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In Siksek's notes The modular approach to Diophantine equations he uses the following result:

Let $p$ be an odd prime. For an elliptic curve $E$ over $\mathbb{Q}$ if $p|E(\mathbb{F}_{\ell})$ then for all but finitely many primes $\ell$ then from Chebotarev's Density Theorem we conclude that $E/\mathbb{Q}$ has a $p$ isogeny.

Since he uses Chebotarev's Density Theorem the proof certainly involves the Galois representation $\rho_{E,\ell}$ but I struggle to see how

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  • $\begingroup$ Rather use $\rho_{E,p}$. Then the Frobenius at the varying $\ell$ has something to do with the question if $p\mid E(\mathbb{F}_{\ell})$. Aim at showing that the image is in a Borel subgroup. $\endgroup$
    – Chris Wuthrich
    Commented Jun 18, 2021 at 16:40
  • $\begingroup$ @JoeSilverman I mean a $p$-isogeny over $\mathbb{Q}$ $\endgroup$
    – Μάρκος Καραμέρης
    Commented Jun 18, 2021 at 16:47
  • $\begingroup$ @ChrisWuthrich How would it help that the image is in a Borel subgroup? $\endgroup$
    – Μάρκος Καραμέρης
    Commented Jun 18, 2021 at 17:23
  • $\begingroup$ The image is in a Borel subgroup if and only if there is a subspace of $E[p]$ fixed by Galois, which is the kernel of an isogeny defined over that field. Your local condition says that the Frobenius at $\ell$ maps to an element of $\operatorname{GL}(E[p])$ with $1$ as an eigenvalue. - I think this should help you to complete this. Sorry for not spelling it all out but I believe it helps you more if you complete it. $\endgroup$
    – Chris Wuthrich
    Commented Jun 18, 2021 at 18:46

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