Kirti Joshi氏は、IUTをperfectoid field に適用しようとしている（＾＾；

https://twitter.com/math_jin
math_jin 10月13日 より
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
I show that one can explicitly construct topologically/geometrically distinguishable data which
provide isomorphic copies (i.e. isomorphs) of the tempered fundamental group of a geometrically connected, smooth, quasi-projective variety over p-adic fields. This is done via Theorem 2.3 and Theorem 2.5. Notably Theorem 2.5 also shows that the absolute Grothendieck
conjecture fails for the class of Berkovich spaces (over algebraically closed perfectoid fields),
arising as analytifications of geometrically connected, smooth, projective variety over p-adic
fields.
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.

The copies provided by Theorem 2.3 and Theorem 2.5 arise from untilts of a fixed algebraically closed perfectoid field of characteristic p > 0 and hence I call these copies untilts of fundamental groups, or more precisely untilts of tempered fundamental groups.

つづく
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