Timeline for $n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 23, 2023 at 12:26 | answer | added | Olivier | timeline score: 2 | |
| Aug 23, 2023 at 11:42 | vote | accept | Stanley Yao Xiao | ||
| Aug 22, 2023 at 23:54 | answer | added | David Zureick-Brown | timeline score: 10 | |
| Aug 22, 2023 at 23:50 | comment | added | R. van Dobben de Bruyn | @YuriZarhin ah right, I was indeed thinking about $n=2$. But when there are trivial counterexamples, there are almost certainly nontrivial ones :) | |
| Aug 22, 2023 at 21:48 | history | edited | Stanley Yao Xiao | CC BY-SA 4.0 |
edited body
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| Aug 22, 2023 at 21:47 | comment | added | Stanley Yao Xiao | Edited in view of @YuriZarhin's comment | |
| Aug 22, 2023 at 16:52 | comment | added | Yuri Zarhin | If $n>2$ then $\mathbf{Q}$ does not contain a primitive $n$th root of unity. In light of the nondegeneracy and Galois equivariance of the Weil pairing, not all points of $E[n]$ are defined over $\mathbf{Q}$. Hence, the ``Dumb example" does not work. | |
| Aug 22, 2023 at 14:28 | comment | added | Chris Wuthrich | Another counter example is a prime $n=p$ of split multiplicative reduction such that $p$ divides the Tamagawa number $c_p$. There is a lot of literature on these questions and googling will reveal some references. | |
| Aug 22, 2023 at 13:47 | comment | added | R. van Dobben de Bruyn | Dumb example: what if $K_n = \mathbf Q$, i.e. all points of $E[n]$ are defined over $\mathbf Q$? | |
| Aug 22, 2023 at 12:09 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |