アレクサンドル・グロタンディーク(Alexander Grothendieck)のアプローチは、固定された射有限群 G に対して有限 G-集合の圏を特徴付ける圏論的性質に関係している。例えば、G として ^Z と表記される群が考えられる。この群は巡回加法群 Z/nZ の逆極限である。あるいは同じことであるが、有限指数の部分群の位相に対する無限巡回群の完備化である。 すると、有限 G-集合は G が商有限巡回群を通して作用している有限集合 X であり、X の置換を与えると特定することができる。
上の例では、古典的なガロア理論との関係は、^Z を任意の有限体 F 上の代数的閉包 F の射有限ガロア群 Gal(F/F) と見なすことである。 すなわち、F を固定する F の自己同型は、 F 上の大きな有限分解体をとるように、逆極限により記述される。幾何学との関係は、原点を取り除いた複素平面内の単位円板の被覆空間として見なすことができる。
https://en.wikipedia.org/wiki/Perfectoid_space Perfectoid space In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p.
A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements.
Perfectoid spaces may be used to (and were invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting equivalence and the almost purity theorem. The notions were introduced in 2012 by Peter Scholze.[1]
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields[1] with residual characteristic p (such as Qp). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge?Tate representation. Hodge?Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the etale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.
Contents 1 General classification of p-adic representations 2 Period rings and comparison isomorphisms in arithmetic geometry 3 Notes 4 References 4.1 Primary sources 4.2 Secondary sources 0028現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 16:48:52.52ID:ncpDqOGk>>27 追加
In algebraic geometry, an etale morphism (French: [etal]) is a morphism of schemes that is formally etale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, etale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the etale topology. The word etale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.[1]
後ろに、Appendix A～Cも付けてあって C.4. On the Prime Number Theorem. C.5. On the Residual Finiteness of Free Groups. とか、基本的な知識の補足もある C.6. Some Lists on Inter-universal Teichmuller Theory とかは、IUTの重要な記号の一覧ですかね
P366 A.3. Hodge-Arakelov-theoretic Comparison Theorem.で ”Note that these can be considered as a discrete analogue of the calculation of Gaussian integral is a Gaussian distribution (i.e., j → j^2) in the cartesian coordinate is a calculation in the polar coordinate ・・・” とか、望月先生の講演ネタで使っていた話の解説もあるな
Cor 3.12は P359 ”Corollary 13.13. (Log-volume Estimates for -Pilot Objects, [IUTchIII, Corollary 3.12]) We write -｜ log(θ)｜∈ R ∪{+∞}” あと P360 ”Then we obtain -｜ log(q)｜＜ -｜ log(θ)｜” で、IUT III Cor3.12 になるけどねｗ（＾＾； （Proof.は、その直後から4ページほどある） 山下サーベイ論文は、それなりに面白いわ（＾＾ 0032粋蕎 ◆C2UdlLHDRI 2020/05/04(月) 17:51:48.12ID:5fT1c3ml 無政府主義57歳が荒らしとったんか？ 0033現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 18:08:49.94ID:ncpDqOGk>>29 補足 https://en.wikipedia.org/wiki/Arakelov_theory In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. Contents 1 Background 2 Results Background Arakelov geometry studies a scheme X over the ring of integers Z, by putting Hermitian metrics on holomorphic vector bundles over X(C), the complex points of X. This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety. Results Arakelov (1974, 1975) defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields. Gerd Faltings (1984) extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context. Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by Gerd Faltings (1991) in his proof of Serge Lang's generalization of the Mordell conjecture. Pierre Deligne (1987) developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov. Arakelov's theory was generalized by Henri Gillet and Christophe Soule to higher dimensions. That is, Gillet and Soule defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soule is the arithmetic Riemann?Roch theorem of Gillet & Soule (1992), an extension of the Grothendieck?Riemann?Roch theorem to arithmetic varieties. The arithmetic Riemann?Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. 0034現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 18:11:20.65ID:ncpDqOGk>>32 粋蕎さん、どうも お元気そうで なによりです（＾＾ 0035１３２人目の素数さん2020/05/04(月) 18:12:06.64ID:Wfaifett ミスター維新消えたの？ 0036現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 18:33:27.13ID:ncpDqOGk メモ https://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) 2016年の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。 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。
