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Let $K$ be a fixed number field and $E$ be any elliptic curve over $K$. When we adjoin to $K$ the $p$-torsion points $E[p]$, we obtain an extension whose Galois group can be embedded in $GL(2, \mathbb{F}_p)$.

Under what conditions (on $E$ and $p$) can we get that $K(E[p])/K$ is finite $p$-extension?

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    $\begingroup$ If and only if ($E(K)[p] = \mathbb{Z}/p\mathbb{Z}$ and $\mu[p]\subset K$) or $K(E[p])=K$. What else would you like? $\endgroup$ – Chris Wuthrich Jan 13 at 18:26
  • $\begingroup$ Serre's Open Image theorem will imply that if $E$ is a non CM curve, given $K$ there can be at most only finitely many primes for which this can happen? For $E$ a CM curve? $\endgroup$ – debanjana Jan 14 at 4:57
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    $\begingroup$ It immediately follows from Chris's comment and Merel's theorem that given K, there can only be finitely many such $p$, even as $E$ varies. $\endgroup$ – Alex B. Jan 14 at 7:18

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