"アラケロフ理論(英語版)(Arakelov theory)"下記ですな 下記では、Faltings、Serge Lang、Mordell conjecture、Deligne、arithmetic Hodge index などなど、重要キーワード満載ですな
(参考) https://en.wikipedia.org/wiki/Arakelov_theory Arakelov theory (抜粋) 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 3 Arithmetic Chow groups 4 The arithmetic Riemann?Roch theorem
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. For this one defines arithmetic Chow groups CHp(X) of an arithmetic variety X, and defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups.
Arakelov's intersection theory for arithmetic surfaces was developed further by Jean-Benoit Bost (1999). The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space {\displaystyle L_{1}^{2}}{\displaystyle L_{1}^{2}}. In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. (引用終り) 0180現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/26(火) 23:09:11.14ID:oYs7jyeH>>179 >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.
(参考) https://en.wikipedia.org/wiki/P-adic_Hodge_theory p-adic Hodge theory (抜粋) 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. 0182現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/27(水) 07:49:43.49ID:qnEhNItW>>181
つづき
Contents 1 General classification of p-adic representations 2 Period rings and comparison isomorphisms in arithmetic geometry
General classification of p-adic representations Let K be a local field with residue field k of characteristic p. In this article, a p-adic representation of K (or of GK, the absolute Galois group of K) will be a continuous representation ρ : GK→ GL(V), where V is a finite-dimensional vector space over Qp. The collection of all p-adic representations of K form an abelian category denoted \mathrm {Rep} _{\mathbf {Q} _{p}}(K)}{\mathrm {Rep}}_{{{\mathbf {Q}}_{p}}}(K) in this article. p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:[2]
{Rep} _{\mathrm {cris} }(K)\subsetneq {Rep} _{st}(K)\subsetneq {Rep} _{dR}(K)\subsetneq {Rep} _{HT}(K)\subsetneq {Rep} _{\mathbf {Q} _{p}}(K)} where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge?Tate representations, and all p-adic representations. In addition, two other categories of representations can be introduced, the potentially crystalline representations Reppcris(K) and the potentially semistable representations Reppst(K). The latter strictly contains the former which in turn generally strictly contains Repcris(K); additionally, Reppst(K) generally strictly contains Repst(K), and is contained in RepdR(K) (with equality when the residue field of K is finite, a statement called the p-adic monodromy theorem).
Period rings and comparison isomorphisms in arithmetic geometry The general strategy of p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called period rings[3] such as BdR, Bst, Bcris, and BHT which have both an action by GK and some linear algebraic structure and to consider so-called Dieudonne modules
D_{B}(V)=(B\otimes _{\mathbf {Q} _{p}}V)^{G_{K}}} (where B is a period ring, and V is a p-adic representation) which no longer have a GK-action, but are endowed with linear algebraic structures inherited from the ring B. In particular, they are vector spaces over the fixed field E:=B^{G_{K}}}E:=B^{{G_{K}}}.[4] This construction fits into the formalism of B-admissible representations introduced by Fontaine. For a period ring like the aforementioned ones B? (for ? = HT, dR, st, cris), the category of p-adic representations Rep?(K) mentioned above is the category of B?-admissible ones, i.e. those p-adic representations V for which
\alpha _{V}:B_{\ast }\otimes _{E}D_{B_{\ast }}(V)\longrightarrow B_{\ast }\otimes _{\mathbf {Q} _{p}}V} is an isomorphism.
This formalism (and the name period ring) grew out of a few results and conjectures regarding comparison isomorphisms in arithmetic and complex geometry:
If X is a proper smooth scheme over C, there is a classical comparison isomorphism between the algebraic de Rham cohomology of X over C and the singular cohomology of X(C) (引用終り) 0184現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/27(水) 07:57:19.21ID:qnEhNItW>>181 補足
p-adic Hodge theory キーワードを拾うと
・The collection of all p-adic representations of K form an abelian category ・and also provides faithful functors to categories of linear algebraic objects that are easier to study. ・where each collection is a full subcategory properly contained in the next.
