Timeline for Tower of moduli spaces in Scholze's theory
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 18, 2019 at 21:55 | comment | added | Ben Heuer | @Zariski93 Yes, I think that description is correct. Here $\eta_n$ now has to be like in D\'efinition 3.1 of Le Halo spectral, with $\alpha=T$. | |
| Apr 18, 2019 at 13:11 | comment | added | Zariski93 | At the end of the day, I think I can say that an $\text{Spf}(R)$-point of $\mathfrak{X}_{\infty,I}$ is a quadruple $(E,\alpha,D_n,\eta_n)$, where $E/R$ is an elliptic curve, $\alpha$ is a level $N$-structure over $E$, $D_n$ is a $p$-divisible group generically anticanonical with all the compatibility you mentioned above. And this interpretation should be suggested by the fact that the equivalence hold at finite level, and the Frobenius morphism under this equivalence, essentially goes to a forgetful functor. What do you think? | |
| Apr 18, 2019 at 13:07 | comment | added | Zariski93 | with level, and $\eta$, so in this case I cannot say the $T$ becomes a $p^\epsilon$. But maybe I'm saying something stupid. What do you think? Btw, I really thank you for such long comments! | |
| Apr 18, 2019 at 13:05 | comment | added | Zariski93 | Dear Ben, your comments are really enlightening, and if I could give you more than one thumb up, I surely would! Btw, your last comment clarifies completely the situation when you consider points with values in a field I guess. In this case, you are really choosing a point in the weight space, hence a character. But what about more general points? I mean, suppose I consider a $\mathbb{Z}_{p^2}[[T]]\langle\frac{p}{T}\rangle$ point of $\mathfrak{X}_{r,I}$ which gives the universal character over the weight space. This is not a physical point, but it should define a family of elliptic curves | |
| Apr 18, 2019 at 11:51 | comment | added | Ben Heuer | @Zariski93 Regarding your second comment: Yes, you're right, AIP use T to define the condition on the Hasse invariant. But when you specialise at a point $w$ of weight space, this sends $T\to p^{\epsilon}$ for some $\epsilon$ depending on $w$. This is related to the fact that so far we haven't talked about the relation between the $r$ of $\mathfrak X_{r}$ in Le Halo Spectral and the $\epsilon$ in the torsion paper: The relation depends on the $p$-adic weight. | |
| Apr 18, 2019 at 11:49 | comment | added | Ben Heuer | Once the $\eta$'s are sorted, one can show the alternative moduli interpretation of $\mathfrak X^{\ast}(p^{-n}\epsilon)$ by comparing the two moduli functors: The transformations sending $(E,\alpha,\eta_n)\to (E/C_n,\alpha/C_n,\eta_n,E[p^n]/C_n)$ and $(E,\alpha,\eta_n,D_n)\to (E/D_n,\alpha,\eta_n)$ should define an equivalence. | |
| Apr 18, 2019 at 11:49 | comment | added | Ben Heuer | If we also worry about $\eta$, I think it might not be possible to reconstruct the $\eta$ of $x=(E,\alpha,\eta)$ from $F(x)=(E',\alpha',\eta')$ and $D_n$, at least not over the locus of supersingular reduction. So instead, in order to say what "$\mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$" is (i.e. the alternative moduli interpretation) one might need to talk about tuples of the form $(E,\alpha,\eta,D_n)$ where $\eta$ is defined in terms of $E/D_n$. I've edited my post to reflect this. | |
| Apr 18, 2019 at 11:47 | comment | added | Ben Heuer | @A.Walker: In answer to your question, I should perhaps say that in all of the above I was focusing on $(E,\alpha,\eta)$ etc and I'll admit I swept the $\eta$'s under the carpet. Sorry -- the post was so long already :). The $\eta$'s make things slightly more complicated, also because they are defined slightly differently for Scholze and Andreatta--Iovita--Pilloni (one uses $\eta Ha^{p^{r+1}}=p$, one $\eta Ha=p^{\epsilon}$, this is a normalisation issue), but to describe the moduli functor, we should of course include them. I didn't mean to imply that you can reconstruct $\eta$ from $D_n$. | |
| Apr 18, 2019 at 11:46 | comment | added | Ben Heuer | Thanks! @Zarisiki93 Regarding your first comment, the things is that a priori we didn't define a formal model of $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$, but of $\mathcal X^{\ast}(p^{-n}\epsilon)$. The Atkin-Lehner isomorphism is then used to show that the formal model of the latter is also a formal model of the former. Re moduli interpretations of this: | |
| Apr 18, 2019 at 11:18 | history | edited | Ben Heuer | CC BY-SA 4.0 |
clarified a few points about the etas which I had previously swept under the carpet.
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| Apr 18, 2019 at 7:03 | comment | added | Zariski93 | an elliptic curve whose Hasse invariant satisfies the Scholze condition. The point is that this open is finer, maybe, as the relation $|\widetilde{\text{Ha}}|^{p^{r+1}}\geq |T|$ is more strict than $|\widetilde{\text{Ha}}|^{p^{r+1}}\geq p$. | |
| Apr 18, 2019 at 7:01 | comment | added | Zariski93 | Oh, maybe I have an idea. In AIP, considering the adic generic fibers of all the spaces involved, you see that for $I=[p,\infty]$, again I think you all understand the notation, we are taking the open inside the adic weight space where $|p|\leq |T^p|$ and the blowing up $\mathcal{X}_{r,I}$ of the modular curve gives the open where $|\widetilde{\text{Ha}}|^{p^{r+1}}\geq |T|$. This last object, by the condition on the valuations of $p$ and $T$, should be an open inside $\mathcal{X}\left(\frac{1}{p^{r+1}}\right)$, so in particular every point of it gives | |
| Apr 18, 2019 at 6:15 | comment | added | A. Walker | Yes, I also don't understand how the moduli description of the adic generic fiber produce such a description at integral level. By the way, really a great answer! Please, clarify me this last fact! | |
| Apr 18, 2019 at 6:12 | comment | added | Zariski93 | Moreover, the definition of Andreatta, Iovita and Pilloni, if I understand it well, provides a global section $\eta$ which makes Hasse invariant dividing the variable $T$. It seems you are really familiar with that paper, so I guess you understand the notation. This seems to me that we cannot first blow up the formal curve and then base change to weight space. Does this fact give any problem in your description? | |
| Apr 18, 2019 at 6:08 | comment | added | Zariski93 | WOW!! I think this is really a great answer! According to me, I only have a couple of questions. First, it seems that this Atkin Lehner isomorphism is only defined rationally. In fact, you always say that the adic generic fibers of these formal schemes are related via Atkin Lehner. Once one assume this iso holds, how can we relate it to the integral model? Maybe one can say that the two integral moduli problems $\mathfrak{X}_{\Gamma_0(p^n)}(\epsilon)$ and $\mathfrak{X}(p^{-n}\epsilon)$ are normalization of the same formal scheme with isomorphic generic fibers? | |
| Apr 18, 2019 at 1:30 | review | First posts | |||
| Apr 18, 2019 at 2:14 | |||||
| Apr 18, 2019 at 1:26 | history | answered | Ben Heuer | CC BY-SA 4.0 |