My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal geometry involved there. Actually, I noticed that there is a paper by Andreatta, Iovita and Pilloni, titled Le halo spectral, which seems to deal with formal integral models of Scholze's towers.
First, if I well understand Scholze, talking about elliptic curves, there is a perfectoid space $\mathcal{X}_{\infty}(\epsilon)$ which gives the "tilda limit" of modular curves $\mathcal{X}_{\Gamma(p^n)}(\epsilon)$ where each $\mathcal{X}_{\Gamma(p^n)}(\epsilon)$ describes open neighborhoods of the ordinary locus of the $\Gamma_1(N)$ modular curve, where the universal elliptic curve coming from pullback is not too supersingular. Actually, the construction of this object is performed by computing the adic generic fiber of the formal scheme $\mathfrak{X}_{\infty}(\epsilon)$ which is the real limit (in the category of formal schemes) of integral models of $\mathcal{X}_{\Gamma_(p^n)}(\epsilon)$ where the maps in the inverse system are given by a lifting of mod $p$-Frobenius.
A very similar construction is performed in chapter $6$ of Andreatta, Iovita and Pilloni's paper, where they construct the integral anticanonical tower $\mathfrak{X}_{\infty}$ exactly in the same way, but working over a basis which is a suitable blowup of an integral model of Coleman's weight space. Now, I'm just wondering whether or not it is possible to interpret these "infinite" level spaces as moduli spaces of elliptic curves plus a new kind of level structure. Somewhere in Scholze's paper it is mentioned that a point of $\mathcal{X}_{\infty}$ over $\text{Spa}(C,\mathcal{O}_C)$, where $C$ is a complete algebraically closed extension of $\mathbb{Q}_p$ corresponds to an elliptic curve over $C$ with a trivialization of its Tate module. Now, why is this true? It's not mentioned in Scholze and I cannot prove it. Moreover, does a similar description hold for different kind of points, e.g. $\text{Spa}(R,R^+)$ with $R$ a perfectoid $\mathbb{Q}_p$-algebra? Moreover, does the same intepretation hold for its formal integral model? And what about the Andreatta, Iovita and Pilloni's tower? Is it true that it parametrizes elliptic curves with $p$-divisible groups playing the role of the canonical subgroup? The point essentially is, does this object gives by pullback a universal elliptic curve? Which kind of level does it have a similar elliptic curve?
Best Answer
Wow, that's eight questions, plus more in the comments -- I don't think I can answer all of them, but I'll try to answer at least a few! :)
First of all, let's fix the setting: It seems to me that you are using three different kinds of level structures, $\Gamma_0(p^n)$, $\Gamma_1(p^n)$ and $\Gamma(p^n)$, and some questions seem to be referring to different ones. Also, there are various possible different meanings of $\mathcal X$ and $\mathfrak X$, since these mean different things in each paper respectively.
So we first have to agree on some uniform setting for your questions: Since you are interest in perfectoid moduli spaces, I suggest we follow Scholze (III.2.2 in the torsion paper) and denote by $\mathfrak X$ the formal completion of the modular curve over $\mathbb Z_p^\mathrm{cyc}$ of some fixed tame level. Denote by $\mathfrak X^{\ast}$ the completion of the compactified modular curve, and by $\mathcal X^{\ast}$ the adic generic fibre. Since you are interested in moduli of elliptic curves, I suggest we now deviate from Scholze's notation and denote by $\mathcal X$ the analytification of the modular curve over $\mathbb Q^{\mathrm{cyc}}$ (rather than the generic fibre of $\mathfrak X$, which is the good reduction locus).
moduli interpretations of the spaces $\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a$, $\mathcal X_{\Gamma_1(p^\infty)}(\epsilon)_a$ and $\mathcal X_{\Gamma(p^\infty)}(\epsilon)_a$
(Most of what I'm going to say in regards to this question can be found in more detail in this related article: https://nms.kcl.ac.uk/ben.heuer/PGp-adMC.pdf.)
Let's follow the torsion paper and start with level $\Gamma_0(p^n)$ and the anticanonical locus $\mathcal X_{\Gamma_0(p^n)}(\epsilon)_a$ of some tame level. This represents the functor which sends a (sheafy) adic space $\mathrm{Spa}(R,R^{+})$ to the set of isomorphism classes of triples $(E,\alpha,D)$ where $E|R$ is an elliptic curve with some condition on the Hasse invariant which ensures that $E$ has a canonical subgroup $C=C(E)\subseteq E[p]$, where $\alpha$ is a tame level structure, and where $D\subseteq E[p^n]$ is an anticanonical cyclic subgroup scheme of rank $p^n$. Here "anticanonical" means $D\cap C=0$.
