Inter-universal geometry と ABC予想 (応援スレ) 44

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420: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/04/26(日) 14:20:57 ID:7O7a3CML

Dupuy先生の予告論文出た（＾＾
意味わからんがｗ（＾＾；

http://www.uvm.edu/~tdupuy/papers.html
[ Taylor Dupuy's Homepage]
[ manuscripts ]
2.The Statement of Mochizuki's Corollary 3.12, Initial Theta Data, and the First Two Indeterminacies, (with A. Hilado)
https://www.dropbox.com/s/8xuj2ws716464vn/initial-theta.pdf?
THE STATEMENT OF MOCHIZUKI'S COROLLARY 3.12, INITIAL
THETA DATA, AND THE FIRST TWO INDETERMINACIES
DRAFT
TAYLOR DUPUY AND ANTON HILADO
Abstract. This paper does not give a proof of Mochizuki's Corollary 3.12. It is the first
in a series of three papers concerning Mochizuki's Inequalities. The present paper concerns
the setup of Corollary 3.12 and the first two indeterminacies, the second [DH20a] concerns
log-Kummer correspondences and ind3, and the third [DH20b] concerns applications to
Diophantine inequalities (in the style of IUT4). These manuscripts are designed to provide
enough definitions and background to give readers the ability to apply Mochizuki's state-
ments in their own investigations. Along the way, we have faithfully simplified a number
of definitions, given new auxillary definitions, and phrased the material in a way to maxi-
mize the dierences between Theorem 1.10 of IUT4 and Corollary 3.12 of IUT3. It is our
hope that doing so will enable creative readers to derive interesting and perhaps unforeseen
consequences Mochizuki's inequality.
Contents
1. Introduction 1
2. Background and Notation 5
3. Fake Adeles, Random Measurable Sets, and Pilot Objects 8
4. Indeterminacies and U 17
5. Global Multiplicative Subspaces, Initial Theta Data, and E11a1 24

つづく

http://rio2016.5ch.net/test/read.cgi/math/1586655469/420

421: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/04/26(日) 14:21:14 ID:7O7a3CML

>>420

1. Introduction
The purpose of this paper and its sequels [DH20b] and [DH20a] is to put Mochizuki's
inequality in a user friendly context for working mathematicians. While these manuscripts
do 1 indicate in some places how certain parts of Mochizuki's constructions work, they do not
attempt to give a proof of [Moc15c, Corollary 3.12]. Moreover we black-box and suppress
the anabelian geometry as much as possible (at some junctures this is simply not possible).

By the end of [DH20b] we will rigorously derive a variant of Theorem 1.10 of [Moc15d],
(an effective version of Szpiro's inequality for elliptic curves in ”initial theta data").
In this, all of the assumptions will be made transparent − including the statement of Corollary 3.12
and how to apply it.

In this manuscript (and its sequels) we work under the hypothesis that
all of Mochizuki's \functorial algorithms" can be expressed in terms of interpretations in the
sense of model theory. We refer the reader to [Hod97, x4.3] for the basics of interpretations
(the more topos-minded readers might be inclined to read [Car18, Definition 6.12] which
provides a more categorical framework). We just mention in passing that for this to work we
need to abandon classical finitary logic and allow for countable conjunctions of formulas and
countably many sorts (this is by default done in the topos theory literature but is atypical
of classical first order model theory literature).

(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1586655469/421

422: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2020/04/26(日) 14:37:28 ID:7O7a3CML

>>420 追加

これも出た
今いｔ意味わからん（＾＾；

http://www.uvm.edu/~tdupuy/papers.html
[ Taylor Dupuy's Homepage]
[ manuscripts ]

１．Log-Kummer Correspondences and The Third Indeterminacy (with A. Hilado) (Appendix of Interpretation Tables Only -- this may be spun into its expository own document)
https://www.dropbox.com/s/7e7roesnirjxxlh/tables-only-live.pdf?
Abstract. This document contains a number of interpretation tables used in Mochizuki's
IUT papers. It is stripped from the Appendix of [DH20].

”Functorial Algorithms" = Interpretations
Mochizuki's theory depends heavily on ”functorial algorithms" which he defines as func-
tors from one category to another. In practice these ”functorial algorithms" are intricate
anabelian reconstructions and we have found that the details of one construction often feed
into later constructions or Theorems i.e. their knowledge ”as a functor" generally tends not
to serve as a good black box. 1

Most of Mochizuki's \functorial algorithms" are interpretations in the sense of Model The-ory2 [Hod97, x4.3]
(see [Car18, Definition 6.12] for a more topos theoretic definition). This
formalism is both convenient and precise for the purposes of discussing IUT.

http://rio2016.5ch.net/test/read.cgi/math/1586655469/422

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