# Points of Scholze's Anticanonical Tower

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?

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.
• Thank you for your answer, really clear! Just a remark. Why does this description hold true only for $(C,\mathcal{O}_C)$ points? What can go wrong if we consider points over a different affinoid adic space? – Zariski93 Oct 2 '18 at 15:22