Lorscheid, Oliver (2016), "A blueprinted view on F1-geometry", in Koen, Thas (ed.), Absolute arithmetic and F1-geometry, European Mathematical Society Publishing House, arXiv:1301.0083 Lorscheid, Oliver (2018b), "The geometry of blueprints part II: Tits-Weyl models of algebraic groups", Forum of Mathematics, Sigma, 6, arXiv:1201.1324
なので、”blueprint”の文献は、Lorscheid, Oliver (2018b)とかの方が適切な気がする
https://arxiv.org/abs/1602.01431 A heuristic for boundedness of ranks of elliptic curves Jennifer Park, Bjorn Poonen, John Voight, Melanie Matchett Wood (Submitted on 3 Feb 2016 (v1), last revised 10 Jul 2018 (this version, v3)) https://arxiv.org/pdf/1602.01431.pdf
>>684 追加参考 ”In particular, the model suggests that all but finitely many elliptic curves over Q have rank <= 21, which would imply that the rank is uniformly bounded.” ”submitted to the Proceedings of the 2018 ICM”だって（＾＾
https://arxiv.org/abs/1711.10112 Heuristics for the arithmetic of elliptic curves Bjorn Poonen (Submitted on 28 Nov 2017 (v1), last revised 30 Nov 2017 (this version, v2))
This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over Q have rank ?21, which would imply that the rank is uniformly bounded.
Comments: 13 pages; submitted to the Proceedings of the 2018 ICM. Version 2 adds some references and corrects some typos
https://arxiv.org/abs/1810.05337 Constraints on symmetry from holography Daniel Harlow, Hirosi Ooguri (Submitted on 12 Oct 2018 (v1), last revised 6 Jun 2019 (this version, v2)) In this letter we use the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence to establish a set of old conjectures about symmetries in quantum gravity. These are that no global symmetries are possible, that internal gauge symmetries must come with dynamical objects that transform in all irreducible representations, and that internal gauge groups must be compact. These conjectures are not obviously true from a bulk perspective, they are nontrivial consequences of the non-perturbative consistency of the correspondence. More details of and background for these arguments are presented in an accompanying paper. https://arxiv.org/pdf/1810.05337.pdf
https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/ Almost all Collatz orbits attain almost bounded values 10 September, 2019 By Terence Tao (抜粋) I’ve just uploaded to the arXiv my paper “Almost all Collatz orbits attain almost bounded values“, submitted to the proceedings of the Forum of Mathematics, Pi. In this paper I returned to the topic of the notorious Collatz conjecture (also known as the {3x+1} conjecture), which I previously discussed in this blog post.
https://arxiv.org/abs/1909.03562 Almost all orbits of the Collatz map attain almost bounded values Terence Tao (Submitted on 8 Sep 2019 (v1), last revised 13 Sep 2019 (this version, v2))
Title: Szpiro's Inequality and Anabelian Constructions Abstract: Szpiro's conjecture relates two important quantities associated to elliptic curves, namely its minimal discriminant and its conductor. Applied to the Frey curve, it implies the abc conjecture of Oesterle and Masser. We give an overview of the inequality proposed by Shinichi Mochizuki and present work done with Taylor Dupuy making explicit the nature and effect of the indeterminacies that play an important role in the approach as well as computations assuming Corollary 3.12 of Mochizuki's IUT III paper.
www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf August 2018 Report by the other participants in the March 2018 discussions Why abc is still a conjecture PETER SCHOLZE AND JAKOB STIX 1.2. Frey curves
851 名前：8, 3-4:30 p.m. Lafayette L307 Anton Hilado, Elliptic Curves and the abc Conjecture In this talk we state the famous "abc conjecture" of Masser and Oesterle, and explain how it can be formulated as a statement involving important quantities related to elliptic curves (Szpiro's conjecture). We give an introduction to Weierstrass equations, reduction types, and the conductor and minimal discriminant of an elliptic curve, which are all needed to state Szpiro's conjecture. We also show how the abc conjecture is related to Fermat's Last Theorem, and introduce the Frey curve, which was used to prove the latter, and relate Szpiro's conjecture to the abc conjecture. Slides http://www.uvm.edu/~unqvnts/Talk%201%20%28Szpiro%27s%20Conjecture%29.pdf Elliptic Curves and the abc Conjecture Anton Hilado University of Vermont October 16, 2018 []
Thursday, September 20, 2018, 3-4:30 p.m. Taylor Dupuy, Mochizuki's Inequality and the ABC Conjecture Mochizuki's approach to the ABC conjecture is to prove an inequality which implies the Szpiro inequality for elliptic curves under certain technical hypotheses called "initial theta data" show that these technical restrictions don't matter. The aim of this talk is to explain exactly what step 1 is all about. Roughly, for an elliptic curve in "initial theta data", Mochizuki's inequality says that the size of one region (encoding one side of Szpiro) is less than the size of a "blurry" region (encoding the other side of Szpiro).