Scholze now proves that there is a perfectoid space $$\mathcal X_{\Gamma_0(p^\infty)}(\epsilon)_a\sim \varprojlim\mathcal X_{\Gamma_0(p^n)}(\epsilon)_a.$$ Since perfectoid tilde-limits satisfy the universal property of the limit for perfectoid test objects, this space represents the functor which sends $\mathrm{Spa}(R,R^{+})$ for any perfectoid $\mathbb Q_p^\mathrm{cyc}$-algebra $R$ to the set of isomorphism classes of $(E,\alpha,D_\infty)$ where $D_\infty = (D_n\subseteq E[p^n])_{n\in\mathbb N}$ is a collection of anticanonical cyclic subgroup schemes with $D_{n+1}[p^n]=D_n$ (See Corollary 3.2 of the above document). So in regards to your question of moduli of $p$-divisible groups, one could call this data an "anticanonical $p$-divisible subgroup of height 1".
Similar results hold for the perfectoid tilde-limits $\mathcal X_{\Gamma_1(p^\infty)}(\epsilon)_a\sim \varprojlim\mathcal X_{\Gamma_1(p^n)}(\epsilon)_a$ and $\mathcal X_{\Gamma(p^\infty)}(\epsilon)_a\sim \varprojlim\mathcal X_{\Gamma(p^n)}(\epsilon)_a$ by the same reasoning: The first represents the functor which sends $\mathrm{Spa}(R,R^{+})$ for any perfectoid $\mathbb Q_p^\mathrm{cyc}$-algebra $R$ to the set of isomorphism classes of $(E,\alpha,\beta: \mathbb Z_p\xrightarrow{\sim} T_pD_\infty(R))$ where $D_\infty$ is an anticanonical $p$-divisible subgroup of height 1 and beta is a trivialisation of its Tate module. The second uses instead isomorphism classes of tuples $(E,\alpha,\gamma: \mathbb Z_p^2\xrightarrow{\sim} T_pE(R))$ where the image of $\gamma(1,0)$ in $E[p](R)$ generates an anticanonical subgroup.
the formal model of the anticanonical tower
As with the last question, we first need to agree on a base: The torsion paper considers an anticanonical tower over $\mathbb Z_p^\mathrm{cyc}$ (whose limit you denote by $\mathfrak X_\infty(\epsilon)$), whereas Le halo spectral basically works over $\mathbb Z_p$ (as you say, they really work relatively to some weight space, which is great because it allows them to construct integral families of modular forms. But I think in order to understand what's going on in terms of moduli, it might be easier if we specialise to a point -- the weight space doesn't change much in that respect). Let's follow Scholze and work over $\mathbb Z_p^\mathrm{cyc}$ if you don't mind, so we simply base-change the constructions of Andreatta--Iovita--Pilloni to $\mathbb Z_p^\mathrm{cyc}$ (their constructions actually require Noetherianess in several places in order to construct normalisations, but once you got the spaces, you may still simply base-change to $\mathbb Z_p^\mathrm{cyc}$. The resulting spaces agree with Scholze's $\mathfrak X^{\ast}(\epsilon)$ up to a normalisation issue).
Now there are arguably two "anticanonical towers", which are isomorphic: The first one, which gives the tower its name, is the tower $$\dots\to\mathcal X^{\ast}_{\Gamma_0(p^2)}(\epsilon)_a\to\mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a\to \mathcal X^{\ast}(\epsilon).$$ The second tower is used in the torsion paper to prove that the above tower has a perfectoid tilde limit $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$: Let's recall how this works. Let $\mathfrak X^{\ast}(\epsilon)$ be like in Scholze's Definition III.2.12. As explained there, (away from the cusps) this represents the functor sending $\mathrm{Spf}(R)$ for $p$-adically complete $\mathbb Z_p^\mathrm{cyc}$-algebras $R$ to the set of isomorphism classes $(E,\alpha,\eta)$ where $E|R$ is an elliptic curve, $\alpha$ is a tame level and $\eta\in \omega_E^{\otimes(1-p)}$ such that $\eta \mathrm{Ha} = p^{\epsilon} \in R/p$. Scholze constructs Frobenius lifts $F:\mathfrak X^{\ast}(p^{-1}\epsilon)\to \mathfrak X^{\ast}(\epsilon)$ which on the level of moduli (away from the cusps) are given by quotienting by the canonical subgroup, i.e. sending $E\mapsto E/C$. In the limit, this gives rise to the space $\mathfrak X^{\ast}(p^{-\infty}\epsilon)=\varprojlim_{{F}} \mathfrak X^{\ast}(p^{-n}\epsilon)$ which is integrally perfectoid. In particular, its adic generic fibre is a perfectoid space.
