Timeline for Tower of moduli spaces in Scholze's theory
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Apr 17, 2019 at 22:39 | comment | added | A. Walker | Yes, I also agree that there should be some problems with simply base changing the universal elliptic curve to the infinite level. What about the possibility of starting with a point $f:\text{Spf}(R)\rightarrow\mathfrak{X}_{r,I}$ and then base change to infinity? It gives a point $\text{Spf}(S)\rightarrow\mathfrak{X_{\infty}}$ which should give an elliptic curve over $S$ by base change of the one over $R$. I guess this curve has all canonical subgroups. | |
| Apr 17, 2019 at 13:40 | comment | added | Zariski93 | ...But then, how to show that an elliptic curve with a subgroup of order $p$ reconstructs also a section $\eta$ which satisfies the equation for $\mathfrak{X}_{r,I}$? | |
| Apr 17, 2019 at 13:40 | comment | added | Zariski93 | But why should this be true? It seems that each level of the anticanonical tower $\mathfrak{X}_{r,I}$ should be isomorphic to $\mathfrak{X}_{\Gamma(p^r)}$, but really I cannot understand why should it be true. We surely have a map sending a couple $(E,\eta)$ as above to the couple $(E,H_r)$, where $H_r$ is the canonical subgroup, and we can clearly use this map to split $\mathfrak{X}_{\Gamma_0(p^r)}$ into a canonical and an anticanonical component. | |
| Apr 17, 2019 at 13:37 | comment | added | Zariski93 | Now, these objects are connected by a lift of Frobenius, which allows to construct towers like $\mathfrak{X}_{\infty,I}^{\text{AIP}}$, which is the $T$-adic formal scheme given as the inverse limit of Frobenius. Now, clearly this lift of Frobenius is not a morphism of formal schemes over $\mathfrak{X}$, as it acts non trivially on $\mathfrak{X}$. So my question is exactly why should a simple base change of the universal elliptic curve work? It seems that in your argument you assume that $\mathfrak{X}_{\infty}^{\text{AIP}}$ is isomorphic to $\mathfrak{X}_{\Gamma(p^\infty)}$ | |
| Apr 17, 2019 at 13:34 | comment | added | Zariski93 | One proves that over this object the universal elliptic curve given by the base change of $\mathfrak{E}^{\text{univ}}$ along the blowing up $\mathfrak{X}_{r,I}\rightarrow\mathfrak{X}$ admits canonical subgroups, essentially of level lower or equal than $r$. Then, over this object, it is possible to define a partial Igusa tower as the normalization of the adic space which generically parametrizes trivializations of the dual canonical subgroup. | |
| Apr 17, 2019 at 13:30 | comment | added | Zariski93 | ...complete and separated $W$-algebra, $T$-torsion free parametrize morphisms $f:\text{Spf}(R)\rightarrow\mathfrak{X}$, which means elliptic curves over $R$ with tame level structure, and sections $\eta\in\text{H}^0(\text{Spf}(R),f^\ast\omega^{p^{r+1}(1-p)})$ which give $\eta\text{Ha}^{p^{r+1}}=T$ mod $p^2$, where $\text{Ha}$ is a local lift of the Hasse invariant. I hope the notation is as clear as possible. | |
| Apr 17, 2019 at 13:27 | comment | added | Zariski93 | Dear Leroy, sorry if I interfere between you and A. Walker, but I though a lot about this question. I tell you my doubts. First, why do you think $\mathfrak{X}_{\infty}^{\text{AIP}}$, the Andreatta-Iovita and Pilloni infinite level modular curve, can be confused with $\mathfrak{X}_{\Gamma(p^{\infty})}$? This is not clear to me. In fact, each $\mathfrak{X}_{r,I}$ which gives the projective limit defining $\mathfrak{X}_{\infty}^{\text{AIP}}$ is a $T$-adic formal scheme over $W:=\text{Spf}\left(\mathbb{Z}_p[[T]]\langle\frac{p}{T}\rangle\right)$ whose $\text{Spf}(R)$-points, for $R$ a normal | |
| Apr 17, 2019 at 10:48 | comment | added | Leeeeroy _Jennnnkins | So about the universal ell curve thing. Working formally, you have $\mathfrak{X}_{\Gamma(p^\infty)} \to \mathfrak{X}$ and a universal (formal) ell curve $\mathfrak{E}_{univ} \to \mathfrak{X}$. So one can take the fibre product $\mathfrak{X}_{\Gamma(p^\infty)} \times \mathfrak{E}_{univ}$ and maybe call this (or maybe its rigid fibre) the universal ell curve at infinite level. (but I worry about doing this as perfectoid spaces, since I don't know if this fibre product is again a perfectoid space). Does something like this not work? | |
| Apr 16, 2019 at 10:59 | comment | added | A. Walker | Essentially, if I have such an $\eta$, I know that canonical subgroup exists and that the generic fiber of its dual is trivial. But on the other way, if I have a trivialization of the dual canonical, how can I get such an $\eta$? | |
| Apr 16, 2019 at 10:48 | comment | added | A. Walker | ..when the connecting morphism is Frobenius. I mean, over $\mathfrak{X}_{r,I}$ there is a universal elliptic curve. May I base change it to an elliptic curve over the infinite level modular curve, and then over Igusa? Do I really get a unique modular curve? And how does the isomorphism of Igusa with $\mathcal{X}_{\Gamma(p^\infty}$ is defined? Doesn't it depend on the base adic space we are dealing with? | |
| Apr 16, 2019 at 10:44 | comment | added | A. Walker | Thank you! Andreatta, Iovita and Pilloni first define $\mathfrak{X}_{\infty}$ as the limit of $X_{r,I}$ along Frobenius, where $\mathfrak{X}_{r,I}$ parametrizes elliptic curves over a blowing up of the weight space with a section $\eta$ such that $\eta\text{Ha}^{p^{r+1}}=T$, where $T$ generates the topology, more or less. Over $\mathfrak{X}_\infty$ they define the Igusa tower as the normalization of the analytic Igusa tower parametrizing trivializations of the dual canonical subgroup. The point is that I cannot see how to get a universal elliptic curve from a system of modular or Igusa curves | |
| Apr 16, 2019 at 10:30 | review | First posts | |||
| Apr 16, 2019 at 10:31 | |||||
| Apr 16, 2019 at 10:28 | history | answered | Leeeeroy _Jennnnkins | CC BY-SA 4.0 |