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。
https://arxiv.org/abs/1010.1540 The Local Langlands correspondence for $\GL_n$ over p-adic fields Peter Scholze 37 pages 7 Oct 2010 We reprove the Local Langlands Correspondence for $\GL_n$ over p-adic fields as well as the existence of ?-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby cycles in certain regular situations. 0037現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 18:36:57.13ID:ncpDqOGk>>35 >ミスター維新消えたの？
性質 一般的に多様体のｌ進コホモロジー群は複素多様体の特異コホモロジー群と似たような性質を持つ。ただ特異コホモロジーは整数もしくは有理数上の加群であるのに対して、ｌ進コホモロジーはｌ進整数もしくはｌ進数上の加群になる。非特異な射影多様体上のｌ進コホモロジーはポアンカレ双対性を満たすほかケネスの公式も満たす。 一方ｌ進コホモロジーは特異コホモロジーと異なり、ガロア群の作用を持つという性質がある。たとえば有理数体上定義された複素多様体のｌ進コホモロジー群は有理数体の絶対ガロア群の作用を持ち、ガロア表現と関係が深い。 (引用終り) 以上 0041現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 20:08:00.85ID:ncpDqOGk woitブログで、David Robertsは、ショルツ先生にバッサリ切られているぞｗ（＾＾； https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=2#comments Latest on abc Posted on April 3, 2020 by woit (抜粋) David Roberts says: April 15, 2020 at 2:45 am @W gosh, thanks! Suppose you take the same argument and present it in two different languages ? one, the standard categorical language, and two, Mochizuki’s language where distinct copies of an isomorphic object are relevant for colimits and other categorical constructions. Assuming no other knowledge of what the argument actually is or how it is written, which language is more likely to conceal a subtle error in calculations or other mistake, and which language is more likely to make such mistakes easier to see? This is tricky: it depends who’s reading it. Who are you envisaging seeing mistakes? I can’t imagine (ignoring the fact this is IUT and tremendously baroque) that someone who’s had a decade of practice with their own idiosyncratic style of working would make mistakes more frequently that someone using the language of the majority, all things being equal, apart from the fact the latter person has more potential external checks and balances. This latter point I think can’t be overemphasised. Andrew Wiles was still speaking the language of his community by the time he emerged with his (first attempt at a) proof of FLT, and even engaged the help of someone else to try to check the subtle parts of the argument before that. This hasn’t happened here…
A bigger problem is the rigid commitment to definitions/structures that are explicitly admitted as being far more general than necessary (*cough* Frobenioids *cough*). This increases the friction for potential eyes on the IUT papers, if you’ll permit me a worrying metaphor mix.
Finally a short answer to David Roberts’ last message: I highly doubt your sentiment that the possibility of doing mistakes is not correlated with how well your language is adapted to the mathematics at hand. (引用終り) 以上 0043１３２人目の素数さん2020/05/04(月) 20:29:39.37ID:wcYBZ0Rw 今年中には掲載されるの？ 0044現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/04(月) 20:32:48.22ID:ncpDqOGk>>43 どうも、コメントありがとう
1.1. イントロダクション. アラケロフ幾何とは，おおまかに言って 体上の代数多様体の代りに，Z 上のスキームと 1 点 ベクトル束の代りに，1 点に計量の入ったベクトル束 を考えるというのものである． Szpiro は アラケロフ幾何について Put metrics at infinity on vector bundles and you will have a geometric intuition of compact varieties to help you. と言っている（[27]）． このことを簡単な場合だが，コンパクトリーマン面と Spec(Z) を対 比させることによって見てみよう． 0047１３２人目の素数さん2020/05/04(月) 22:37:32.07ID:XBGxFPW5>>45 「査読はSSレポート（2018）の後、２年かけてされたので、 SSレポートは否定されたと見て良い」
>Peter Scholze, who everyone thinks is the greatest mathematician of this generation, says he cannot deduce 3.12 (which is the ABC conjecture, in paper #4) from 3.11 (a summary of the first 3 ABC papers) in Mochizuki’s papers.
>Koshikawa had a similar problem, and when he asked Mochizuki about it, the latter responded that the deduction is self-evident. ーーーーーーーーーーーーーーーーーーーーーーーーーーーーーーーー 13１３２人目の素数さん2018/06/09(土) 20:16:27.19ID:lShLrM9+ >>9 貼るならself-evidentのくだりが削除されてないやつを貼れよ無能 0061現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/05(火) 07:55:25.31ID:dnbV/fKk>>60 コメントありがとう 興味深く読みました
”>Koshikawa had a similar problem, and when he asked Mochizuki about it, the latter responded that the deduction is self-evident.”