I will explain what these regions are and how they relate to Szpiro explicitly. In particular we will discuss "indeterminacies", "q-pilots", "theta-pilots", and "initial theta data". Later in the semester we will discuss the anabelian constructions that go into the "blurry construction". This talk is supposed to set up future talks down the road. Much of this project of making these inequalities explicit is joint work with Anton Hilado.
Thursday, October 18, 2018, 3-4:30 p.m. Taylor Dupuy, Log Volume Computations We are going to continue discussing Mochizuki's inequality. In particular we will discuss the indeterminacies Ind1,Ind2,Ind3 and start in on the log-volume computations which give rise to a version of Szpiro's inequality for elliptic curves sitting in initial theta data.
Thursday, November 15, 2018, 3-4:30 p.m. Taylor Dupuy, More Log Volume Computations We will perform computations similar to the computations in IUT4 using Mochizuki's Inquality (Corollary 3.12 of IUT3) and the definitions of the indeterminacies therein to give a Szpiro-type inequality for Elliptic Curves in initial theta data (Theorem 1.10 of IUT4).
＜Frey-Hellegouarch curve＞ https://ncatlab.org/nlab/show/Mochizuki%27s+proof+of+abc Mochizuki's proof of abc Last revised on March 14, 2013 at 19:26:24. See the history of this page for a list of all contributions to it.
The infinite set of counterexamples. They come from Masser’s paper Masser: Note on a conjecture of Szpiro,Asterisque* 1990, as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over Q whose conductor N and minimal discriminant Δ satisfy
Take the pair (E,L), where E/Q is a semistable elliptic curve with (say, for the sake of simplifying) rational 2-torsion (i.e., a Frey-Hellegouarch curve) of minimal discriminant Δ and conductor N (square-free). Assume that:
（参考） https://en.wikipedia.org/wiki/Frey_curve Frey curve (抜粋) Frey curve or Frey?Hellegouarch curve is the elliptic curve y^2=x(x-a^l)(x+b^l) associated with a (hypothetical) solution of Fermat's equation a^l+b^l=c^l. The curve is named after Gerhard Frey.
History Yves Hellegouarch (1975) came up with the idea of associating solutions (a,b,c)}(a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve. If ? is an odd prime and a, b, and c are positive integers such that a^l+b^l=c^l, then a corresponding Frey curve is an algebraic curve given by the equation y^2=x(x-a^l)(x+b^l) or, equivalently y^2=x(x-a^l)(x-c^l). This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q.
(Gerhard Frey 1982) called attention to the unusual properties of the same curve as Hellegouarch, which became called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama?Shimura?Weil conjecture implies Fermat's Last Theorem. However, his argument was not complete. In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama?Shimura?Weil conjecture implies Fermat's Last Theorem. []
（参考：これ分り易いかも） https://inference-review.com/article/fukugen Fukugen Ivan Fesenko Published on September 28, 2016 in Volume 2, Issue 3. On Shinichi Mochizuki’s Inter-universal Teichmuller Theory (抜粋) THE SUNLIGHT is strong in Kyoto, even in winter. In December of 2014, I visited Shinichi Mochizuki at the Research Institute for Mathematical Sciences to discuss his inter-universal Teichmuller theory (IUT).1 A distinguished mathematician and a leading figure in anabelian geometry, Mochizuki first made his papers about IUT available at the end of August, 2012. Their study has proved challenging.
A term that is frequently used in mathematical discussions about anabelian geometry and IUT is fukugen, which may be translated as restoration or as reconstruction, and which, like so many words in a foreign language, cannot be truly translated.
It must be used without translation. But isn’t this true of mathematics itself?
IUT contributes to a new view of the numbers. This may sound as if Mochizuki had announced, rather than executed, a program in pure mathematics. But IUT yields proofs of several outstanding problems in number theory: the strong Szpiro conjecture for elliptic curves, Vojta’s conjecture for hyperbolic curves, and the Frey conjecture for elliptic curves.
And it settles the famous Oesterle?Masser or abc conjecture.2
The abc conjecture is easy to state and difficult to prove. Prime numbers are defined in terms of multiplication in the ring of integers.
