Inter-universal geometry と ABC予想 (応援スレ) 49

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460(2): 現代数学の系譜雑談 ◆yH25M02vWFhP 2020/10/14(水)16:44 ID:WB0JVdoR(3/6)調 AAS
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
https://twitter.com/5chan_nel (5ch newer account)

461(1): 現代数学の系譜雑談 ◆yH25M02vWFhP 2020/10/14(水)16:44 ID:WB0JVdoR(4/6)調 AAS
>>460
つづき

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.
This note began as a part of another note, [Jos20a], which I put into a limited circulation some time in July 2020, outlining my own approach to some constructions of [Moc12a;Moc12b; Moc12c; Moc12d]. Peter Scholze immediately, but gently, pointed out that the section of [Jos20a], from which the present note is extracted, needed some details.
At that time I was readying another note, [Jos20b], for wider circulation and addressing the issue noted by
Scholze took longer and on the way I was able to substantially strengthen and clarify my results
(which appear here). So ultimately I decided that it would be best to publish the present note
separately (while preparation of [Jos20a] continued). My thanks are due to Peter Scholze, and
also to Yuichiro Hoshi, Emmanuel Lepage, and Jacob Stix, for promptly providing comments,
suggestions or corrections.

2 The main theorem

Lemma 2.1. Let K be a valued field and let R ⊂ K be the valuation ring. The following
conditions are equivalent:
(1) K is an algebraically closed field, complete with respect to a rank one non-archimedean
valuation and with residue characteristic p > 0.
(2) K is an algebraically closed, perfectoid field.
Proof. A perfectoid field has residue characteristic p > 0 and is complete with respect to a rank one valuation.
(引用終り)
以上

467(3): 現代数学の系譜雑談 ◆yH25M02vWFhP 2020/10/14(水)21:03 ID:qOwFO4Cy(4/11)調 AAS
>>460 追加

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)

This note began as a part of another note, [Jos20a], which I put into a limited circulation some time in July 2020, outlining my own approach to some constructions of [Moc12a;Moc12b; Moc12c; Moc12d]. Peter Scholze immediately, but gently, pointed out that the section of [Jos20a],
from which the present note is extracted, needed some details.
At that time I was readying another note, [Jos20b], for wider circulation and addressing the issue noted by
Scholze took longer and on the way I was able to substantially strengthen and clarify my results
(which appear here). So ultimately I decided that it would be best to publish the present note
separately (while preparation of [Jos20a] continued). My thanks are due to Peter Scholze, and
also to Yuichiro Hoshi, Emmanuel Lepage, and Jacob Stix, for promptly providing comments,
suggestions or corrections.

[Jos20a] Kirti Joshi. “On Mochizuki’s log-link . . .”. In: (2020). In preparation.
[Jos20b] Kirti Joshi. “The absolute Grothendieck conjecture is false for Fargues-Fontaine curves”. In: (2020). Preprint. URL: https://arxiv.org/abs/2008.01228.
(引用終り)

”At that time I was readying another note, [Jos20b], for wider circulation and addressing the issue noted by
Scholze took longer and on the way I was able to substantially strengthen and clarify my results
(which appear here). ”とあるな。見落としていたな（＾＾；

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