Completion of the proof The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
Deligne's second proof Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallee Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.
Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallee Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.
https://en.wikipedia.org/wiki/Lefschetz_pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.
It has been shown that Lefschetz pencils exist in characteristic zero. They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds. It has also been shown that Lefschetz pencils exist in characteristic p for the etale topology.
Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them. (引用終り) 以上 0339現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/07(土) 19:40:56.78ID:H2e5WMAT>>336
(>>306) 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 October 2019 Abstract. (抜粋) n the present paper, estimates arising from these multiradial algorithms for splitting monoids of LGP-monoids are applied to verify various diophantine results which imply, for instance, the so-called Vojta Conjecture for hyperbolic curves, the ABC Conjecture, nd the Szpiro Conjecture for elliptic curves.
（>>323） 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 October 2019 (抜粋) P57 Remark 2.3.3. Corollary 2.3 may be thought of as an effective version of the Mordell Conjecture. 0355現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/11(水) 17:44:13.83ID:W0aIOzhV メモ
・ブルース・クライナーとジョン・ロット, Notes on Perelman's Papers（2006年5月） ペレルマンによる幾何化予想についての証明の細部を解明・補足 ・朱熹平と曹懐東、A Complete Proof of the Poincare and Geometrization Conjectures - application of the Hamilton- Perelman theory of the Ricci flow（2006年7月、改訂版2006年12月） ペレルマン論文で省略されている細部の解明・補足 ・ジョン・モーガンと田剛、Ricci Flow and the Poincare Conjecture（2006年7月） ペレルマン論文をポアンカレ予想に関わる部分のみに絞って詳細に解明・補足
まあ、IUTの人たちは説明責任を果たさないといけないね（＾＾ 0362現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/11(水) 21:19:43.41ID:6dBaZfdC なお、素人のドテ勘だが、IUTは成立しているんだろうと思っている （不成立にしては、IUTに関わる人が、大杉だ） 0363現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/11(水) 23:31:24.62ID:6dBaZfdC メモ https://arxiv.org/abs/1512.04389 https://arxiv.org/pdf/1512.04389.pdf Semi-galois Categories I: The Classical Eilenberg Variety Theory Takeo Uramoto (Submitted on 14 Dec 2015 (v1), last revised 20 Jan 2017 (this version, v4)) 0364現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/12(木) 07:34:32.59ID:aEgA7HUg>>363 追加
メモ http://www.math.is.tohoku.ac.jp/~uramoto/preprints/preprint_christol.pdf Semi-galois Categories II: An arithmetic analogue of Christol’s theorem Takeo Uramoto Graduate School of Information Sciences, Tohoku University February 12, 2018 (抜粋) 5.2 Canonicity of Eilenberg theory for DFAs and its geometric extension Geometric extension of Eilenberg theory
参考文献 [3] https://books.google.co.jp/books?id=_H7PuKhJHZkC&;printsec=frontcover&dq=Szamuely+Galois+groups+and+fundamental+groups&hl=ja&sa=X&ved=0ahUKEwjGv5nFirDmAhXDdXAKHUErAO4Q6AEIKTAA#v=onepage&q=Szamuely%20Galois%20groups%20and%20fundamental%20groups&f=false Galois Groups and Fundamental Groups Cambridge University Press 著者: Tamas Szamuely 2009
[12] https://arxiv.org/abs/0906.3146 https://arxiv.org/pdf/0906.3146.pdf Λ-RINGS AND THE FIELD WITH ONE ELEMENT JAMES BORGER 2009 Abstract. The theory of Λ-rings, in the sense of Grothendieck’s Riemann? Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.
Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumeration formulas in linear algebra over finite fields Fq as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geometry over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, following Tits [60], to call this mythical base F1, the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F1, that Spec Z ?F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function fields.1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes.
