# finite $p$ extensions on adjoining $p$-torsion points of an elliptic curve

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?

• 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? – Chris Wuthrich Jan 13 at 18:26
• 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? – debanjana Jan 14 at 4:57
• 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. – Alex B. Jan 14 at 7:18