>>751 ＜Frey curve＞ https://en.wikipedia.org/wiki/Abc_conjecture#cite_ref-1 abc conjecture (抜粋) The precise statement is given below. The abc conjecture originated as the outcome of attempts by Oesterle and Masser to understand the Szpiro conjecture about elliptic curves.[1]
Citations [1] https://www.maths.nottingham.ac.uk/plp/pmzibf/notesoniut.pdf Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF), European Journal of Mathematics, 1 (3): 405?440, doi:10.1007/s40879-015-0066-0. (抜粋) P4 1.3. Conjectural inequalities for the same property. (a) the effective Mordell conjecture ? a conjectural extension of the Faltings?Mordell theorem which involves an effective bound on the height of rational points of the curve C over the number field K in the Faltings theorem in terms of data associated to C and K, (b) the Szpiro conjecture, see below, (c) the Masser?Oesterle conjecture, a.k.a. the abc conjecture (whose statement over Q is well known^6 , and which has an extension to arbitrary algebraic number fields, see Conj. 14.4.12 of [6]), (d) the Frey conjecture, see Conj. F.3.2(b) of [15], (f) arithmetic Bogomolov?Miyaoka?Yau conjectures (there are several versions).
The Szpiro conjecture was stated several years before^7 the work of Faltings, who learned much about the subject related to his proof from Szpiro. Using the Frey curve^8, it is not difficult to show that (c) and (d) are equivalent and that they imply (b), see e.g. see sect. F3 of [15] and references therein. Using Belyi maps as in 1.1, one can show the equivalence of (c) and (a). For the equivalence of (c) and (e) see e.g. Th. 14.4.16 of [6] and [47]. For implications (e) ⇒ (f) see [48].
Footnote ^8 y^2 = x(x+a)(x?b) where a,b,a+b are non-zero coprime integers
脚注 3^D. Goldfeld, Modular forms, elliptic curves, and the ABC-conjecture. www.math.columbia.edu/~goldfeld/ DORIAN GOLDFELD Professor Mathematics Columbia University New York www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf MODULAR FORMS, ELLIPTIC CURVES AND THE ABC?CONJECTURE Dorian Goldfeld? Dedicated to Alan Baker on the occasion of his sixtieth birthday （1999） (抜粋) P7 Consider the Frey?Hellegouarch curve EA,B : y^2 = x(x - A)(x + B). A minimal model for EA,B has discriminant (ABC)^2・2^-s and conductor N・2^-t for certain absolutely bounded integers s, t, (see Frey [F1]). Plugging this data into Szpiro’s conjecture immediately shows the equivalence.
追加 https://en.wikipedia.org/wiki/Dorian_M._Goldfeld (抜粋) Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in analytic number theory and automorphic forms at Columbia University.
Professional career Goldfeld received his B.S. degree in 1967 from Columbia University. His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia. He has held positions at the University of California at Berkeley (Miller Fellow, 1969?1971), Hebrew University (1971?1972), Tel Aviv University (1972?1973), Institute for Advanced Study (1973?1974), in Italy (1974?1976), at MIT (1976?1982), University of Texas at Austin (1983?1985) and Harvard (1982?1985). Since
870 名前：1985, he has been a professor at Columbia University.[1]
His work on the Birch and Swinnerton-Dyer conjecture includes the proof of an estimate for a partial Euler product associated to an elliptic curve,[12] bounds for the order of the Tate?Shafarevich group[13]
Awards and honors In 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the prizes in Number Theory, for his solution of Gauss' class number problem for imaginary quadratic fields. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley. []
この日本文は不正確 正確には、下記英文 なお、下記(a, b, c, n)は (a, b, c, p)が正確かもね、y^2 = x (x - a^p)(x + b^p)だからね https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem Fermat's Last Theorem (抜粋) Contents 2.5 Connection with elliptic curves
Ribet's theorem for Frey curves Main articles: Frey curve and Ribet's theorem In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution (a, b, c) for exponent p > 2, then it could be shown that the semi-stable elliptic curve (now known as a Frey-Hellegouarch[note 3]) y^2 = x (x - a^p)(x + b^p) would have such unusual properties that it was unlikely to be modular.[122] This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey observed that a proof of the Taniyama?Shimura?Weil conjecture might also simultaneously prove Fermat's Last Theorem.[123] By contraposition, a disproof or refutation of Fermat's Last Theorem would disprove the Taniyama?Shimura?Weil conjecture.
In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of 4 numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama?Shimura?Weil conjecture. Therefore if the latter were true, the former could not be disproven, and would also have to be true.
Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem ? or at least to prove it for the types of elliptical curves that included Frey's equation (known as semistable elliptic curves). This was widely believed inaccessible to proof by contemporary mathematicians.[121]:203?205, 223, 226 Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was plausible but did not go as far as giving a full proof. The missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was identified by Jean-Pierre Serre who also gave an almost-complete proof and the link suggested by Frey was finally proved in 1986 by Ken Ribet.[124] (引用終り) 以上
（英文） https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem Wiles's proof of Fermat's Last Theorem (抜粋) Contents 1 Precursors to Wiles' proof 1.1 Fermat's Last Theorem and progress prior to 1980 1.2 The Taniyama?Shimura?Weil conjecture 1.3 Frey's curve 1.4 Ribet's theorem 1.5 Situation prior to Wiles' proof 2 Andrew Wiles 3 Announcement and subsequent developments 3.1 Announcement and final proof (1993?1995) 3.2 Subsequent developments 4 Summary of Wiles' proof 5 Mathematical detail of Wiles proof 5.1 Overview 5.2 General approach and strategy 5.3 3-5 trick 5.4 Structure of Wiles's proof 5.5 Overviews available in the literature 6 References
>>748 ＞Roughly, for an elliptic curve in "initial theta data", Mochizuki's inequality says that the size of one region (encoding one side of Szpiro) is less than the size of a "blurry" region (encoding the other side of Szpiro).
"blurry" region って、IUTの正式用語なの？ｗ（＾＾；
（>>747より） www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf August 2018 Report by the other participants in the March 2018 discussions Why abc is still a conjecture PETER SCHOLZE AND JAKOB STIX (抜粋) P10の最後 We voiced these concerns in this form at the end of the fourth day of discussions. On the fifth and final day, Mochizuki tried to explain to us why this is not a problem after all. In particular, he claimed that up to the “blurring” given by certain indeterminacies the diagram does commute; it seems to us that this statement means that the blurring must be by a factor of at least O(l^2) rendering the inequality thus obtained useless. (引用終り)
とあって、SCHOLZEは、“blurring”を攻撃しているように見えたんだ で、Mochizukiが苦し紛れの口頭説明に、“blurring”を言い出したと見ていたんだけどねぇ（原論文は、もちろん読んでないからさｗ） へー、"blurry" region って、IUTの正式用語なの？へー ｗ（＾＾；
で、やっぱ、Mochizuki氏が苦し紛れの口頭説明に、“blurring”を言い出したみたい でも、>>748のTaylor Dupuy Thursday, September 20 unQVNTS (Vermont) 2018では ”Mochizuki's inequality says that the size of one region (encoding one side of Szpiro) is less than the size of a "blurry" region (encoding the other side of Szpiro).” ”I will explain what these regions are and how they relate to Szpiro explicitly. In particular we will discuss "indeterminacies", "q-pilots", "theta-pilots", and "initial theta data". Later in the semester we will discuss the anabelian constructions that go into the "blurry construction". なんだね
同僚のAnton Hilado氏 と ”This talk is supposed to set up future talks down the road. Much of this project of making these inequalities explicit is joint work with Anton Hilado.”
Anton Hiladoさん、フィリピンからか Hiladoは、平戸かもしらんね F1体について解説している記事があるな（＾＾；
https://ahilado.wordpress.com/author/ahilado/ THEORIES AND THEOREMS Math and Physics for Everyone AUTHOR: ANTON HILADO
https://ahilado.wordpress.com/about/ About My name is Anton Hilado. I am a mathematics Ph.D. student at the University of Vermont under Taylor Dupuy. Before that, I completed my master’s degree in physics from the University of the Philippines, where my research was on mathematical physics under Eric Galapon.
Taylor Dupuy先生、2019年にIUTのPDFを２つ作っているね 内容は、殆ど同じかも ダウンロード↓のマークが、右上にあって、PDFをダウンロードできた（＾＾
両PDFとも、冒頭に ”WARNINGS!!! Normalizations/constant/index mistakes happen!! Double check before applying statements from these slides! We use simplifying assumptions! Look at references at the end. Many people have helped me understand these things and deserve academic credit!” みたいに出てくるのは、 Scholzeに対する反論の意味かいな？（＾＾；
https://www.uvm.edu/~tdupuy/talks.html [ Taylor Dupuy's Homepage]
