# Moduli interpretation of Fargues-Fontaine curve

The Fargues-Fontaine curve is, in his schematic version, a noetherian regular scheme $X$ of dimension 1 associated to a pair $(E,F)$, where $E$ is a local field (i.e. complete w.r.t. a discrete valuation with finite residual field $k$, or, more concretely, a finite extension of $\mathbb{Q}_p$ or $k((t))$) and $F$ an algebrically closed non-archimedean field of characteristic $p$. It has also an adic version, more easily constructed (once you know adic spaces) using some versions of the rings of $p$-adic periods $B$ introduced by Fontaine some time ago. Some people call it the fundamental curve of $p$-adic Hodge theory. See History of Fargues-Fontaine curve or the papers in Fargues homepage for more information.

There are some other objects in arithmetic geometry, like Jacobians of curves, Néron models, modular curves, Shimura varieties, etc, that have constructions more or less difficult, but most of the time you only need to know its existence and its moduli interpretation (or some kind of universal property) to use it or even to prove some results for it.

My question is it there is some moduli interpretation or universal property verified by the Fargues-Fontaine curve (in schematic or adic version) that can be used as a black-box as in the previously mentioned cases?

• The original reason to consider this geometric object (as far as I am aware) was asking about the "untilts" of a perfectoid field E.g. see section 4, in particular theorem 4.1 of Scholze's 2014 ICM article. – Stiofáin Fordham Jul 3 '18 at 15:50
• For $C$ an algebraically closed field, perfectoid field of characteristic $p$, the closed points of the schematic Fargues--Fontaine curve $X_C$ bijectively correspond to Frobenius equivalence classes of characteristic 0 untilts of C. See Theorem 2.5.4 of [Weinstein's 2017 AWS notes] (math.bu.edu/people/jsweinst/AWS2017.pdf). – Jackson Morrow Jul 3 '18 at 16:21