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In http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183555477, Ribet gives a proof of Serre's open image theorem for elliptic curves using results from Faltings' proof of the Mordell conjecture. His argument is also repeated at http://mathoverflow.net/questions/108169/modern-proof-of-serres-open-image-theorem.

Unfortunately, I don't understand how he uses Faltings' Theorem to arrive at the punchline.

  1. What does it mean for $\rho_{\ell, E}$ to be semisimple? I thought a semisimple representation was a direct sum of irreducibles, but Shafarevich's Theorem already guarantees that $V_{\ell}$ is irreducible.

  2. What is the significant of the fact that $\text{End}_{\mathfrak{g}_{\ell}}(V_{\ell}) \simeq \mathbb{Q}_{\ell}$ (in the case of a non-CM elliptic curve)?

To summarize, how do these two results rule out the case of a non-split Cartan?

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