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This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze.

In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps defined over $\text{Spf(}\mathbb{Z}_p^{\text{cycl}})$ between the integral models of strict neighborhoods of the ordinary locus for the modular curve. Then, he computes the projective limit along Frobenius of these maps, and he gets a formal scheme over $\text{Spf}(\mathbb{Z}_p^{\text{cycl}})$, whose generic fiber is perfectoid and is called the anticanonical tower of modular curves. Later on in his exposition, he writes that the $(C,\mathcal{O}_C)$-points of this perfectoid space (where $C$ is the completion of an algebraic closure of $\mathbb{Q}_p$, and $\mathcal{O}_C$ is its ring of integers) parametrize elliptic curves over $C$ with a trivialization of their Tate module.

First, I do not see why this construction provides a unique elliptic curve! First, I would like to say that the construction provides a projective system of elliptic curves over $C$, where every elliptic curve has $p^n$-torsion trivialized (for $n$ becoming bigger along the tower), where the maps defining the projective system are quotient by the canonical subgroup. But why is such a kind of system the same as a unique elliptic curve with Tate module trivialized?

Second question, does a similar description hold for the integral anticanonical tower? Is it true that an $R$ point of the anticanonical tower, where $R$ is a complete and flat (maybe normal) $\mathbb{Z}_p^{\text{cycl}}$-algebra, gives a family of elliptic curves over $R$ with a trivialization (at least a generic trivialization) of its Tate module seen as a sheaf? Thank you for any kind of suggestion!

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  • $\begingroup$ If nobody answers in the next day or two, you might consider cross-posting this to MathOverflow. $\endgroup$ – André 3000 Dec 26 '18 at 19:59
  • $\begingroup$ Strange, I supposed to have posted it on Mathoverflow first! Let me do this, and thank you very much! $\endgroup$ – Zariski93 Dec 28 '18 at 16:43

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