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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459: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/14(水) 10:26:48.53 ID:WB0JVdoR

>>443より
http://swc.math.arizona.edu/aws/1998/98Frankenhuysen.pdf
THE ABC CONJECTURE IMPLIES ROTH’S THEOREM AND MORDELL’S CONJECTURE
MACHIEL VAN FRANKENHUYSEN
これ結構面白いわ（＾＾

(抜粋)
1. Introduction
in §5, we formulate Mordell’s conjecture and ‘effective Mordell’. §6.3 is devoted
to Bely??’s construction of an algebraic function which is ramified over 0, 1 and ∞
alone [1]. The application of this construction to P1
yields Roth’s theorem, §6.4,
and the application to a curve C of genus 2 or higher yields Mordell’s conjecture,§6.7.

Both Roth’s theorem and Mordell’s conjecture are theorems, see [10, 16] and
[2,4,6,7,21] respectively, and from this point of view it seems uninteresting to have
conditional proofs of these theorems, depending on the ABC conjecture, whose
validity is still unknown. However, the proofs of these theorems using ABC are
much simpler and transparent, and point out very clearly the relationship between
the theory of Diophantine approximation and the theory of points on curves of high
genus. More importantly, using ABC, one can prove considerably stronger versions
of the two theorems. Specifically, ABC implies effective Mordell (see §5.1), and a
certain stronger form of the ABC conjecture implies a certain refinement of Roth’s
theorem (see §4.1).
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1600350445/459

460: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/14(水) 16:44:09.24 ID:WB0JVdoR

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)

http://rio2016.5ch.net/test/read.cgi/math/1600350445/460

461: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/14(水) 16:44:51.73 ID:WB0JVdoR

>>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

http://rio2016.5ch.net/test/read.cgi/math/1600350445/461

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