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https://arxiv.org/pdf/2010.05748.pdf
Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups
Kirti Joshi October 13, 2020
(抜粋)
1 Introduction

The existence of distinctly labeled copies of the tempered fundamental groups is, as far as
I understand, crucial to [Moc12a; Moc12b; Moc12c; Moc12d], but produced in loc. cit. by
entirely different means (for more on this labeling problem see Section 3). Let me also say at
the onset that Mochizuki’s Theory does not consider passage to complete algebraically closed
fields such as Cp and so my approach here is a significant point of departure from Mochizuki’s
Theory . . . and the methods of this paper do not use any results or ideas from Mochizuki’s
work. Nevertheless the results presented here establish unequivocally that isomorphs of tempered (and ´etale) fundamental groups, of distinguishable provenance, exist and can be explicitly
constructed.

An important consequence of these results is Corollary 3.1, which provides a function from
a suitable Fargues-Fontaine curve to the isomorphism class of the tempered fundamental group of a fixed variety (as above) which provides a natural way of labeling the copies obtained here by closed points of a suitable Fargues-Fontaine curve.
In the last section of the paper I show that there is an entirely analogous theory of untilts of topological fundamental groups of connected Riemann surfaces.

3 Untilts of tempered fundamental groups
The results of the preceding section can be applied to the problem of producing labeled copies
of the tempered fundamental groups. A simple example of the labeling problem is the following: let G be a topological group isomorphic to the absolute Galois group of some p-adic field.
In this case one can ask if there are any distinguishable elements in the topological isomorphism class of G with the distinguishing features serving as labels.