Timeline for Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2019 at 23:28 | vote | accept | AZMEH | ||
| Jul 5, 2019 at 18:46 | comment | added | Filippo Alberto Edoardo | Thanks, corrected. | |
| Jul 5, 2019 at 18:46 | history | edited | Filippo Alberto Edoardo | CC BY-SA 4.0 |
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| Jul 5, 2019 at 17:15 | comment | added | Will Sawin | This is much better. Indeed the key point is to use the main theorem of CM. In your equation in part 1, I don't think you mean to view $H^1(E(\mathbb C), \mathbb Q_\ell)$ as a $\mathbb Q_\ell$-module and then tensoring with $K_\ell$, as that produces a rank $2$ $K_\ell$-module. Instead I think you want to view it as a $K_\ell$-module directly by the action of $K$, and dualize there. This also affects the last line. In part 2, it doesn't make sense to tensor over $\mathbb Q_\ell$ with $K$ . - you want to tensor over $\mathbb Q$. | |
| Jul 5, 2019 at 16:59 | history | edited | Filippo Alberto Edoardo | CC BY-SA 4.0 |
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| Jul 3, 2019 at 0:22 | comment | added | AZMEH | @WillSawin, Thank you very much for pointing that out. | |
| Jul 2, 2019 at 15:48 | comment | added | Filippo Alberto Edoardo | @WillSawin You're certainly right, I will write something more reasonable as soon as I have time. Thank you. | |
| Jul 2, 2019 at 15:47 | history | edited | Filippo Alberto Edoardo | CC BY-SA 4.0 |
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| Jul 1, 2019 at 23:51 | comment | added | Will Sawin | The Tate module only splits as a sum of two modules when $E$ is CM, CM is crucial here. | |
| Jul 1, 2019 at 23:50 | comment | added | Will Sawin | This answer is wrong or something is very strange. $H^1 ( E_{\overline{K}}, \mathbb Q)$ has no interesting Galois action, so it can't be isomorphic to the dual of the Tate module. | |
| Jun 25, 2019 at 5:58 | history | edited | Filippo Alberto Edoardo | CC BY-SA 4.0 |
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| Jun 24, 2019 at 16:24 | comment | added | AZMEH | Thank you very much for your answer. | |
| Jun 24, 2019 at 16:24 | vote | accept | AZMEH | ||
| Jul 3, 2019 at 0:21 | |||||
| Jun 24, 2019 at 16:21 | history | answered | Filippo Alberto Edoardo | CC BY-SA 4.0 |