Timeline for Supersingular curves over $\mathbb{F}_q$ and the splitting of $p$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 27, 2021 at 15:07 | comment | added | Will Sawin | @Rdrr One can probably avoid them by being more careful: Check that the endomorphism algebra mod $\ell$ acts faithfully on the $\ell$-torsion and therefore is a matrix algebra over $\mathbb Z/\ell$, and then work out enough of the theory of quaternion algebras where splitting means this mod $\ell$ phenomenon to run the argument. | |
| Jul 26, 2021 at 17:01 | comment | added | Rdrr | Limits in the categorical sense; the limits involved in the definitions of $\mathbb{Q}_p = \lim \mathbb{Z}/p^n$ and $T_p(E)= \lim E[p^n]$. | |
| Jul 26, 2021 at 16:57 | comment | added | Will Sawin | @Rdrr What do you mean by "limits" in this context? | |
| Jul 26, 2021 at 16:28 | comment | added | Rdrr | @WillSawin This is essentially the proof in Lang's Elliptic functions. I was hoping for a proof without limits, but one may not exists. This seems to be a good way to link the characteristic zero info (the splitting of $p$) with the mod p information (the p-torsion). | |
| Jul 24, 2021 at 15:11 | comment | added | Yuri Zarhin | Your suggestion works just fine. I may only add that since $E$ is a simple abelian variety, its endomorphism algebra has no zero divisors and therefore cannot be isomorphic to the matrix algebra of size 2 over the rationals. | |
| Jul 24, 2021 at 14:54 | comment | added | Will Sawin | How do you check that the endomorphism algebra in the supersingular case is a division algebra over $\mathbb Q_p$? I sketched one way in the comments, but it's not (to me) obvious. | |
| Jul 24, 2021 at 14:41 | history | edited | Yuri Zarhin | CC BY-SA 4.0 |
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| Jul 24, 2021 at 14:16 | history | answered | Yuri Zarhin | CC BY-SA 4.0 |