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To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius equivalence, and is the solution to a classification problem of some sort. I have a few very basic questions

  • Although the curve $X^{\text{FF}}_{E,F}$ is unique for different choices of $(E,F)$ and also we have the fact that $H^1(X^{\text{FF}}_{E,F},\mathcal{O}_{X^{\text{FF}}_{E,F}})=0$, is there a notion of topological classification for such curves in terms of pairs $(E,F)$ or something analog to topological classification in the usual case? For example, is there any meaning to the change of the topology of the curve as we change either $E$ or $F$ or both?

  • Knowing that one of the results of Fargues and Fontaine is showing the smoothness of this curve, can one define analogs of differential forms and say integration theory over such curves for general $F$ or e.g. when $F$ is algebraically-closed? e.g. is there a canonical bundle given that $X^{\text{FF}}_{E,F}$ fails to satisfy the Riemann-Roch Theorem or a Haar measure similar to the case of $p$-adic numbers?

Good references on these topics would be highly appreciated.

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    $\begingroup$ Replacing $F$ by another perfectoid field with the same tilt will give you an isomorphic Fargues-Fontaine curve - the difference will amount to picking a different "base point" (which corresponds to picking an untilt of that tilt). In that sense, the Fargues-Fontaine curve depends not on $F$ but on its tilt. At the level of Scholze's diamonds this curve also has a very simple description as a product of diamonds $\mathrm{Spd}F^\flat\times\mathrm{Spd}E/\varphi^{\mathbb Z}$. If not for the quotient there would be an obvious relation for base change, but with it, it's less clear. $\endgroup$
    – Wojowu
    Feb 14, 2022 at 14:36

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