https://www.dropbox.com/s/b63figtzrg86u5t/IUTFAQ.pdf? Spring 2019, University of Tennesse Knoxville, Barrett Lectures A User's Guide to Mochizuki's Inequality
https://www.dropbox.com/s/lye9hgeqqgpqsy6/IUTFAQ.pdf?dl=0 Spring 2019, Rice, AGNT Seminar Explicit Computations in IUT
上記以外に、PDFリンクなしの表題もある Fall 2018, unQVNTS (Three Talks) Mochizuki's Inequalities Spring 2018, UConn, CTNT Summer School Mochizuki's Inequalities Spring 2017, Purdue, Automorphic Forms Seminar Indeterminacies in IUT Summer 2016, IUT Summit, RIMS Kyoto Introduction to IUT2. Multiiradiality. Spring 2016, University of Copenhagen, Number Theory Seminar Some Constructions Used in Mochizuki's IUT
>>770より SSレポートで www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf August 2018 Report by the other participants in the March 2018 discussions Why abc is still a conjecture PETER SCHOLZE AND JAKOB STIX (抜粋) P10の最後 We voiced these concerns in this form at the end of the fourth day of discussions. On the fifth and final day, Mochizuki tried to explain to us why this is not a problem after all. In particular, he claimed that up to the “blurring” given by certain indeterminacies the diagram does commute; it seems to us that this statement means that the blurring must be by a factor of at least O(l^2) rendering the inequality thus obtained useless. (引用終り)
（参考）（下記で“blur”で検索掛けたがヒットなし） www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf REPORT ON DISCUSSIONS, HELD DURING THE PERIOD MARCH 15 ? 20, 2018, CONCERNING INTER-UNIVERSAL TEICHMULLER THEORY (IUTCH) Shinichi Mochizuki February 2019
https://www.dropbox.com/s/b63figtzrg86u5t/IUTFAQ.pdf? Spring 2019, University of Tennesse Knoxville, Barrett Lectures A User's Guide to Mochizuki's Inequality
このPDFで September 10, 2019 という日付で P29に People Who Have Explained Things to Me: Fesenko, Hoshi, Joshi,Kedlaya, Lepage, Mochizuki, Mok, Saidi, Scanlon, Thakur, Voloch
>>808 ＞Before that, I completed my master’s degree in physics from the University of the Philippines, where my research was on mathematical physics under Eric Galapon.
Anton Hiladoちゃん、もと物理屋 ”where my research was on mathematical physics under Eric Galapon”なんだ（＾＾ 主流じゃんか（＾＾；
例えば、>>810より下記のページご参照（＾＾ （注：PDFスライドは、全体で86枚あるが、ページ付けは30ページである） ”Results with
933 名前：Hilado”と、Hilado氏の成果を強調しているんだ そして、P17で、Scholze-Stixの議論にも触れている（Taylor Dupuy先生は、勿論望月支持でしょうね） https://www.dropbox.com/s/b63figtzrg86u5t/IUTFAQ.pdf? Spring 2019, University of Tennesse Knoxville, Barrett Lectures A User's Guide to Mochizuki's Inequality
P5 Results with Hilado Theorem (Dupuy-Hilado, Mochizuki) Assume IUT3 Corollary 3.12. There exists some constant A0 such that for every prime ` we have
P6 Computations with Hilado Bounds showing dependence on the eld of `-torsion: Theorem (Dupuy-Hilado, Mochizuki) Assume IUT3 Corollary 3.12. If K = F(EF0 [`]) then we have
P17 Scholze-Stix Naive Relation between Pq and Pθ: Scholze-Stix assert that this relation is the only relation. Mochizuki interprets what they say as a congruence.
P19 (People looking at the slides online: look at the expanded discussion around these points both documents.) []
（再録） https://www.dropbox.com/s/b63figtzrg86u5t/IUTFAQ.pdf? Spring 2019, University of Tennesse Knoxville, Barrett Lectures A User's Guide to Mochizuki's Inequality P17 Scholze-Stix Naive Relation between Pq and Pθ: Scholze-Stix assert that this relation is the only relation. Mochizuki interprets what they say as a congruence. (引用終り)
P17に、こんな式がある deg_lgp(Pθ)=(l(l+1)/12)deg_Fo(Pq)
Scholzeは、”this relation is the only relation.” Mochizukiは、”interprets what they say as a congruence”だと
congruenceを、下記 ”Pq and Pθ”が、”【数学】 (2 図形の)合同”みたいな意味で 使っているのかな？？ 詳しいことは、さっぱり分かりませんが（＾＾；
www.uvm.edu/~tdupuy/anabelian/VermontNotes_20.pdf (抜粋) KUMMER CLASSES AND ANABELIAN GEOMETRY JACKSON S. MORROW Date: April 29, 2017. ABSTRACT. These notes comes from the Super QVNTS: Kummer Classes and Anabelian geometry. Any virtues in the notes are to be credited to the lecturers and not the scribe; however, all errors and inaccuracies should be attributed to the scribe. That being said, I apologize in advance for any errors (typo-graphical or mathematical) that I have introduced.
