https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage] 3.Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12, (with A. Hilado) 0209現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/07(木) 07:41:32.75ID:K7FsfJ9N>>208 関連
Acknowledgementsに、Kiran Kedlaya、Emmanuel Lepage、Chung Pang Mok、Thomas Scanlon 達の名前が挙がっている ”Preparatory Center for Research in Next-Generation Geometry located at RIMS”も、挙がっているね
https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage]論文集 https://arxiv.org/pdf/2004.13108.pdf 3.Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12, (with A. Hilado) Date: April 30, 2020. (抜粋) P4 Acknowledgements. This article is very much indebted to many previous expositions of IUT including (but not limited to) [Fes15, Hos18, Ked15, Hos15, Sti15, Mok15, Moc17, Yam17, Hos17, Tan18, SS17]. The first author also greatly benefitted from conversations with many other mathematicians and would especially like to thank Yuichiro Hoshi for helpful discussions regarding Kummer theory and his patience during discussions of the theta link and Mochizuki’s comparison; Kirti Joshi for discussions on deformation theory in the context of IUT; Kiran Kedlaya for productive discussions on Frobenioids, tempered fundamental groups, and global aspects of IUT; Emmanuel Lepage for helpful discussions on the p-adic logarithm, initial theta data, aut holomorphic spaces, the log-kummer correspondence, theta functions and their functional equations, tempered fundamental groups, log-structures, cyclotomic synchronization, reconstruction of fundamental groups, reconstruction of decomposition groups, the ”multiradial representation of the theta pilot object”, the third indeterminacy, the second indeterminacy, discussions on Hodge Theaters, labels, and kappa coric functions, and discussions on local class field theory;
Shinichi Mochizuki for his patience in clarifying many aspects of his theory ? these include discussions regarding the relationship between IUT and Hodge Arakelov theory especially the role of ”global multiplicative subspaces” in IUT, discussions on technical hypotheses in initial theta data; discussions on Theorem 3.11 and ”(abc)-modules”, discussions on mono-theta environments and the interior and exterior cyclotomes, discussions of the behavior of various objects with respect to automorphisms and providing comments on treatment of log-links and the use of polyisomorphisms, discussions on indeterminacies and the multiradial representation, discussions of the theta link, discussions on various incarnations of Arakelov Divisors, discussions on cyclotomic synchronization; Chung Pang Mok for productive discussions on the p-adic logarithm, anabelian evaluation, indeterminacies, the theta link, and hodge theaters; Thomas Scanlon for discussions regarding interpretations and infinitary logic as applied to IUT and anabelian geometry. We apologize if we have forgotten anybody.
The research discussed in the present paper profited enormously from the generous support of the International Joint Usage/Research Center (iJU/RC) located at Kyoto Universities Research Institute for Mathematical Sciences (RIMS) as well as the Preparatory Center for Research in Next-Generation Geometry located at RIMS. (引用終り) 以上 0211１３２人目の素数さん2020/05/07(木) 08:25:03.69ID:A9ti5Rfb>>207
（参考） http://www.kurims.kyoto-u.ac.jp/~motizuki/research-japanese.html 望月 過去と現在の研究 ・南出新氏による、IUTeichにおける明示的な不等式に関する講演のスライドを掲載 http://www.kurims.kyoto-u.ac.jp/~motizuki/Minamide%20---%20Explicit%20estimates%20in%20inter-universal%20Teichmuller%20theory%20(in%20progress).pdf Explicit estimates in inter-universal Teichm¨uller theory (in progress) (joint work w/ I. Fesenko, Y. Hoshi, S. Mochizuki, and W. Porowski) Arata Minamide RIMS, Kyoto University November 2, 2018
§0 Notations
V(F)non: the set of nonarchimedean places of F V(F)arc: the set of archimedean places of F V(F) def = V(F)non ∪ V(F)arc
§1 Introduction Main theorem of IUTch: There exist “multiradial representations”? i.e., description up to mild indeterminacies in terms that make sense from the point of view of an alien ring structure ? of the following data:
⇒ As an application, we obtain a diophantine inequality.
P7 Theorem (ABC Conjecture for number fields) Note: We do not know the constant “C(d, ?)” explicitly. For instance, it is hard to compute noncritical Belyi maps explicitly!