https://www.math.columbia.edu/~woit/wordpress/?p=11723 David Roberts says: May 3, 2020 at 5:52 pm Hi Peter, I think it worth including *in the pdf document* a mention that the discussion continued, and give a link back. You mentioned that there was content before the first comment you included, so why not also say the discussion continued? Not everyone reading that document will come to it by a link from your blog post where you mention this. I say this merely from the point of view of having a coherent scholarly record (whatever one’s view of the various positions). 0065現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/05(火) 08:18:06.15ID:dnbV/fKk>>57 >だいたい柏原さんはIUT論文を読んだのかね？ >絶対に読んでないと思うぞ。 >完全に彼の専門外だしね。
”論文を読む”を、どう定義するかも問題だが それは、未定義としてｗ（＾＾ 全く読んでないってこともないでしょう ”Masaki Kashiwara, head of the team that examined the professor’s theory”（下記）を信じればね 自分が、「正しい」と確信を持てる程度には、読んだのでは？
“There are a number of new notions and it was hard to understand them,” Masaki Kashiwara, head of the team that examined the professor’s theory, said at a news conference. 0066現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/05(火) 08:37:08.31ID:dnbV/fKk>>55 >Akshay VenkateshはABC予想はまだ証明されてないって意見に同意しているんだよ。 >英語を勉強して出直してこい。ボケ。
この発言のすぐ後に、有名なTerence Tao saysがあって、当時そちらしか見ていなかったんだｗ ”Terence Tao says: December 18, 2017 at 2:46 pm Thanks for this. I do not have the expertise to have an informed first-hand opinion on Mochizuki’s work, but on comparing this story with the work of Perelman and Yitang Zhang you mentioned that I am much more familiar with, one striking difference to me has been the presence of short “proof of concept” statements in the latter but not in the former, by which I mean ways in which the methods in the papers in question can be used relatively quickly to obtain new non-trivial results of interest (or even a new proof of an existing non-trivial result) in an existing field. In the case of Perelman’s work, already by the fifth page of the first paper Perelman had a novel interpretation of Ricci flow as a gradient flow which looked very promising,・・”
IUT IV で、Venkateshは、P8謝辞とP36の２箇所に出てきます（＾＾ （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 (抜粋) P8 Acknowledgements: In addition, I would like to thank Kentaro Sato for useful comments concerning the set-theoretic and foundational aspects of the present paper, as well as Vesselin Dimitrov and Akshay Venkatesh for useful comments concerning the analytic number theory aspects of the present paper.
P36 Remark 1.10.6. On the other hand, it was pointed out to the author by A. Venkatesh that in fact it is not difficult to modify the construction of these examples of abc sums given in [Mss] so as to obtain similar asymptotic estimates to those obtained in [Mss] [cf. the discussion of Remark 1.10.5, (ii)], even without taking into account the contributions at the prime 2. 0069I couldn't agree less2020/05/05(火) 09:42:16.12ID:zuve741i>>67 >なるほど、”I couldn’t agree more.”は、慣用句か
Venkatesh氏が、>>66の ”The ABC conjecture has (still) not been proved Posted on December 17, 2017 by Persiflage” に登場して、１行コメントをして行った ”agree”って、何に対して agreeしているか？ も明言していない。だから、文脈から推察するしかないのです
さて、Venkatesh氏はこの後2018年にフィールズ賞受賞して、フェセンコ先生のDR生だったという また、>>68に引用したが、望月 IUT IV に名前が挙がっているように、IUT IV の組合わせ論に 意見を寄せているのですよ
ついでに 雪江明彦 「代数学３」を見たけど、分離閉包は扱われていなかった Coxのガロア本も、分離閉包は無かったな アルティン本も、記憶では無かった思う Van der van der Waerden ”Moderne Algebra”は、手元に無いが（読んでもいない チラ見のみ）、多分ないんじゃないかな？（＾＾
（参考） https://books.google.co.jp/books?id=oVfzBgAAQBAJ&;printsec=frontcover&dq=Moderne+Algebra+van+der+Waerden&hl=ja&sa=X&ved=0ahUKEwjTxPTQ-ZvpAhXRBIgKHfIcC-YQ6AEINzAB#v=onepage&q=Moderne%20Algebra%20van%20der%20Waerden&f=false Moderne Algebra 著者: Bartel Eckmann L. Van der van der Waerden、 1937 0087１３２人目の素数さん2020/05/05(火) 14:29:42.99ID:kGB3j7wQ ベンカデシュはフェセンコの弟子なの？ 0088１３２人目の素数さん2020/05/05(火) 14:30:10.16ID:kGB3j7wQ そこまでは知らなんだ 0089現代数学の系譜 雑談 ◆e.a0E5TtKE 2020/05/05(火) 14:41:16.39ID:dnbV/fKk>>85 >ショルツ、ヴェンカテシュと同じく、2018年にフィールズ賞を受賞した >コーチェル・ビルカーは、フェセンコの弟子だそうだ >ちなみにクルド人で、本名は Fereydoun Derakhshani
