion, so $Ny’ =Ny$. Because this is true for all $y’$, $Ny$ is Galois-invariant and thus is in $E(k)$.

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    $\begingroup$ More generally, Lawrence (Proc. Amer. Math. Soc.) has proved that a countable abelian group carrying a discrete norm is free. Since $E(\mathbb{Q}^{\mathrm{ab}})$ has the strong Bogomolov property on the canonical height, the same result follows. Similarly, say, $E(\mathbb{Q}^{\mathrm{tr}}) / \mathrm{tors}$ is the direct sum of countably many copies of $\mathbb{Z}$. There was a paper ( arxiv.org/pdf/1408.4915.pdf ) on this by Grizzard, Habegger and Pottmeyer. $\endgroup$ – Vesselin Dimitrov Jul 6 '18 at 20:29
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    $\begingroup$ @VesselinDimitrov Thanks for pointing that out. Specifically, the paper you link credits the resolution of the original question to Iwasawa. $\endgroup$ – Will Sawin Jul 6 '18 at 20:52
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Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer.

Amoroso and Dvornicich discovered (A lower bound on the height in abelian extensions, JNT 2000) that the absolute logarithmic height on $\mathbb{G}_m(\bar{\mathbb{Q}})$ is bounded below by the absolute positive constant $(\log{5}) /12$ on the subset $\mathbb{G}_m(\mathbb{Q}^{ab}) \setminus \mu_{\infty}$ of abelian points minus the torsion. A simplified proof of this result (with a weakened constant) is in chapter four of Bombieri and Gubler's Heights in Diophantine Geometry. The problem of finding the best possible constants (the infimum and the limit infimum of the canonical height on abelian non-torsion points) has remained unsolved.

Lawrence had proved in 1984 that a countable abelian group with a discrete norm is isomorphic to the direct sum $ \bigoplus \mathbb{Z}$ of copies of $\mathbb{Z}$. Since Amoroso-Dvornicich implies that the canonical height descends to a discrete norm on the abelian group $\mathbb{G}_m(\mathbb{Q}^{\mathrm{ab}}) / \mu_{\infty}$, this recovers Iwasawa's old result about the freeness of $(\mathbb{Q}^{\mathrm{ab}})^{\times} / \mathrm{tors}$.

The elliptic analog of the Amoroso-Dvornicich theorem was proved, following the original $\mathbb{G}_m$ blueprint, by Baker (Lower bounds for the canonical height on elliptic curves over abelian extensions, 2002) and Silverman (A lower bound for the canonical height on elliptic curves over abelian extensions, 2003). The higher dimensional case was done by them jointly. The proof of this result also includes Ribet's result you cited as a particular case.

In particular, since this implies again that the canonical height descends to a discrete norm on the abelian group $E(\mathbb{Q}^{\mathrm{ab}}) / \mathrm{tors}$, it follows again that the latter group is isomorphic to $\bigoplus \mathbb{Z}$.

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As for the torsion subgroup, Michael Chou has classified the possible subgroups that may occur as $E(\mathbb{Q}^{\text{ab}})_{\text{tors}}$ for an elliptic curve $E/\mathbb{Q}$. You can find a preprint here.

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