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I'm currently studying Gross' paper about modular elliptic curves. Lemma 4.3 is quite difficult for me to understand. It states that an elliptic $E$ has no $p$-torsion rational over $K_n$, the ring class field of order $n$. The only thing I seem to know by now is that there is a well defined representation $\rho:\text{Gal}(\mathbb{Q}(E_p)/\mathbb{Q})\to \text{GL}_2(\mathbb{Z}/p\mathbb{Z})$.

Any insights on this, or references recommended? Thanks in advance.

defacto
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  • There’s a proof given. What part of it are you stuck at/unclear about? It would also be more polite to define the terms you are using rather than make people go to a link. – Arkady Aug 23 '22 at 00:16
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    Does this answer help at all? The point is that $\rho$ contains the trivial representation if and only if $E$ has $p$-torsion. Or is your confusion at another part of the proof? – Mathmo123 Aug 23 '22 at 06:29
  • Thanks for the answers. Gross splits the problem in two cases (due to Mazur), $E_p(K_n)=\mathbb{Z}/p\mathbb{Z}$ and $E_p(K_n)=(\mathbb{Z}/p\mathbb{Z})^2$. Supposing $E$ has $p$-torsion, I understand that $\rho$ contains the trivial representation. I struggle with the assertions that follow and why they contradict the hypothesis. – defacto Aug 23 '22 at 11:43

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