Timeline for Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Feb 18, 2011 at 1:37 | comment | added | Felipe Voloch | I thought Bogomolov proved something about homotheties and not a full open image theorem. I'll have a look, thanks. | |
| Feb 17, 2011 at 9:42 | comment | added | Olivier | Dear Felipe, I was referring to Sur l'algébricité des représentations l-adiques (C.R.A.S 290 F.Bogomolov). There are several results of Serre from the 80s, mostly found in letters to other people, which also cover these kind of results. | |
| Feb 17, 2011 at 0:24 | comment | added | Felipe Voloch | @Olivier: What's the result of Bogomolov you've alluded to? I'd be interested to see. | |
| Feb 16, 2011 at 22:43 | vote | accept | Felipe Voloch | ||
| Feb 16, 2011 at 22:30 | answer | added | Pete L. Clark | timeline score: 15 | |
| Feb 16, 2011 at 22:11 | comment | added | Olivier |
This seems very true. Let $L_{\mathfrak p}$ be a finite extension of $\mathbb Q_{p}$ containing all the traces of the Frobenius morphisms acting on $T_{p}A$. Assume that $A$ has no CM. Then there are no quadratic character $\eta$ such that $V=T_{p}A\otimes L_{\mathfrak p}$ is isomorphic to $V\otimes\eta$. This implies that $\operatorname{End}_{L_{\mathfrak p}[G_{K}]}$ is equal to $L_{\mathfrak p}$ by Frobenius reciprocity and this in turn implies that $V$ is irreducible. Does that sound good to you or am I missing something? Didn't Bogomolov proved the open image theorem you want anyway?
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| Feb 16, 2011 at 21:30 | history | asked | Felipe Voloch | CC BY-SA 2.5 |