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

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 49 (1002ﾚｽ)
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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（＾＾；

469(1): 現代数学の系譜雑談 ◆yH25M02vWFhP 2020/10/14(水)21:09 ID:qOwFO4Cy(5/11)調 AAS
>>467
追加

Kirti Joshi氏はすげーな

https://arxiv.org/pdf/2008.01228.pdf
The Absolute Grothendieck Conjecture is false for
Fargues-Fontaine Curves
Kirti Joshi
August 5, 2020
(抜粋)
2 The main theorem
Let F be an algebraically closed perfectoid field of characteristic p > 0. Let E be a p-adic field
i.e. E/Qp is a finite extension. Following [Jos20], I say that two p-adic fields are anabelomorphic if there exists a topological isomorphism GE ? GE′ of their absolute Galois groups;

480(5): 現代数学の系譜雑談 ◆yH25M02vWFhP 2020/10/15(木)11:23 ID:esT5rSCm(2/6)調 AAS
>>467 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)
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.

502(2): 現代数学の系譜雑談 ◆yH25M02vWFhP 2020/10/16(金)21:18 ID:w2Iu7oDW(1/3)調 AAS
>>467 補足

https://arxiv.org/pdf/2010.05748.pdf
Untilts of fundamental groups: construction of labeled
isomorphs of fundamental groups
Kirti Joshi
October 13, 2020
（翻訳）＜www.DeepL.com/Translator（無料版）の翻訳に手を入れた＞
このノートは、私が2020年7月頃に限定的に回覧した別のノート[Jos20a]の一部として始まったもので、[Moc12a;Moc12b;Moc12c;Moc12d]のいくつかの解釈に対する私自身のアプローチを概説したものである。
Peter Scholzeはすぐに、しかし優しく、このノートが抜粋されている[Jos20a]の部分には詳細が必要だと指摘した。
そこで、私は別のノート[Jos20b]を準備し、より広く回覧するために私は自分の結果を大幅に強化し、Scholzeが指摘した問題への対処に時間をかけて、明確にすることができました（それがここに掲載されています）。
そのため，最終的には（[Jos20a] の準備を続けながら）本ノートとは別に出版した方が良いと判断しました．Peter Scholze氏と、コメント、提案、修正を迅速に提供してくれた星裕一氏、Emmanuel Lepage氏、Jacob Stix氏に感謝します。
（訳の終り）

　これから分かることは、
・Kirti Joshi ノート[Jos20a]を、「限定的に回覧」（which I put into a limited circulation）した。
・別のノート[Jos20b]を、「より広く回覧」（for wider circulation）
・Peter Scholze氏、Jacob Stix氏に感謝します（My thanks are due to Peter Scholze, Jacob Stix,for promptly providing comments,suggestions or corrections.）
・だから、今年４月から５月のwoitブログの時からは、大分議論は変わってきて、Kirti Joshi の仕事が、IUTとパーフェクトイドとを融合する役割をしているように思う
・Peter Scholze氏とJacob Stix氏とも、Kirti Joshi の仕事を認めている

今後も、こういう動きが、どんどん出てくるように思います（＾＾

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