Points of Scholze's Anticanonical Tower

Question

In his paper "On torsion of locally symmetric spaces", Schole introduces a perfectoid space, which is the anticanonical tower. He claims that points coming from $\text{Spa}(C,C^+)$, where $C$ is a complete algebraically closed extension of $\mathbb{Q}_p$ parametrize abelian varieties with a polarization and a trivialization of the Tate module. I really cannot see why this holds true. So, consider such a point. How do I get a unique abelian variety instead of a projective system of abelian varieties? And second, how do I get the splitting?

Answer 1

Scholze's object is a limit (in some rather delicate sense) of objects at finite level. The $n$-th of these finite-level objects is a moduli space for abelian varieties $A$ with a polarization $\lambda$ and a symplectic basis $B$ of their $p^n$-torsion (satisfying some "anticanonical" condition but never mind that for now).

The key is that the map from the n-th layer to the $(n-1)$-th layer sends $(A, \lambda, B)$ to $(A, \lambda, B \bmod p^{n-1})$. The abelian variety doesn't change: you just throw away some information about the level structure.

I'm assuming that Scholze's "tilde-limits" are well-behaved enough that a $C$-point of the limit is the same thing as a compatible collection of $C$-points of the spaces at each finite level. So a $C$-point of the infinite-level object consists of a single polarized abelian variety $(A, \lambda)$, together with a collection of bases $B_n$ of its $p^n$ torsion for every $n$, with the $B_n$ being compatible in the obvious sense; and it's pretty clear that this is the same data as a trivialisation of the Tate module as a $\mathbf{Z}_p$-module.