Elliptic curve prime power isogenies

Question

Let $E/K$ be an elliptic curve, where $K$ is a number field. Let $p>2$ be a prime number.

Is there any bound on $n$ such that $E$ has a $K$-rational cyclic $p^n$-isogeny?

Can it have such isogeny for infinitely many $n$?

Answer 1

If $\phi\colon E\to E'$ is a rational cyclic $p^n$-isogeny with kernel $E[\phi]$, then $E[\phi]$ is a $G_K = \mathrm{Gal}(\overline K/K)$ stable subgroup of $E[p^n]$. It follows that $E[p^n]$ is reducible as $G_K$-module.

Assume that $E$ is not CM. In particular, by a theorem of Serre/Shafarevich, the Tate module $\varprojlim_n E[p^n]$ is irreducible as a $G_K$-module. It follows that $E[p^n]$ must be irreducible when $n$ is large enough. In particular, $E$ cannot have a rational $p^n$-isogeny for infinitely many $n$.

As for a uniform bound depending on $K$, one probably exists, although it might be hard to prove - hopefully someone more knowledgeable than me can answer!

Except when $K = \mathbb Q$, the analogous question asking for a uniform bound depending only on for the largest prime for which there exists a -isogeny is an open problem!