The most obvious way of doing this is to consider weaker algebraic structures than rings (commutative, as always), such as commutative monoids, and to try using them as the affine building blocks for a more rigid theory of algebraic geometry. This has been pursued in a number of papers, which I will cite below. Another natural approach is motived by the following question, first articulated by Soul´e [57]: Which rings over Z can be defined over F1? Less set-theoretically, on a ring over Z, what should descent data to F1 be? The main goal of this paper is to show that a reasonable answer to this question is a Λ-ring structure, in the sense of Grothendieck’s Riemann?Roch theory [31]. More precisely, we show that a Λ-ring structure on a ring can be thought of as descent data to a deeper base in the precise sense that it gives rise to a map from the big ´etale topos of Spec Z to a Λ-equivariant version of the big ´etale topos of Spec Z, and that this deeper base has many properties expected of the field with one element. Not only does the resulting algebraic geometry fit into the supple formalism of topos theory, it is also arithmetically rich?unlike the category of sets, say, which is the deepest topos of all. For instance, it is closely related to global class field theory, complex multiplication, and crystalline cohomology. (引用終り) 以上 0407現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/14(土) 19:19:28.35ID:s6Tab8iq メモ https://mathsoc.jp/publication/tushin/index18-2.html 「数学通信」第１８巻第２号目次 2013 https://mathsoc.jp/publication/tushin/1802/abe-saito.pdf 阿部知行氏の平成 25 年度文部科学大臣表彰 若手科学者賞受賞に寄せて 東京工業大学理工学研究科理学研究流動機構 斎藤 秀司 (抜粋) 阿部氏に出会ったのは，私が 2004 年に東大数理に赴任して最初の年，彼がまだ大学 3 年生の時である．Deligne の Weil 予想の証明 (Weil II) を勉強したいので付き合ってほし いと個人的に頼みを受けた．ご存知の方も多いかと思うが，この論文は EGA はもちろん SGA といった代数幾何の最先端理論を駆使した難解な論文であり，こんな論文を大学３ 年生がはたしてどれほど理解できるのかといぶかりながら始めたセミナーであった．が， 彼の理解度は驚くほど深く，その類まれなる才能には目を見張らされた (東大に赴任した ばかりだったので，東大生はみなこんなにできるのかと驚愕したのだが，これについて は私の思い過ごしであることがのちに判明している)．修士課程に入って私が指導教官と なってからも最先端の理論を次々に吸収していく様は見事というしかない．このような逸 材を「研究指導」の名のもとに私の狭量な数学のなかに制約することは憚れる思いであっ たのだが，そうこうするうちに「数論的 D 加群」という私の専門を逸脱した研究テーマ を自分で勝手に見つけてくれた．私は立場上は大学院指導教官ではあるが，彼との数学交 流において多くを学ばせてもらっているのは私の方であると感じている． 阿部氏の研究のエッセンスを抜き出して表現すると，大局的な視野に立った問題意識， 問題の本質を見通す深い洞察力，そこから湧き上がる着想を実現する強力な計算力，そし て忘れてならないのは，阿部氏の数学に脈々と流れる豊かな感性である．阿部氏の数学に は，多くの優れた業績に共通する芸術的ともいえる美的感覚がある．実は，阿部氏はピア ニストとしても人並み外れた才能を持っており，彼の音楽的感性が数学にも表現され恩恵 をもたらしているのだろう．
http://www.kurims.kyoto-u.ac.jp/~motizuki/research-japanese.html 過去と現在の研究 望月新一 http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html 2018年3月、数理研で行なわれたIUTeichに関する議論の関連文書 http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf 2018年3月、数理研で行なわれたIUTeichに関する議論を纏めた報告書 REPORT ON DISCUSSIONS, HELD DURING THE PERIOD MARCH 15 ? 20, 2018, CONCERNING INTER-UNIVERSAL TEICHMULLER THEORY (IUTCH) ¨ Shinichi Mochizuki February 2019
P1 §1. The present document is a report on discussions held during the period March 15 ? 20, 2018, concerning inter-universal Teichm¨uller theory (IUTch). These discussions were held in a seminar room on the fifth floor of Maskawa Hall, Kyoto University, according to the following schedule: ・ March 15 (Thurs.): 2PM ? between 5PM and 6PM, ・ March 16 (Fri.): 10AM ? between 5PM and 6PM, ・ March 17 (Sat.): 10AM ? between 5PM and 6PM, ・ March 19 (Mon.): 10AM ? between 5PM and 6PM, ・ March 20 (Tues.): 10AM ? between 5PM and 6PM.
P41 §17. The fundamental misunderstandings of IUTch discussed in the present report may be summarized as a failure to understand the following central aspects of IUTch:
P42 §18. In the context of the present report, it is important to recall that (Vrf1) IUTch has been checked, verified, read and reread, and orally exposed in detail in seminars in its entirety countless times since the release of preprints on IUTch in August 2012 by a collection of mathematicians (not including myself) involved in this line of research. (For instance, Fesenko estimates, in the most recent updated version of §3.1 of his survey [Fsk],that IUTch has been verified at least 30 times.) This collection of mathematicians has (together with me) also been actively involved in detailed discussions and dialogues with mathematicians who have any questions concerning IUTch.
Ｐ43 Indeed, at numerous points in the March discussions, I was often tempted to issue a response of the following form to various assertions of SS (but typically refrained from doing so!): Yes! Yes! Of course, I completely agree that the theory that you are discussing is completely absurd and meaningless, but that theory is completely different from IUTch!
