Timeline for Canonical models of Shimura varieties for GL2
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jun 14, 2022 at 6:58 | history | edited | David Loeffler | CC BY-SA 4.0 |
Fixed a small typo (thanks KennyLau for spotting this)
|
| S Jun 14, 2022 at 6:58 | history | suggested | Kenny Lau | CC BY-SA 4.0 |
I fixed what I assumed to be a small typo.
|
| Jun 14, 2022 at 2:25 | review | Suggested edits | |||
| S Jun 14, 2022 at 6:58 | |||||
| Apr 24, 2017 at 13:22 | answer | added | Olivier | timeline score: 1 | |
| Apr 24, 2017 at 11:37 | comment | added | François Brunault | @DavidLoeffler The Galois action on the cusps is discussed by Schappacher-Scholl in "Beilinson's theorem on modular curves" section 3. Unfortunately I'm not familiar enough with Deligne's work, so I cannot be certain that they use the Deligne canonical model. | |
| Apr 24, 2017 at 10:50 | comment | added | David Loeffler | @FrançoisBrunault I know how Galois acts on the cusps of my two models of $Y_1(N)$. Do you have a reference for the action of Galois on the cusps of the Deligne canonical model? | |
| Apr 24, 2017 at 10:09 | comment | added | François Brunault | Would the action of Galois on the cusps be sufficient to distinguish these two models? The set of cusps (seen as complex points) is a certain double quotient of $\mathrm{GL}_2(\hat{\mathbf{Z}})$, and the group $\mathrm{Aut}(\mathbf{C}/\mathbf{Q})$ acts by multiplying on the left by $\begin{pmatrix} \chi & 0 \\ 0 & 1 \end{pmatrix}$ where $\chi$ is the cyclotomic character. | |
| Apr 24, 2017 at 6:19 | comment | added | David Loeffler | Even if you use right actions throughout, there are still two conventions, differing by the automorphism $g \mapsto (\det g)^{-1} g$ of $GL_2(\hat{\mathbf{Z}})$. | |
| Apr 24, 2017 at 1:11 | comment | added | Jeremy Rouse | I think this is a matter of whether you consider the action of $GL_2$ to be on the left (matrix times column vector) or the right (row vector times matrix). The convention that Shimura takes is to have the matrices act on the right (which is maybe a little bit unusual). With this convention the 2nd would be $Y_{1}(N)$ and the first would be the other one. David Zureick-Brown and I discuss this issue a bit in Section 2 of our preprint here. | |
| Apr 23, 2017 at 21:08 | comment | added | Keerthi Madapusi | Seems to me it's a question of whether you're asking for trivializations of the homology or cohomology of the elliptic curve. At least, by Deligne's conventions in his Bourbaki article (which I'm sure has consistent signs), it's homology that's being trivialized, so I think what I said is the correct convention. I must confess though that I'm not completely convinced yet. | |
| Apr 23, 2017 at 20:33 | comment | added | David Loeffler | @KeerthiMadapusiPera Both are quotients of Y(N), but to get something out of this one has to know how to identify the canonical model of Y(N) with the moduli space of elliptic curves with full level N structure. There's more than one possible convention for how to do this. (Kato's article in Asterisque 295 gives a model for Y(N) for which the quotient by the first subgroup classifies points of order N; but I'm pretty sure Kato's model is not the canonical model, because the Galois action on the connected components differs by a sign from the one on p109 of Milne's Shimura Varieties notes.) | |
| Apr 23, 2017 at 20:22 | comment | added | Keerthi Madapusi | Both are quotients of the (disconnected) curve with full level $N$, which parameterizes elliptic curves $E$ with basis $(e_1,e_2)$ for $E[N]$. Giving a point of the quotient by the first compact open amounts to saying that Galois acts on $e_1$ by a (non-trivial) character, while the second amounts to saying that Galois fixes $e_1$. So it appears to me that the first gives you the $\mu_N$ model while the second gives the usual one. | |
| Apr 23, 2017 at 19:52 | history | asked | David Loeffler | CC BY-SA 3.0 |