(a) p 進 Teichm¨uller 理論:(1993 年〜1996 年) この理論は、複素数体上の双曲的リーマン面に対する Koebe の上半平面に よる一意化や、そのモジュライに対する Bers の一意化の p 進的な類似と見る こともでき、また Serre-Tate の通常アーベル多様体に対する標準座標の理論の 双曲曲線版と見ることもできる。詳しくは、 A Theory of Ordinary p-adic Curves や An Introduction to p-adic Teichm¨uller Theory をご参照下さい。
(b) p 進遠アーベル幾何:(1995 年〜1996 年) この理論の代表的な定理は、「劣 p 進体」(= p 進局所体上有限生成な体の部 分体)上の相対的な設定において、双曲的曲線への任意の多様体からの非定数 的な射と、それぞれの数論的基本群の間の開外準同型の間に自然な全単射が存 在するというものである。詳しくは、 The Local Pro-p Anabelian Geometry of Curves をご参照下さい。
(c) 楕円曲線の Hodge-Arakelov 理論:(1998 年〜2000 年) この理論の目標は、複素数体や p 進体上で知られている Hodge 理論の類似 を、数体上の楕円曲線に対して Arakelov 理論的な設定で実現することにある。 代表的な定理は、数体上の楕円曲線の普遍拡大上のある種の関数空間と、楕円 曲線の等分点上の関数からなる空間の間の、数体のすべての素点において計量 と(ある誤差を除いて)両立的な全単射を主張するものである。この理論は、 古典的なガウス積分 ∫ ∞ ?∞ e?x2 dx = √π の「離散的スキーム論版」と見ることもできる。詳しくは、 A Survey of the Hodge-Arakelov Theory of Elliptic Curves I, II をご参照下さい。
・Inter-universal Teichm¨uller theory I: Hodge-Arakelov-theoretic aspects (2009 年に完成(?)予定) p 進 Teichm¨uller 理論における曲線や Frobenius の、「mod pn」までの標準持ち上 げに対応する IU 版を構成する。 ・Inter-universal Teichm¨uller theory II: limits and bounds (2010 年に完 成(?)予定) 上記の「mod pn」までの変形の n を動かし、p 進的極限に対応する「IU 的な極 限」 を構成し、pTeich における Frobenius 持ち上げの微分に対応するものを計算 する。 (引用終り) 以上 0196132人目の素数さん2019/11/28(木) 22:40:49.02ID:lvt0VL8R 4050 しろ@hu_corocoro 11月27日 苦節6ヶ月、初満点&一等賞です! https://twitter.com/hu_corocoro/status/1199593474128896000 https://twitter.com/5chan_nel (5ch newer account) 0197現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/28(木) 23:10:56.96ID:QdpmOFrx>>196 おめでとうございます 凄いですね(^^ 0198現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/28(木) 23:48:43.22ID:QdpmOFrx メモ貼る https://www.youtube.com/watch?v=Rz5g-plyuAg Peter Scholze - The geometric Satake equivalence in mixed characteristic 7,685 回視聴?2017/04/13
Institut des Hautes Etudes Scientifiques (IHES) チャンネル登録者数 2.91万人 Seminaire Paris Pekin Tokyo / MArdi 11 avril 2017
In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting. 0199現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/28(木) 23:52:37.94ID:QdpmOFrx>>198 >Satake equivalence
”The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirkovi? and Kari Vilonen (2007).” ”which is a fortiori an equivalence of tannakian categories (Ginzburg 2000).”
https://en.wikipedia.org/wiki/Satake_isomorphism Satake isomorphism (抜粋) Jump to navigationJump to search In mathematics, the Satake isomorphism, introduced by Ichir? Satake (1963), identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirkovi? and Kari Vilonen (2007).
Statement Classical Satake isomorphism Let {\displaystyle G}G be a semisimple algebraic group, {\displaystyle K}K be a non-Archimedean local field and {\displaystyle O}O be its ring of integers. It's easy to see that {\displaystyle Gr=G(K)/G(O)}{\displaystyle Gr=G(K)/G(O)} is grassmannian.
which is a fortiori an equivalence of tannakian categories (Ginzburg 2000). 0201現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/29(金) 00:19:47.46ID:KnsCfpdu>>200 >which is a fortiori an equivalence of tannakian categories (Ginzburg 2000).
In mathematics, a Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory.
The name is taken from Tannaka?Krein duality, a theory about compact groups G and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups.
The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group {}^{L}G} of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G. This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with {}^{L}G}.
Extensions Wedhorn (2004) has established partial Tannaka duality results in the situation where the category is R-linear, where R is no longer a field (as in classical Tannakian duality), but certain valuation rings. Duong & Hai (2017) showed a Tannaka duality result if R is a Dedekind ring.
Iwanari (2014) has initiated the study of Tannaka duality in the context of infinity-categories. 0203現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/29(金) 00:29:29.74ID:KnsCfpdu>>202 >Iwanari (2014) has initiated the study of Tannaka duality in the context of infinity-categories.
