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cll
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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.

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.

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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cll
cll
  • 2.3k
  • 10
  • 30

Points of elliptic curves over cyclotomic extensions

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.