Timeline for Equivalence of algebraic and topological monodromy representations?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 12, 2016 at 21:13 | comment | added | Aaron Landesman | @abx That seems to answer it since all the representations are given by the action of $\pi_1$ on $H^1$, and then one can pass between the topological $H^1$ and etale $H^1$. | |
| Jul 12, 2016 at 6:54 | comment | added | abx | For a smooth curve $C$, there are canonical isomorphisms $H^1(C,\mathbb{Z})\otimes \mathbb{Z}_{\ell}\rightarrow H^1_{et}(C,\mathbb{Z}_{\ell})$ and $H^1_{et}(JC,\mathbb{Z}_{\ell})\rightarrow H^1_{et}(C,\mathbb{Z}_{\ell})$. | |
| Jul 12, 2016 at 3:39 | comment | added | Venkataramana | @Qiaochu Yuan: the action on $l$ division points is exactly the action on the pro-l completion of the fundamental group of the abelian variety in question. | |
| Jul 12, 2016 at 3:37 | comment | added | Qiaochu Yuan | @Venkataramana: that doesn't seem like the hard part of the question. Of course if we profinitely complete the first representation we get something that looks like the second representation. But why is it in fact the second representation? | |
| Jul 12, 2016 at 3:06 | comment | added | Venkataramana | I think this is just an elementary statement that if a group $\Gamma$ acts on $\mathbb {Z}^{2g}$, then the profinite completion $\widehat{\Gamma}$ acts on the profinite completion $\widehat{\mathbb{Z}}^{2g}$. | |
| Jul 12, 2016 at 1:14 | history | asked | Aaron Landesman | CC BY-SA 3.0 |