# Reference request for equivalence of reducibility of $\ell$-adic Galois representation and isogeny of degree $\ell$ on elliptic curve

Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$. For each prime $$\ell$$, the action of $$\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$$ on $$E[\ell]$$ (the group of $$\ell$$-division points of $$E$$) defines a representation $$\rho=\rho_\ell:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \mathrm{GL}(2,\mathbb{F}_\ell).$$

Where can I read the proof of:

$$\rho$$ is reducible if and only if $$E$$ admits an isogeny of degree $$\ell$$

• References are given in the book "Eisenstein Series and Automorphic Representations" by Fleig et al., page $448$. Commented May 5, 2020 at 12:44
• I have improved your title - titles should be more specific and informative than "what book for this theorem", which could describe just about anything. Please check to see that this new title accurately describes your question. Commented May 5, 2020 at 20:43
• @KReiser Thank you Commented May 5, 2020 at 21:33
• I don't know of a reference, but isn't reducibility of $\rho$ equivalent to $E$ having a cyclic subgroup $C$ of order $\ell$ that is defined over $\Bbb{Q}$ (= stable under the action of the Galois group)? The related isogeny being just $E\to E/C$? I'm afraid I'm not an expert on this, so may be I'm missing something major? Commented May 6, 2020 at 5:22