The relation to the anticanonical tower is that on the level of adic spaces over $\mathbb Q_p^\mathrm{cyc}$, there is a natural "Atkin-Lehner" isomorphism $$\varphi_n:\mathcal X^{\ast}(p^{-n}\epsilon)\to \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a, \quad E\mapsto (E/C_{n},E[p^n]/C_{n})$$ where $C_n\subseteq E[p^n]$ is the rank $p^n$ canonical subgroup. Its inverse is given by sending $(E,D)\mapsto E/D$. One can now check on the level of moduli that for different $n$, these give a comparison isomorphism between the anticanonical tower and the Frobenius tower:
$\require{AMScd}$ \begin{CD} \dots @>>> \mathcal X^{\ast}_{\Gamma_0(p^2)}(\epsilon)_a @>>> \mathcal X^{\ast}_{\Gamma_0(p)}(\epsilon)_a @>>> \mathcal X^{\ast}(\epsilon)\\ @AAA @AA\varphi_2A @AA\varphi_1A @|\\ \dots @>>> \mathcal X^{\ast}(p^{-2}\epsilon) @>>> \mathcal X^{\ast}(p^{-1}\epsilon) @>>> \mathcal X^{\ast}(\epsilon) \end{CD}
We may thus see the tower of morphisms $F:\mathfrak X^{\ast}(p^{-(n+1)}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$ as a formal model of the anticanonical tower. In particular, we may see $\mathfrak X^{\ast}(p^{-\infty}\epsilon)$ as a canonical formal model for $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$. Alternatively, I think this should imply that we can regard $\mathfrak X^{\ast}(p^{-n}\epsilon)$ as representing (away from the cusps) tuples $(E,\alpha,D_n)$ where $D_n\subseteq E[p^n]$ is a cyclic rank $p^n$ subgroup scheme which is generically anticanonical (its special fibre may well be canonical).
So what is the moduli interpretation of a $\mathrm{Spf}(R)$-point of $\mathfrak X^{\ast}(p^{-\infty}\epsilon)$ (away from the cusps) where $R$ is a complete $\mathbb Z_p^\mathrm{cyc}$-algebra? One answer is that, by definition, it is the data of $(E_0,E_1,E_2,\dots,\alpha, (\eta_n)_{n\in\mathbb N})$ where $(E_0,\alpha,\eta_0)$ is like before, $E_{n+1}/C(E_{n+1})=E_n$ for all $n$, and the $\eta_n$ are compatible under $F$. Alternatively, by the above tower this should be equivalent to the data of $(E,\alpha, (\eta_n)_{n\in\mathbb N},D_\infty)$ where $E:=E_0$ and $D_\infty=(D_n)_{n\in\mathbb N}$ is a generically anticanonical $p$-divisible subgroup of $E[p^\infty]$ of height 1. Here $D_n$ is defined as the kernel of the dual isogeny to $E_n\to E_0$, so that $E_n=E_0/D_n$, and the $\eta_n\in \omega_{E_n}^{\otimes(1-p)}$ are as before.
the integral model for $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$ of Andreatta--Iovita--Pilloni
Now to the spaces in Le halo spectral, I'll try to elaborate on Leeeeroy_Jennnnkins' answer and answer a question raised in the comments. (If you allow another plug, most of this can be found in more detail in \S 4 of https://arxiv.org/pdf/1902.03985.pdf).
Andreatta--Iovita--Pilloni go further than what you denote by "$\mathfrak X_\infty(\epsilon)$": They also consider the Igusa schemes $\mathfrak I\mathfrak G_n(p^{n}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$ which relatively represent the choice of a trivialisation $\mathbb Z/p^n\mathbb Z\to C_n^\vee$, namely morphisms which are an isomorphism over the ordinary locus. They show that the Frobenius isogeny lifts to a "Frobenius" morphism $F:\mathfrak I\mathfrak G_{n+1}\to \mathfrak I\mathfrak G_n$ and form the "Igusa curve at infinite level" which in order to be consistent with my notation I should probably denote by $\mathfrak I\mathfrak G_{\infty}(p^{-\infty}\epsilon)=\varprojlim_{F}\mathfrak I\mathfrak G_{n}(p^{-n}\epsilon)$.
