Irreducibility of residual Galois representations attached to an elliptic curve

Question

Let $E$ be a given elliptic curve over a number field $F$. For each prime $p$, one has the Galois representation $\mathrm{Gal}(\bar{F}/F)\to GL(E[p])$ where $\bar{F}$ is a fixed algebraic closure of $F$. Is $E[p]$ irreducible for infinitely many prime $p$? I suspect this might be well-known and elementary in the theory of Galois representations, but I failed find any literature explain this.

Answer 1

As Will Sawin points out, the following is mostly an answer to the question "Is it true that the set of primes such that $\rho_p$ is reducible is finite?" which is related but strictly stronger than the question actually asked.

That question admits a negative answer, but a positive answer if $E$ has no complex multiplication. If $E$ has no complex multiplication (that is to say if $\operatorname{End}_F(E)=\mathbb Z$), a good deal more is actually true.

Theorem (Serre, 1972): Assume $E/F$ is without complex multiplication. Then the Galois representation $\rho_p:\operatorname{Gal}(\bar{F}/F)\longrightarrow\operatorname{Aut}(E[p])$ is surjective for all $p$ except a finite number.

The fact that all $p$ except possibly a finite number are such that $\rho_p$ is irreducible, not to mention the fact that there are infinitely many $p$ such that $\rho_p$ is irreducible, is significantly easier as it follows quite rapidly from the theorem of Shafarevich on the finiteness of the set of isomorphism classes of $F$-isogenous elliptic curves (see Abelian $\ell$-adic Representations and Elliptic Curves Jean-Pierre Serre, 1968).

However, if $E$ has complex multiplication by a quadratic imaginary subfield $K$ of $F$, consider $p$ a prime splitting as $\pi\bar{\pi}$ in $K$. Then $E[\pi]$ is a set of $F$-rational points of order $p$. In that case, there is a thus an infinite number of prime such that $\rho_p$ is reducible. As Will Sawin points out, even in that case, the representation $\rho_p$ is nevertheless irreducible for infinitely many $p$. Indeed, for $p$ inert in $K$, $\rho_p$ is a representation in the unit group of a free module of rank 1 over the localization at $p$ of the order $\operatorname{End}_F(E)$ and the fact that such a representation is surjective for almost all such $p$ follows.

On that topic, the articles of Agnès David and Nicolas Billerey are recommended.