# Mod $p$-representation ($p$-torsion points ) of elliptic curve (over number fields) with CM can be irreducible?

Let $$E/K$$ be an elliptic curve with complex multiplication and let $$E[p]$$ be the group of $$p$$-torsion points of $$E$$ where $$K$$ is a number field and $$p$$ is a prime number.

My question is: as a $$(\mathbb{Z}/p\mathbb{Z})[G_K]$$-representation, does there exist a case such that $$E[p]$$ is irreducible? ($$G_K$$ denotes the absolute Galois group of $$K$$.)

remark:

1. When $$E/K$$ has no CM, Serre's theorem implies $$E[p]$$ is irreducible for almost all $$p$$.

2. When $$E/K$$ has CM over $$K$$, $$E[p]$$ can never be absolutely irreducible, at least for the case that $$End_K(E)\cong$$ full ring of integers of an imaginary quadratic field.

## 1 Answer

Certainly. It will (typically) be irreducible for primes where the curve has supersingular reduction. Indeed, for such primes, the p-torsion as a module over the endomorphism ring $$O$$ is isomorphic to $$O/p \simeq \mathbf{F}_{p^2}$$, and the image will (for all but finitely many such primes) will be all of $$\mathbf{F}^*_{p^2}$$ thought of as a subgroup of $$\mathrm{GL}_2(\mathbf{F}_{p})$$ via the isomorphism of groups $$\mathbf{F}_{p^2} \simeq (\mathbf{F}_{p})^2$$ (The image will be the so-called “non-split Cartan). This is certainly always irreducible - for example, reducible subgroups have order dividing the order $$p(p-1)^2$$ of the Borel.

An alternative way to think about it: reducibility for almost all $$p$$ implies the existence of $$p$$ isogenies for almost all $$p$$. Since the isogeny class of an elliptic curve is finite, this implies that for any infinite set of primes $$S$$ where p-isogenies exist there will be an endomoprhism of degree $$pq$$ with distinct primes $$p$$ and $$q$$ in $$S$$. (By pigeonhole principle the two isogenies are both from E to the same E’ then take one followed by the dual isogeny of the other.) But their only exist endomorphisms of E of orders which are norms in $$O$$ which forces $$p$$ and $$q$$ to split in the quadratic field.