P8 Goal of this joint work: Under certain conditions, we prove (＊) directly [i.e., without applying the theory of noncritical Belyi maps] to compute the constant “C(d, ?)” explicitly. Technical Difficulties of Explicit Computations (i) We cannot use the compactness of “K” at the place 2 ⇒ We develop the theory of ´etale theta functions so that it works at the place 2 (ii) We cannot use the compactness of “K” at the place ∞ ⇒ By restricting our attention to “special” number fields, we “bound” the archimedean portion of the “height” of the elliptic curve “Eλ”
P9 §2 Theta Functions
P10 Now we have the following sequence of log tempered coverings:
P11 ? Next, we recall the def’n of the theta function Θ¨ .
P14 We want to develop the theory of Θ functions in the case of p = 2. ⇒ In this work, instead of “2-torsion points”, we consider 6-torsion points of X(K)!
P15 §3 Heights First, we recall the notion of the Weil height of an algebraic number.
P21 §5 Expected Main Results Expected Theorem (Effective ABC for mono-complex number fields)
Expected Corollary (Application to Fermat’s Last Theorem) ∃ explicitly computable n0 ∈ Z?3 s.t. if n ? n0, then no triple (x, y, z) of positive integers satisfies x^n + y^n = z^n (引用終り) 以上 0216１３２人目の素数さん2020/05/07(木) 11:51:51.75ID:A9ti5Rfb>>213
（参考） http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html 望月 最新情報 2012年08月30日 ・（論文）新論文を掲載： Inter-universal Teichmuller Theory I: Construction of Hodge Theaters. Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation. Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice. Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations. 0220１３２人目の素数さん2020/05/07(木) 13:13:57.71ID:AZEZWtke>>218 どうも コメントありがとう
Long proofs The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. ・1799 The Abel?Ruffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. ・1963 Odd order theorem by Feit and Thompson was 255 pages long, which at the time was over 10 times as long as what had previously been considered a long paper in group theory. ・1964 Resolution of singularities Hironaka's original proof was 216 pages long; it has since been simplified considerably down to about 10 or 20 pages. ・2000 Lafforgue's theorem on the Langlands conjecture for the general linear group over function fields. Laurent Lafforgue's proof of this was about 600 pages long, not counting many pages of background results. ・2003 Poincare conjecture, Geometrization theorem, Geometrization conjecture. Perelman's original proofs of the Poincare conjecture and the Geometrization conjecture were not lengthy, but were rather sketchy. Several other mathematicians have published proofs with the details filled in, which come to several hundred pages. ・2004 Classification of finite simple groups. The proof of this is spread out over hundreds of journal articles which makes it hard to estimate its total length, which is probably around 10000 to 20000 pages. 0230現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 07:57:31.21ID:g/NZ4Ytw>>227 >面白いけどムズカシすぎて無理。
4章＝IUT IVだね、きっと そして、P67のSection 3の下記引用部分だね （参考） http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: ¨ LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 P67 Section 3: Inter-universal Formalism: the Language of Species In the following discussion, we shall work with various models - consisting of “sets” and a relation “∈” - of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].
