Timeline for About isogenies of abelian varieties
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2010 at 14:53 | vote | accept | zcqc | ||
| Nov 3, 2010 at 15:12 | answer | added | VA. | timeline score: 9 | |
| Nov 1, 2010 at 18:11 | history | edited | Francesco Polizzi |
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| Nov 1, 2010 at 16:02 | answer | added | Francesco Polizzi | timeline score: 5 | |
| Nov 1, 2010 at 15:31 | comment | added | BCnrd | [Above assumed $G \ne 1$; i.e., $\phi$ not principal.] "Hard" step in justifying list of $k$-simple $G_0$ over general $k$ is local-local case when char$(k) = p > 0$. Enough to show local-local $G_0$ of $p$-power order with vanishing Frob. and Ver. are precisely $\alpha_p^N$. Pf #1: $F = 0$ makes $G_0$ "equivalent" to its commutative $p$-Lie algebra, on which $p$-operator is linearization of ${\rm{Lie}}(V_G)$. Pf #2: Dieudonne theory gives result over perfect closure, so want $\alpha_p^N$ over $k$ has no nontriv. "fppf forms". Aut functor is ${\rm{GL}}_N$, so use (non-comm.) fppf Hilbert 90. | |
| Nov 1, 2010 at 15:12 | comment | added | BCnrd | Here is "why" proof at end of Mumford's book works only over sep. closed $k$ (better than alg. closed). Over any field $k$ always exists polarization $\phi$; seek to "modify" it to be principal. The finite $k$-gp scheme $G = \ker \phi$ contains a $k$-simple $G_0$, and wish to induct on order of $G$ using list of $k$-simple $G_0$. Over any $k$, list is: simple etale $E$ (irred. finite Galois modules), and for ${\rm{char}}(k) = p > 0$ also $\alpha_p$ and dual $E'$ of simple etale $E$ of $p$-power order. For sep. closed $k$, can say $E = \mathbf{Z}/(\ell)$ for prime $\ell$. That's "why". | |
| Nov 1, 2010 at 14:38 | history | asked | zcqc | CC BY-SA 2.5 |