Now how does this compare to Scholze's spaces? The short exact sequence of group schemes $$0\to C_n\to E[p^n]\to E[p^n]/C_n\to 0$$ shows that the Weil pairing canonically identifies $C_n^{\vee}$ with $E[p^n]/C_n$. Thus the Igusa tower equivalently parametrises trivialisations $\mathbb Z/p^n\mathbb Z\to E[p^n]/C_n$. But under the above "Atkin-Lehner" isomorphism, $E[p^n]/C_n$ is the corresponding anticanonical subgroup of $E/C_n$. This means that on the adic generic fibre, this isomorphism lifts to a canonical isomorphism
\begin{CD} \mathfrak I\mathfrak G_n(p^{-n}\epsilon)^{\mathrm{ad}}_{\eta} @>\sim>>\mathcal X^{\ast}_{\Gamma_1(p^n)}(\epsilon)_a \\ @AAA @AAA \\ \mathcal X^{\ast}(p^{-n}\epsilon) @>\sim>> \mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a. \end{CD}
In particular, this means that $\mathfrak I\mathfrak G_n(p^{-n}\epsilon)$ is a formal model of $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$. In the limit, it follows that $$\mathfrak I\mathfrak G_\infty(p^{-\infty}\epsilon)^{\mathrm{ad}}_{\eta}=\mathcal X^{\ast}_{\Gamma_1(p^\infty}(\epsilon)_a.$$ So this gives you a canonical formal model of $\mathcal X^{\ast}_{\Gamma_1(p^\infty)}(\epsilon)_a$. Its moduli interpretation (away from the cusps) may be given in terms of tuples $(E,\alpha,(\eta_n)_{n\in\mathbb N},\beta:\mathbb Z_p\to T_pD_\infty)$ where $(E,\alpha,(\eta_n)_{n\in\mathbb N},D_\infty)$ is like above, and beta is a morphism that becomes an isomorphism over the ordinary locus.
Finally, if you are interested in integral models for the full level modular curve $\mathcal X^{\ast}_{\Gamma(p^\infty)}$, you may want to have a look at Lurie's preprint http://www.math.harvard.edu/~lurie/papers/LevelStructures1.pdf.
universal elliptic curves
There are different universal elliptic curves over $\mathfrak X^{\ast}(p^{-\infty}\epsilon)$, and the "right" one depends on your choice of moduli interpretation. Looking at the above comparison map of towers again, as you say, we get a different "universal elliptic curve" by pullback along any $\mathfrak X^{\ast}(p^{-\infty}\epsilon)\to \mathfrak X^{\ast}(p^{-n}\epsilon)$. This is the universal $E_n$ in the moduli description in terms of data $(\alpha,\eta,E_0,E_1,E_2,\dots)$. Alternatively, the moduli interpretation in terms of $(E,\alpha,\eta,D_\infty)$ suggest to look at the pullback $\mathfrak E_\infty^{\mathrm{univ}}$ of the universal elliptic curve $\mathfrak E^{\mathrm{univ}}$ along $\mathfrak X^{\ast}(p^{-\infty}\epsilon)\to \mathfrak X^{\ast}$.
Can we make sense of the adic generic fibre of $\mathfrak E_\infty^{\mathrm{univ}}\to \mathfrak X^{\ast}(p^{-\infty}\epsilon)$? Yes:
The adic generic fibre of $\mathfrak E_\infty^{\mathrm{univ}}$ can be described as the fibre product of the relatively smooth rigid space $(\mathfrak E^{\mathrm{univ}})^{\mathrm{ad}}_{\eta}\to \mathcal X^{\ast}(\epsilon)$ with the perfectoid space $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)\to \mathcal X^{\ast}(\epsilon)$. I think this should exist as a sousperfectoid (hence sheafy) adic space $\mathcal E_\infty^{\mathrm{univ}}$.
Is it a perfectoid space over $\mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)_a$? No:
Fibre products of perfectoid spaces are perfectoid, but if you take the fibre product with any point $\mathrm{Spa}(\mathbb Q_p^\mathrm{cyc})\to \mathcal X^{\ast}_{\Gamma_0(p^\infty)}(\epsilon)$, you will get the analytification of an elliptic curve $E^{an}\to \mathrm{Spa}(\mathbb Q_p^{\mathrm{cyc}})$ which is certainly not perfectoid. If you want something perfectoid, I think it would be reasonable to guess that $\varprojlim_{[p]} \mathcal E_\infty^{\mathrm{univ}}$ is perfectoid -- this is true over the good reduction locus, but as far as I know, it's not currently known whether it's true over the whole space.