CONTENTS 1. On Mochizuki’s approach to Diophantine inequalities Lecturer: Kiran Kedlaya . . . . . . 2 2. Why the ABC Conjecture? Lecturer: Carl Pomerance . . . . . 3 3. Kummer classes, cyclotomes, and reconstructions (I/II) Lecturer: Kirsten Wickelgren . . . . . 3 4. Kummer classes, cyclotomes, and reconstructions (II/II) Lecturer: David Zureick-Brown . . . . . 6 5. Overflow session: Kummer classes Lecturer: Taylor Dupuy . . . . . 8 6. Introduction to model Frobenioids Lecturer: Andrew Obus . . . . . 11 7. Theta functions and evaluations Lecturer: Emmanuel Lepage . . . . . . 13 8. Roadmap of proof Notes from an email from Taylor Dupuy . . . . 17 References . . . . . . 19
6. INTRODUCTION TO MODEL FROBENIOIDS LECTURER: ANDREW OBUS
By way of introduction, Mochizuki loosely defines a Frobenioid as a category theoretic abstraction of divisors or line bundles on a geometry object. Our main example will be an abstract category which encodes etale coverings and information concerning divisors.
https://en.wikipedia.org/wiki/Pi (抜粋) British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[82]
面白いところを抜粋（＾＾； www.kurims.kyoto-u.ac.jp/~narutaka/lists.html 小沢 登高 Narutaka OZAWA コメント付き論文リスト (抜粋) [5] Almost completely isometric embeddings between preduals of von Neumann algebras. J. Funct. Anal., 186 (2001), 329--341. doi:10.1006/jfan.2001.3796 dvi これもNgが出した問題。99年10月にやった。あっちを叩けばこっちが出っ張るという状況の中、3週間ぐらい集中した。それまでの証明はすべて気合一発でやっていたが、これは方針を立ててひとつずつ証明していった。Kirchbergと無関係なネタもこれが初めて。
[6] On the set of finite-dimensional subspaces of
1026 名前：preduals of von Neumann algebras. C. R. Acad. Sci. Paris Ser. I Math., 331 (2000), 309--312. doi:10.1016/S0764-4442(00)01646-3 dvi また気合一発で証明してしまった。きれいな結果ではあるが、何かの役に立つことはなさそう。
[7] Amenable actions and exactness for discrete groups. C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 691--695. doi:10.1016/S0764-4442(00)00248-2 math.OA/0002185 Paris第六大学に滞在していたとき、Guentner-Kaminkerのプレプリントを読んでいて、彼らの重要な見落としに気が付いた。この論文のおかげで、作用素空間の外でも名前が売れた。数学においても、競争という側面を無視するわけにはいかない。
[10] An application of expanders to B(\ell_2)\otimes B(\ell_2). J. Funct. Anal., 198 (2003), 499--510. doi:10.1016/S0022-1236(02)00107-6 math.OA/0110151 01年8月末にTAMUから東京に戻って数日後、Pisierから某氏がQWEP予想を解いたと主張していることを知らされる。さっそくプレプリントをダウンロードして読んでみたら、実際にはさらに強い(疑わしい)主張が述べてあった。反例を探すこと一ヵ月でようやく見つかった。 反例の構成法をいじったところ、思いがけず別の問題が解けたので、論文にした。数学というものは、勢いさえあれば何らかの結果にぶつかるものだと思った。
[18] A note on non-amenability of B(\ell_p) for p=1,2. Internat. J. Math., 15 (2004), 557--565. doi:10.1142/S0129167X04002430 math.FA/0401122 正確な時期は思い出せないが、01年の秋ぐらいに東京でやったはず。01年春にReadが示した定理を、夏のTAMU研究集会でPisierが分かりやすく解説してくれた。 Pisierに勧められてそれを応用してみたところ、Connesの定理(1978)に作用素環の知識を必要としない簡単な証明をつける事ができた。 さらに新しい結果に結びつくかもしれないと思って放置しておいたが、某氏から某誌に論文を投稿して欲しいと頼まれたので、引っ張り出した。Banach環の研究者向けに分かりやすく書いたつもり。 (追記：09年秋になって、Banach空間論のある標準的な知識を合わせれば追い求めていた結果に到達できるということを知らされた。人に出し抜かれたのはこれが初めてだ。)
[39] Noncommutative real algebraic geometry of Kazhdan's property (T). J. Inst. Math. Jussieu, 15 (2016), 85--90. doi:10.1017/S1474748014000309 arxiv:1312.5431 普段は晩酌をして、その後は睡眠の妨げとなるようなことはしないのだが、研究集会懇親会翌日の休肝日に夕食後も気になる問題を弄んでいたら、思いがけないほど当たり前に解けてしまった。 翌朝早くに目が覚めたので始発で研究所に行き即論文にした。夜に示されたことは日の光に曝されて蒸発するのが大抵なので、不安だったのだ。 以前からコンピュータを使って定理を示す数学に興味を持っていたが、今回ようやく関係するであろう仕事が出来た。とはいえ自分は実装に関してはまるで素人なのだ。 (引用終り) 以上
IUTその４をざっと眺めてみた（＾＾； １） ”the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A. ” みんな、Theorem Aで証明可能？（＾＾
1056 名前：Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations. PDF NEW !! (2019-10-31)
P3 Theorem A. (Diophantine Inequalities) Then, relative to the notation of [GenEll] [reviewed in the discussion preceding Corollary 2.2 of the present paper], one has an inequality of “bounded discrepancy classes” Thus, Theorem A asserts an inequality concerning the canonical height [i.e., “htωX(D)”], the logarithmic different [i.e., “log-diffX”], and the logarithmic conductor [i.e., “log-condD”] of points of the curve UX valued in number fields whose extension degree over Q is ? d . In particular, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A. We refer to [Vjt] for a detailed exposition of these conjecture
P55 Remark 2.3.1. We take this opportunity to correct some unfortunate misprints in [GenEll].
