# Irreducibility of residual Galois representations attached to an elliptic curve

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.

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.

• The question asks whether it is irreducible for infinitely many p, which is true for CM as long as you don't demand it be absolutely irreducible. Nov 3, 2017 at 21:25
• Oh yeah, you're right. I should edit. Nov 3, 2017 at 21:26