References https://arxiv.org/abs/1409.3321 Iwanari, Isamu (2014), Tannaka duality and stable infinity-categories, arXiv:1409.3321, doi:10.1112/topo.12057 Comments: The final version. Published in Journal of Topology, Wiley 2018
弱い絆については、米スタンフォード大学のマーク・グラノヴェッター博士の論文『The Strength of Weak Ties』で紹介されています。日本では「弱い紐帯(ちゅうたい)の強み」と紹介されることが多いようです。筆者なりに意訳すれば、「有益な情報をもたらすのはあまり親しくない人、言い換えれば弱い絆の人」だということです。
ここ誤訳やね 原文は下記 ”for example in Kurt Godel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory.”
The model-theoretic viewpoint has been useful in set theory; for example in Kurt Godel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory. (引用終り) 0224現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 11:24:10.70ID:id6ENHqe>>223 追加
M Most recent [M3] About certain aspects of the study and dissemination of Shinichi Mochizuki's IUT theory
https://www.maths.nottingham.ac.uk/plp/pmzibf/rapg.pdf ABOUT CERTAIN ASPECTS OF THE STUDY AND DISSEMINATION OF SHINICHI MOCHIZUKI’S IUT THEORY IVAN FESENKO ‘Phil: Do you ever have deja vu, Mrs. Lancaster? Mrs. Lancaster: I don’t think so, but I could check with the kitchen.’ (Groundhog Day) (抜粋) This text communicates in a compact form some of factual information related to the study of Sh. Mochizuki’s IUT theory1 and its dissemination, as well as various aspects of the situation around IUT.
The number of mathematicians able to write expert reports on the IUT papers exceeds the number of such reports on previous major breakthrough papers at the time of their publication. 2020 will be a special RIMS year with 4 international workshops on anabelian geometry, combinatorial anabelian geometry and IUT. Some mathematicians have tried to study IUT on the own, but have not been able to proceed far. In particular, none of number theorists who made their own breakthrough decades ago have apparently managed to advance in their study of IUT. In contrast, there are several young researchers who in the course of several years of hard study of IUT, have asked interesting questions and contributed to new original developments. There were few people, all lacking any expertise even in anabelian geometry, active in applying efforts to produce ignorant critical remarks about IUT. Their online remarks and debates were devoid of any valid math evidence of faults in the theory. Some were active in spreading fake information that might have affected some mathematicians in other areas. All of these unprofessional behaviour should be strongly rejected.
3.3. An attempt to study IUT by two German mathematicians and ethical issues. In 2013?2017 not a single concrete mathematical remark indicating a serious problem in IUT was produced. This did not prevent some cheap irresponsible talk. Since 2014 P. Scholze kept talking publicly at various workshops about faults in IUT.12 Eventually Scholze visited RIMS, together with J. Stix, in March 2018, just for 5 days.13 After the meeting, Scholze and Stix came with their caricature version of IUT based on their oversimplification of IUT in which they identify all isomorphic rings and ‘forget’ about the fundamental role of automorphism groups. In particular, the two German mathematicians deny the use of anabelian geometry and infinitely many theatres in IUT.14 Initially, Scholze and Stix intended to put their report about the meeting online. However, after reading Mochizuki’s report on their report, see especially its sect. 17-18 15 and these comments16, they completely changed their mind in July 2018 and stopped to be interested to post their own report. They eventually agreed to let the author of IUT to include their report on his pages. The author of IUT formulated several questions to the German mathematicians in his report that may have helped them to appreciate how erroneous was their take on IUT. The second version of their report did not address most of comments of Mochizuki on their first report. The second version of their report also included new incorrect statements such as a blunder in classical height theory and a fundamental misunderstanding of one of Faltings work.
No mathematicians are known to support the superficial take of Scholze?Stix on IUT. Their short lived study of IUT17 stands in shark contrast with the deep study of it by the other mathematicians mentioned above, who asked/made many good questions, remarks and comments. If one does not apply appropriate efforts to study the area of a fundamentally new theory, one does not become an expert in it, whatever one’s own different area of specialisation is and achievements in it. Of course, it is still possible to contribute useful questions, comments, remarks in relation to more conventional parts of the theory, e.g. those that came in 2012 from two analytic number theorists. To make a mistake in one’s mathematical study is rather normal, especially when one tries to understand a complex theory going much deeper than standard research. However, to publicly talk about faults in another theory for several years without ever having any valid evidence of the faults is irresponsible. The failure of those two German mathematicians should not stop serious researchers to study IUT. The failure of Mrs. Lancaster to understand the question does not in any way imply anything negative about the question.