P63 Remark 2.3.4. Various aspects of the theory of the present series of papers are substantially reminiscent of the theory surrounding Bogomolov’s proof of the geometric version of the Szpiro Conjecture, as discussed in [ABKP], [Zh]. Put another way, these aspects of the theory of the present series of papers may be thought of as arithmetic analogues of the geometric theory surrounding Bogomolov’s proof. Alternatively, Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint [perhaps even a sort of Rosetta stone!], for understanding substantial portions of the theory of the present series of papers. The author would like to express his gratitude to Ivan Fesenko for bringing to his attention, via numerous discussions in person, e-mails, and skype conversations between December 2014 and January 2015, the possibility of the existence of such fascinating connections between Bogomolov’s proof and the theory of the present series of papers. We discuss these analogies in more detail in [BogIUT].
P64 In the following discussion, we shall refer to this geometry as the Schwarz-theoretic geometry of D. Perhaps the most fundamental difference between the proofs of Parshin and Bogomolov lies in the fact that (PB1) Whereas Parshin’s proof revolves around estimates of displacements arising from actions of elements of the fundamental group on a certain two dimensional complete [Kobayashi] hyperbolic complex manifold by means of the holomorphic geometry of the Kobayashi distance, i.e., in effect, the Schwarz-theoretic geometry of D, Bogomolov’s proof [cf. the review of Bogomolov’s proof given in [BogIUT]] revolves around estimates of displacements arising from actions of elements of the fundamental group on a one-dimensional real analytic manifold [i.e., a universal covering of a copy of the unit circle S1] by means of the real analytic
1059 名前：symplectic geometry of the upper half-plane. Here, it is already interesting to note that this fundamental gap, in the case of results over complex function fields, between the holomorphic geometry applied in Parshin’s proof of the Mordell Conjecture and the real analytic symplectic geometry applied in Bogomolov’s proof of the Szpiro Conjecture is highly reminiscent of the fundamental gap discussed in Remark 2.3.3, (iii), in the case of results over number fields, between the arithmetically holomorphic nature of the proof of the Mordell Conjecture given in [Falt] and the “arithmetically quasi-conformal” nature of the proof of the Szpiro Conjecture [cf. Corollary 2.3] via inter-universal Teichm¨uller theory given in the present series of papers. That is to say,
P67 Section 3: Inter-universal Formalism: the Language of Species
(†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.
P68 Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that it may be possible to justify the stance of ignoring such issues in the context of the present series of papers ? at least from the point of view of establishing the validity of various “final results” that may be formulated in ZFC-models ? by invoking the work of Feferman [cf. [Ffmn]]. Precise statements concerning such issues, however, lie beyond the scope of the present paper [as well as of the level of expertise of the author!]. In the following discussion, we use the phrase “set-theoretic formula” as it is conventionally used in discussions of axiomatic set theory [cf., e.g., [Drk], Chapter 1, §2], with the following proviso: In the following discussion, it should be understood that every set-theoretic formula that appears is “absolute” in the sense that its validity for a collection of sets contained in some universe V relative to the model of set theory determined by V is equivalent, for any universe W such that V ∈ W, to its validity for the same collection of sets relative to the model of set theory determined by W [cf., e.g., [Drk], Chapter 3, Definition 4.2].
Remark 3.1.2. (ii) One interesting point of view that arose in discussions between the author and F. Kato is the following. The relationship between the classical approach to discussing mathematics relative to a fixed model of set theory ? an approach in
>>932 つづき ５） a [small] category は、集合論ZFC内の speciesでやれる？
P71 Remark 3.1.4. Note that because the data involved in a species is given by abstract set-theoretic formulas, the mathematical notion constituted by the species is immune to, i.e., unaffected by, extensions of the universe ? i.e., such as the ascending chain V0 ∈ V1 ∈ V2 ∈ V3 ∈ ... ∈ Vn ∈ ... ∈ V that appears in the discussion preceding Definition 3.1 ? in which one works. This is the sense in which we apply the term “inter-universal”. That is to say, “inter-universal geometry” allows one to relate the “geometries” that occur in distinct universes.
