Timeline for Points of infinite level modular curve
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 9, 2018 at 6:45 | comment | added | rime | Yes, you are right! Here the tower is an inverse limit given essentially by modular curves which, at each step, trivialize the torsion, and the transition morphisms are not given by Frobenius. So, do yo u think it's not possible to get a kind of moduli interpretation for the anticanonical tower? | |
| Jul 9, 2018 at 5:22 | comment | added | naf | I looked at the lemma in Scholze that you mentioned. The space he is considering there, i.e. $\mathcal{X}_{\Gamma(p^{\infty})}$, is not the inverse limit of the anticanonical tower (which is an open in the space denoted by $\mathcal{X}_{\Gamma_0(p^{\infty})}$ in the paper). | |
| Jul 8, 2018 at 6:53 | comment | added | rime | Btw, I agree this is not a moduli interpretation since, I think, it only holds for geometric points, but I still want to understand why is it true at this level and moreover, is there a moduli interpretation for this perfectoid space? I think no. | |
| Jul 8, 2018 at 6:51 | comment | added | rime | Yes! Look at the proof of Lemma III.3.6. There, after a reduction argument in which he passes to a rank 1 point, he says that such a point corresponds to an abelian variety with a trivialization of the Tate module. You may also have a look to Ana Caraiani's notes for AWS2017, they are titled "Lecture notes on perfectoid Shimura Varieties". In these notes, at page 41, she declares that geometric points of the infinite level modular curve have such a moduli interpretation. But really I don't see the connection. Thanks! | |
| Jul 8, 2018 at 6:29 | comment | added | naf | When one takes the inverse limit of the anticanonical tower one does get a perfectoid space but this is not the moduli of elliptic curves with a trivialisation of the Tate module. So I assume you are misinterpreting what Scholze says: can you point to the precise statement that you are referring to? | |
| Jul 7, 2018 at 14:35 | answer | added | skd | timeline score: 4 | |
| Jul 7, 2018 at 13:33 | history | asked | rime | CC BY-SA 4.0 |