つづき The various ZFC-models that we work with may be thought of as [but are not restricted to be!] the ZFC-models determined by various universes that are sets relative to some ambient ZFC-model which, in addition to the standard axioms of ZFC set theory, satisfies the following existence axiom [attributed to the “Grothendieck school” ? cf. the discussion of [McLn], p. 193]: (†G) Given any set x, there exists a universe V such that x ∈ V . We shall refer to a ZFC-model that also satisfies this additional axiom of the Grothendieck school as a ZFCG-model. This existence axiom (†G) implies, in particular, that: (引用終り) 0232現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 08:06:13.09ID:g/NZ4Ytw>>230 >俺はZFC公理系の公理の数が9個であるとする論文は見たことがないが、そうする一般向け解説は時-見るから、望月は専門外(笑)なので間違えてる可能性の方が高いと結論せざるを得ない
１．ZFC公理系の公理の数が9個ではなく、 今の論文では、”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” だから、普通には、ZFが9個でしょ？ それは、下記のZF wikipedia の9個と合う ２．かつ、” - cf., e.g., [Drk], Chapter 1, §3]”と書いてあるから、” [Drk], Chapter 1, §3]”をチェックしての発言なのかな？ 自分は[Drk]をチェックする気が無いけどｗ ３．だから、ZFが9個で、ZFCなら10個って話かな？ 元の2012年版の記憶で書いているのかな？ 意味不明ですね（＾＾；
Contents 1 History 2 Axioms 2.1 1. Axiom of extensionality 2.2 2. Axiom of regularity (also called the axiom of foundation) 2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension) 2.4 4. Axiom of pairing 2.5 5. Axiom of union 2.6 6. Axiom schema of replacement 2.7 7. Axiom of infinity 2.8 8. Axiom of power set 2.9 9. Well-ordering theorem 3 Motivation via the cumulative hierarchy 0233現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 08:15:40.86ID:g/NZ4Ytw>>231 >Grothendieck school as a ZFCG-model. This existence axiom (†G) implies, in particular, that:
Contents 1 History 2 Axioms 2.1 1. Axiom of extensionality 2.2 2. Axiom of regularity (also called the axiom of foundation) 2.3 3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension) 2.4 4. Axiom of pairing 2.5 5. Axiom of union 2.6 6. Axiom schema of replacement 2.7 7. Axiom of infinity 2.8 8. Axiom of power set 2.9 9. Well-ordering theorem 3 Motivation via the cumulative hierarchy (引用終り) 0237１３２人目の素数さん2020/05/08(金) 10:27:04.19ID:enp/+yz7>>236 >”axiom”を１個と数えれば
ほいよ >>236 あなたも、ここで論陣を張りたければ、まずは事実を確認してくださいね まずは、望月氏 IUT IVが引用している >>230の”cf., e.g., [Drk], Chapter 1, §3”を見ましょうね （P85 Bibliography [Drk] F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). です）
V で作業する代わりに、可算推移モデル M と (P,≤,1) ∈ Mを考える。 ここで言うモデルというのはZFCの十分多くの”有限個”の公理を満たすものを言う。 推移性というのは x ∈ y ∈ M ならば x ∈ Mとなることである。 0241１３２人目の素数さん2020/05/08(金) 10:59:47.25ID:qXGvfbUV>>238 >あなたも、ここで論陣を張りたければ、まずは事実を確認してくださいね >まずは、望月氏 IUT IVが引用している >>230の”cf., e.g., [Drk], Chapter 1, §3”を見ましょうね
F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). 静岡大学附属図書館と 新潟大学附属図書館とがヒットしますね（＾＾；
アマゾン/Set-Theory-Introduction-Foundations-Mathematics/dp/0720422795 Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics) (英語) ハードカバー ? 1974/10/1 F. R. Drake (著)
Set theory : an introduction to large cardinals | 静岡大学附属図書館 ...opac.lib.shizuoka.ac.jp ? opacid Google Books. ブックマーク済み. Set theory : an introduction to large cardinals ... North-Holland, 1974; 形態: xii, 351 p. ; 23 cm; 著者名: Drake, F. R. (Frank ... シリーズ名: Studies in logic and the foundations of mathematics ; v. 76 ... 410.8/179/76.
Set theory : an introduction to large cardinals | 新潟大学附属図書館 ...opac.lib.niigata-u.ac.jp ? opc ? recordID ? catalog.bib 9780720422795 [0720422795] (North-Holland) CiNii Books Webcat Plus Google Books; シリーズ名: Studies in logic and the foundations of mathematics ; v. 76 ... 0242１３２人目の素数さん2020/05/08(金) 11:09:04.02ID:qXGvfbUV>>240 >＞私は、[Drk]に何を書いてあるかは知らない >じゃ、調べたら？
>>239より 望月氏は ”[i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice - cf., e.g., [Drk], Chapter 1, §3].” これ、”- cf., e.g., [Drk], Chapter 1, §”を、「[Drk]には、こう書いてあるけれども」と、軽く読めば良いんじゃない？（＾＾
”Well-ordering theorem”は、最初 Zermeloは定理だと考えていたのですね で、下記のように、１階述語論理では、選択公理や Zorn's Lemmaと equivalentだと （ここまでは 学部生でも常識でしょうね） しかし、２階述語論理では、strictly stronger than the axiom of choice だと なるほどね（＾＾；
（参考） https://en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem (抜粋) "Zermelo's theorem" redirects here. For Zermelo's theorem in game theory, see Zermelo's theorem (game theory). Not to be confused with Well-ordering principle.