P72 Example 3.2. Categories. The notions of a [small] category and an isomorphism class of [covariant] functors between two given [small] categories yield an example of a species. That is to say, at a set-theoretic level, one may think of a [small] category as, for instance, a set of arrows, together with a set of composition relations, that satisfies certain properties; one may think of a [covariant] functor between [small] categories as the set given by the graph of the map on arrows determined by the functor [which satisfies certain properties]; one may think of an isomorphism class of functors as a collection of such graphs, i.e., the graphs determined by the functors in the isomorphism class, which satisfies certain properties. Then one has “dictionaries” 0-species ←→ the notion of a category 1-species ←→ the notion of an isomorphism class of functors at the level of notions and a 0-specimen ←→ a particular [small] category a 1-specimen ←→ a particular isomorphism class of functors at the level of specific mathematical objects in a specific ZFC-model. Moreover, one verifies easily that species-isomorphisms between 0-species correspond to isomorphism classes of equivalences of categories in the usual sense. つづく
>>933 つづき ６） “∈-loops” a ∈ b ∈ c ∈ ... ∈ a が、the notion of a “species”で、正則性公理に反せずに実現できる？（＾＾；
P74 Remark 3.3.1. (i) One well-known consequence of the axiom of foundation of axiomatic set theory is the assertion that “∈-loops” a ∈ b ∈ c ∈ ... ∈ a can never occur in the set theory in which one works. On the other hand, there are many situations in mathematics in which one wishes to somehow “identify” mathematical objects that arise at higher levels of the ∈-structure of the set theory under consideration with mathematical objects that arise at lower levels of this ∈-structure. In some sense, the notions of a “set” and of a “bijection of sets” allow one to achieve such “identifications”. That is to say, the mathematical objects at both higher and lower levels of the ∈-structure constitute examples of the same mathematical notion of a “set”, so that one may consider “bijections of sets” between those sets without violating the axiom of foundation. In some sense, the notion of a species may be thought of as a natural extension of this observation. That is to say, the notion of a “species” allows one to consider, for instance, speciesisomorphisms between species-objects that occur at different levels of the ∈-structure of the set theory under consideration ? i.e., roughly speaking, to “simulate ∈-loops” ? without violating the axiom of foundation. Moreover, typically the sorts of species-objects at different levels of the ∈-structure that one wishes to somehow have “identified” with one another occur as the result of executing the mutations that arise in some sort of mutation-history
>>934 つづき ７）（最後） IUTを考えた動機”the original motivations”と、最後には”Corollary 3.12.”の説明か（＾＾；
P76 Remark 3.3.2. One somewhat naive point of view that constituted one of the original motivations for the author in the development of theory of the present series of papers is the following. In the classical theory of schemes, when considering local systems on a scheme, there is no reason to restrict oneself to considering local systems valued in, say, modules over a finite ring. If, moreover, there is no reason to make such a restriction, then one is naturally led to consider, for instance, local systems of schemes [cf., e.g., the theory of the “Galois mantle” in [pTeich]], or, indeed, local systems of more general collections of mathematical objects. One may then ask what happens if one tries to consider local systems on the schemes that occur as fibers of a local system of schemes.
Example 3.5. Absolute Anabelian Geometry. (i) Let S be a class of connected normal schemes that is closed under isomorphism [of schemes]. Suppose that there exists a set ES of schemes describable by a set-theoretic formula with the property that every scheme of S is isomorphic to some scheme belonging to ES .
P82 Remark 3.6.3
P84 Here, we observe in passing that the “apparently horizontal
1064 名前：arrow-related” issue discussed in (H2) of simultaneous realization of “label-dependent” and “labelfree” mathematical objects is reminiscent of the vertical arrow portion of the bicoricity theory of [IUTchIII], Theorem 1.5 ? cf. the discussion of [IUTchIII], Remark 1.5.1, (i), (ii); Step (vii) of the proof of [IUTchIII], Corollary 3.12. (引用終り) 以上 []
思うに 「”the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A. ” みんな、Theorem Aで証明可能」 ということを、”正しい” と認めさせるべし！！（＾＾
(引用開始) 思うに 「”the ABC Conjecture, and the Szpiro Conjecture for elliptic curves all follow as special cases of Theorem A. ” みんな、Theorem Aで証明可能」 ということを、”正しい” と認めさせるべし！！（＾＾ そうすれば、「IUTを勉強しなくちゃ！」という人が増えるよ (引用終り)
IUTが、the ABC Conjecture, and the Szpiro Conjecture たちを証明するのに失敗しているとすれば くその役にも立たないIUTなんて、だれも習熟したいとは思わないだろう やる気の無い人を、習熟させるなんてことはムリゲーだろう