4. Developments. Several are mentioned above. The book by F. Kato about IUT provides more general information about various features of IUT to the wider audience. This book was in the list of top twenty bestselling books in all subject areas on amazon.co.jp and was awarded the Yaesu prize. There are new developments related to IUT, in different directions. Four international workshops on anabelian geometry and IUT are organised during a special RIMS Project Research year on Expanding Horizons of Inter-universal Teichmuller Theory in 2020?2021 18, supported by the new Center for Research in Next-Generation Geometry. (引用終り) 以上 0229現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 13:46:25.15ID:id6ENHqe>>228 補足
(引用開始) Initially, Scholze and Stix intended to put their report about the meeting online. However, after reading Mochizuki’s report on their report, see especially its sect. 17-18 15 and these comments16, they completely changed their mind in July 2018 and stopped to be interested to post their own report. They eventually agreed to let the author of IUT to include their report on his pages. The author of IUT formulated several questions to the German mathematicians in his report that may have helped them to appreciate how erroneous was their take on IUT. The second version of their report did not address most of comments of Mochizuki on their first report. The second version of their report also included new incorrect statements such as a blunder in classical height theory and a fundamental misunderstanding of one of Faltings work. (引用終り) 0231現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 13:54:40.29ID:id6ENHqe>>228 補足
(引用開始) To make a mistake in one’s mathematical study is rather normal, especially when one tries to understand a complex theory going much deeper than standard research. However, to publicly talk about faults in another theory for several years without ever having any valid evidence of the faults is irresponsible. The failure of those two German mathematicians should not stop serious researchers to study IUT. The failure of Mrs. Lancaster to understand the question does not in any way imply anything negative about the question. (引用終り) 0232現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 13:59:21.82ID:id6ENHqe>>226 補足 >https://www.maths.nottingham.ac.uk/plp/pmzibf/rapg.pdf >ABOUT CERTAIN ASPECTS OF THE STUDY AND DISSEMINATION OF SHINICHI MOCHIZUKI’S IUT THEORY >IVAN FESENKO
IVAN FESENKO先生、IUTに対して自信満々 P. Scholze-J. Stixについては、一刀両断でばっさり切っている
http://people.maths.ox.ac.uk/kimm/ Webpage of Minhyong Kim Professor of Number Theory Joint Head of Oxford Number Theory Research Group (with Ben Green) Fellow of Merton College
Contents 1 Precise formulation of the theorem 2 History 2.1 Early proof attempts 2.2 Proof by computer 2.3 Simplification and verification 3 Summary of proof ideas 4 False disproofs 5 Three-coloring 6 Generalizations 7 Relation to other areas of mathematics 8 Use outside of mathematics 0242現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 18:08:30.28ID:id6ENHqe メモ https://inference-review.com/article/a-crisis-of-identification Inference A Crisis of Identification David Michael Roberts Published on March 1, 2019 in Volume 4, Issue 3. (抜粋) David Michael Roberts is a Research Associate at Adelaide University’s Institute for Geometry and Applications.
Formalizing Theorem 3.11 of IUT, whose statement runs to more than five pages, is Herculean.
In the absence of a formal proof, the scruples expressed by Scholze and Stix gave nonexperts something to hold on to. “I received unsolicited emails from people whom I knew in quite distant parts of the world,” Conrad remarked, and “[e]ach of them told me that they had worked through the IUT papers on their own and were able to more or less understand things up to a specific proof where they had become rather stumped.”22 The specific proof was, of course, that of Corollary 3.12.
For all that, there are a small number of mathematicians who have intensely studied Mochizuki’s work, and affirm quite emphatically that it is correct.23 Mochizuki himself remarked that
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 [emphasis original].24 0243現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 18:19:03.52ID:id6ENHqe>>242
https://www.math.columbia.edu/~woit/wordpress/?p=10560 Not Even Wrong Scholze and Stix on the Mochizuki Proof Posted on September 20, 2018 by woit 0245現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/12/01(日) 20:56:18.94ID:id6ENHqe>>244 さて、
<IUTの現状分析> 1.2012年のIUT論文4つが完成以来、いまだ成否定まらず ・特記として、2018年9月のScholze and Stixの誤りだという指摘と、それへの反論があった ・1)IUT成立派(RIMS以外にも)と、2)IUT不成立派(国際的には、Scholze and Stix以外に何人か) ・3)中間派:この中でも、IUTに好意的な人達が何人かいる。来年のIUTワークショップの1本目に参加表明している人達 2.来年IUTのシンポジュームを打って、4本のワークショップが企画されている ・多分、IUT成立派は、これを最大限利用して、IUT成立の国際的合意を得たいだろう (果たして)