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).[1][2]
History
It turned out, though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo?Fraenkel axioms is sufficient to prove the other, in first order logic (the same applies to Zorn's Lemma). In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.[7]
There is a well-known joke about the three statements, and their relative amenability to intuition:
The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?[8] 0246１３２人目の素数さん2020/05/08(金) 11:44:00.27ID:qXGvfbUV>>243-244 私は、別に望月先生を擁護をする気はないけど、あなたの言うことは、本筋からずれているよね
追加 これ、現代では大分見直しされているようですね（＾＾； https://en.wikipedia.org/wiki/Axiom_schema_of_specification Axiom schema of specification (抜粋) In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Godel considered it the most important axiom of set theory.
Relation to the axiom schema of replacement The axiom schema of separation can almost be derived from the axiom schema of replacement. For this reason, the axiom schema of specification is often left out of modern lists of the Zermelo?Fraenkel axioms. However, it's still important for historical considerations, and for comparison with alternative axiomatizations of set theory, as can be seen for example in the following sections. Unrestricted comprehension Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other Zermelo?Fraenkel axioms (but not the axiom of extensionality, the axiom of regularity, or the axiom of choice) then became necessary to make up for some of what was lost by changing the axiom schema of comprehension to the axiom schema of specification ? each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy, i.e. it is a special case of the axiom schema of comprehension. 0253現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 13:52:17.90ID:qXGvfbUV>>251 粋蕎さん、どうも コテ抜けてたなｗ（＾＾； お元気そうですねｗ
どうも 全く同意 基礎論のプロ数学者の専門家がいうならともかくも ド素人がイチャモン付けるなら せめて 原典の>>241 F. R. Drake, Set Theory: an Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, North-Holland (1974). くらいは、当たってからにしてくれよ、おい って話ですねｗ（＾＾； 0255１３２人目の素数さん2020/05/08(金) 14:12:40.79ID:t31dz+7K>>254 海外でも随分格下の人とか専門外の人がいちゃもんつけてるのがな。Scholzeの肩に乗りたいだけの人とか。情けない。 0256現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 14:13:49.22ID:qXGvfbUV>>249
（参考） http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: ¨ LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020 P67 In the present §3, we develop ? albeit from an extremely naive/non-expert point of view, relative to the theory of foundations! ? the language of species. Roughly speaking, a “species” is a “type of mathematical object”, such as a “group”, a “ring”, a “scheme”, etc. 0282１３２人目の素数さん2020/05/08(金) 16:23:09.96ID:enp/+yz7 望月は、欅坂４６の「サイレントマジョリティ」の歌詞が IUTを考案した自分の心情と合致しているとブログに書いてるが
https://ncatlab.org/nlab/show/species species (抜粋) Contents 1. Idea 2. Definition 1-categorical 2-categorical (∞,1)-categorical Operations on species Sum Cauchy product Hadamard product Dirichlet product Composition product 3. In Homotopy Type Theory Operations on species Coproduct Hadamard product Cauchy product Composition 4. Properties Cardinality 5. Variants
1. Idea A (combinatorial) species is a presheaf or higher categorical presheaf on the groupoid core(FinSet), the permutation groupoid. A species is a symmetric sequence by another name. Meaning: they are categorically equivalent notions.
https://en.wikipedia.org/wiki/Combinatorial_species Combinatorial species (抜粋) In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are (finite) graphs, permutations, trees, and so on; each of these has an associated generating function which counts how many structures there are of a certain size.
Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species.
The category of species is equivalent to the category of symmetric sequences in finite sets.[1]
.Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species. (引用終り) 以上 0285現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 17:32:18.62ID:qXGvfbUV>>276 追加
.Category theory provides a useful language for the concepts that arise here, but it is not necessary to understand categories before being able to work with species. ↓ １行削除 なんか同じ行で、ダブっているね（＾＾； 0287現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/08(金) 17:56:35.55ID:qXGvfbUV>>266 追加 > 17，8の時にSGA読んでたとか凄いな。望月さん。
