# Moduli interpretation of the integral anticanonical tower

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!

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