Timeline for Determining existence of a $p$-isogeny from $p|E(\mathbb{F}_{\ell})$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 18, 2021 at 18:46 | comment | added | Chris Wuthrich | 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. | |
| Jun 18, 2021 at 17:23 | comment | added | Μάρκος Καραμέρης | @ChrisWuthrich How would it help that the image is in a Borel subgroup? | |
| Jun 18, 2021 at 16:47 | comment | added | Μάρκος Καραμέρης | @JoeSilverman I mean a $p$-isogeny over $\mathbb{Q}$ | |
| Jun 18, 2021 at 16:47 | history | edited | Μάρκος Καραμέρης | CC BY-SA 4.0 |
added 11 characters in body
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| Jun 18, 2021 at 16:40 | comment | added | Chris Wuthrich | 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. | |
| Jun 18, 2021 at 16:25 | review | First posts | |||
| Jun 18, 2021 at 18:10 | |||||
| Jun 18, 2021 at 16:24 | history | asked | Μάρκος Καραμέρης | CC BY-SA 4.0 |