TITLE: Rational Isogenies of Prime Degree
QUESTION [10 upvotes]: Dear MO Community, 
Let $N$ be a prime, and let $X_0(N)$ be the classical modular curve over $\mathbb{Q}$. We know ([1]) that, if there are noncuspidal points in $X_0(N)(\mathbb{Q})$, then $N \in$ {${ \mbox{primes } \leq 19} $} $ \cup $ {37,43,67,163}. 
The basic question of this post is:

Are there similar lists of primes when $\mathbb{Q}$ is replaced by a number field $K$? That is, if we fix a general number field $K$, can we determine the primes $N$ for which $X_0(N)(K)$ has noncuspidal points?

Perhaps in this generality the question is hard, so suppose we restrict from now on to imaginary quadratic $K$. Then [1] gives an approach to the question, but with the following snags:

We need to construct an "optimal" quotient of $J_0(N)$, call it $A$, such that $A(K)$ has Mordell-Weil rank 0;
We must restrict ourselves to primes $N$ which are inert in $K$. 

Actually, I don't think (1) is a big problem; provided $N \> 48h(K)^3 + 1$, we can take $A = \widetilde{J}$, the Eisenstein quotient. [Speculation : if we took the "winding quotient" instead, maybe we can lower that bound...]
When $N$ splits, then we can construct "CM points" on $X_0(N)(\mathbb{C})$, but usually they will not be defined over $K$, and even if they are, there will only be a handful of them. 

Question: For which $N$ that splits in $K$ do we have points on $X_0(N)(K)$ that are neither cuspidal nor CM? Is there a way to systematically find these points?

By "systematically", I guess I mean something like the 'isogeny character' approach of [1], where the hunt for the $N$s comes down to when certain congruences are satisfied mod $N$. 
Many thanks. 
[1]: Mazur, B. "Rational isogenies of prime degree", Inventiones Mathematicae, 1978

REPLY [9 votes]: You can see also the recent paper of A. David on ArXiv: Critères d'irréductibilité pour une courbe elliptique semi stable sur un corps de nombres. A. David gives a uniform bound for irreducibility of mod p Galois representations of semi stable elliptic curves over number field K provided some condition (namely that there is not any CM elliptic curve with everywhere good reduction on K). 
But the general question of isogenies over a number field is hard. The "Mazur-Kamienny" style method to study this should be to look at the morphism from the d-th symmetric power of X_0(p) to an optimal quotient of J_0(p). (See also Kamienny works for torsion points on number fields).