P44 Nevertheless, the March discussions were productive in the sense that they yielded a valuable first glimpse at the mathematical content of the misunderstandings that underlie criticism of IUTch (cf. the discussion of §3). In the present report, we considered various possible causes for these misunderstandings, namely: (PCM1) lack of sufficient time to reflect deeply on the mathematics under discussion (cf. the discussion in the final portions of §2, §10); (PCM2) communication issues and related procedural irregularities (cf.(T6), (T7), (T8)); (PCM3) a deep sense of discomfort, or unfamiliarity, with new ways of thinking about familiar mathematical objects (cf. the discussion of §16; [Rpt2014], (T2); [Fsk], §3.3).
On the other hand, the March discussions were, unfortunately, by no means sufficient to yield a complete elucidation of the logical structure of the causes underlying the misunderstandings summarized in §17.
略歴： 2004年03月 東京工業大学 理学部 数学科 卒業 2004年04月 京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 入学 2006年03月 京都大学大学院 理学研究科 修士課程 数学・数理解析専攻 修了 修士論文： Fundamental groups of log configuration spaces and the cuspidalization problem PDF 2006年04月 京都大学大学院 理学研究科 博士課程 数学・数理解析専攻 進学 2006年04月～2007年03月 日本学術振興会 特別研究員（DC1） 2007年04月 京都大学 数理解析研究所 基礎数理研究部門 助教 2009年07月 京都大学 数理解析研究所 博士学位 (論文博士) 取得 学位論文： Absolute anabelian cuspidalizations of configuration spaces of proper hyperbolic curves over finite fields PDF 2011年12月 京都大学 数理解析研究所 無限解析研究部門 講師 2011年12月 2011年度井上研究奨励賞受賞 2017年12月 京都大学 数理解析研究所 無限解析研究部門 准教授 0430現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/15(日) 14:10:38.76ID:BvQtIPz4 メモ
Gくんのサーベイは出版されず Hくんのは3つ出版された http://www.kurims.kyoto-u.ac.jp/~gokun/ http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc2019Jul5.pdf RIMS K?oky?uroku Bessatsu Bx (201x), 000?000 A proof of the abc conjecture after Mochizuki. preprint. last updated on 8/July/2019. on the footnote (* FAQ on Inter-universal Teichmuller Theory) Abstract We give a survey of S. Mochizuki’s ingenious inter-universal Teichm¨uller theory and explain how it gives rise to Diophantine inequalities. The exposition was designed to be as self-contained as possible.
https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/244678 B76 On the examination and further development of inter-universal Teichmuller theory Mono-anabelian Reconstruction of Number Fields (On the examination and further development of inter-universal Teichmuller theory) Hoshi, Yuichiro (2019-08) 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, B76: 1-77 宇宙際Teichmuller理論入門(On the examination and further development of inter-universal Teichmuller theory) 星, 裕一郎 (2019-08) 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, B76: 79-183
https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/244674 B72 Algebraic Number Theory and Related Topics 2015 続・宇宙際Teichmuller理論入門 (Algebraic Number Theory and Related Topics 2015) 星, 裕一郎 (2018-12) 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, B72: 209-307 0431現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/15(日) 19:28:07.41ID:BvQtIPz4 メモ http://taro-nishino.blogspot.com/2019/12/blog-post077.html TARO-NISHINOの日記 数論の賢人 12月 12, 2019 (抜粋) Quanta Magazine誌に始めてショルツ博士が登場した"The Oracle of Arithmetic"を今回紹介します。勿論、もっと以前から数学界では有名な人でしたが、一般大衆を読者層とするオンライン科学ジャーナルにおいては始めての登場だったのではないかと思います。 これを最初に読んだ時の私の率直な感想を書くと、ショルツ博士はあの若さで数学的業績も圧倒的なら、あの若さで人柄も素晴らしいと思いました。後日フィールズ賞等を受賞し、世界を引っ張るリーダと呼ばれるのは当然のことなのかも知れません。 以下にその私訳を載せておきます。
数論の賢人 2016年06月28日 Erica Klarreich
28歳でピーター・ショルツは数論と幾何学の間の深い繋がりを明らかにしつつある。
2010年、びっくりさせる噂が数論コミュニティに行き渡り、Jared Weinsteinに届いた。どうやら、ボン大学の或る学生が数論における一つの不可解な証明に捧げられた288ペィジの本"Harris-Taylor" [訳注: 2001年01月にプリストン大学出版部から出版された、Michael HarrisとRichard Taylor共著の有名な本The Geometry and Cohomology of Some Simple Shimura Varietiesのこと] をたった37ペィジに再構成する論文を書いたようだ。22歳の学生ピーター・ショルツは証明の最も複雑な部分の一つ(それは数論と幾何学の間の広範囲にわたる繋がりを扱っている)を回避する方法を発見していた。 "そんなに若い誰かがとても革命的なことを成し遂げていたことは本当にすごかった。非常に屈辱的だった"と現在はボストン大学にいる34歳の数論学者Weinsteinは言った。
