Timeline for Reduction "modulo $p$" of $\mathfrak{p}$-torsion points of CM elliptic curves
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 10, 2016 at 7:50 | comment | added | Hugo Chapdelaine | There should still exist some kind of uniform proof involving the theory of formal groups... | |
| Jun 10, 2016 at 7:49 | comment | added | Hugo Chapdelaine | In order to treat the case where $p O_K=\mathfrak{p}\bar{\mathfrak{p}}$ one may use the following ad-hoc observation: The map $E----> E/E[\mathfrak{p}] \pmod{\wp}$ is the Frobenius post composed with an isomorphism with kernel $E[\mathfrak{p}]$. S2 follows from that. | |
| Jun 10, 2016 at 7:13 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| Jun 9, 2016 at 22:08 | comment | added | Hugo Chapdelaine | ....Therefore, $\tilde{E}(O_L/\wp)$ has size $p^n+1$ which is coprime to $p$. So essentially it remains to treat the special case where $p$ splits in $K$. | |
| Jun 9, 2016 at 22:02 | comment | added | Hugo Chapdelaine | Now that I'm thinking about it, if $\mathfrak{p}\cap \mathbf{Z}=p\mathbf{Z}$ is inert in $K$ then, for $p\geq 5$, the eigenvalues of $Fr_{\mathfrak{p}}$ are associated algebraic numbers which must differ by a sign, and therefore the coefficien $Tr(Fr_{\mathfrak{p}})=a_{\mathfrak{p}}=0$. Therefore, $\tilde{E(O_L/\wp)$ has size p^2+1 which is coprime to $p$. | |
| Jun 9, 2016 at 16:45 | history | edited | Hugo Chapdelaine |
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| Jun 9, 2016 at 10:27 | comment | added | Hugo Chapdelaine | If E is defined over $Q$ and has CM by $K$, then E will be supersingular at p if and only if p is inert in K. Does it help ? | |
| Jun 9, 2016 at 10:22 | history | edited | Hugo Chapdelaine | CC BY-SA 3.0 |
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| Jun 9, 2016 at 10:17 | comment | added | Jeff Yelton | Maybe I'm missing something, but it seems to me that $\mathrm{ker}(\pi) \supseteq E[\mathfrak{p}]$ implies that $E$ has supersingular reduction at the prime lying above $\mathfrak{p}$, which is not necessarily the case. | |
| Jun 9, 2016 at 9:51 | history | asked | Hugo Chapdelaine | CC BY-SA 3.0 |