余接層の定義 M を滑らかな多様体とし M × M を M の自身とのカルテジアン積とする。対角写像 Δ は M の点 p を M × M の点 (p, p) に送る。Δ の像は対角線 (diagonal) と呼ばれる。{\displaystyle {\mathcal {I}}}{\mathcal {I}} を対角線上消える M × M 上の滑らかな関数の芽の層とする。 このとき商層 {\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}{\mathcal {I}}/{\mathcal {I}}^{2} はより高次の項を法として対角線上消える関数の同値類からなる。余接層はこの層の M への引き戻し（英語版）である。
\Gamma T^{*}M=\Delta ^{*}({\mathcal {I}}/{\mathcal {I}}^{2}). テイラーの定理によって、これは M の滑らかな関数の芽の層に関して加群の局所自由層である。したがってそれは M 上のベクトル束、余接束 (cotangent bundle) を定義する。
講演タイトル On Chern class inequalities for surfaces in positive characteristic アブストラクト I will explain my proof of the inequality $c_1^2\leq 5c_2$ for a class of smooth, projective surfaces over algebraically closed fields of characteristic $p>0$. My approach is based on a study of slopes of Frobenius morphism on crystalline cohomology of $X$ and of the de Rham-Witt complex of $X$. In particular my methods do not require any lifting hypothesis.
つづく 0121現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/23(土) 13:52:55.17ID:iKDSmfWl>>120 つづき 下記論文を、 27 Aug 2019に投稿しているね (this is also inspired by Mochizuki's results)などと記されている https://arxiv.org/abs/1906.06840 Mochizuki's anabelian variation of ring structures and formal groups Kirti Joshi (last revised 27 Aug 2019 (this version, v2)) (抜粋) I show that there is a universal formal group (over a suitable (non-zero) ring) which is equipped with an action of the multiplicative monoid O? of non-zero elements of the ring of integers of a p-adic field. Lubin-Tate formal groups also arise from this universal formal group. If two p-adic fields have isomorphic multiplicative monoids O? then the additive structure of one arises from that of the other by means of this universal formal group law (in a suitable manner). In particular if two p-adic fields have isomorphic absolute Galois groups then it is well-known that the two respective monoids O? are isomorphic and so this construction can be applied to such p-adic fields. In this sense this universal formal group law provides a single additive structure which binds together p-adic fields whose absolute Galois groups are isomorphic (this anabelian variation of ring structure is studied and used extensively by Shinichi Mochizuki). In particular one obtains a universal (additive) expression for any non-zero p-adic integer (in a given p-adic field) which is independent of the ring structure of the p-adic field (this is also inspired by Mochizuki's results). These ideas extend to geometric situations: for a smooth curve X/K there is a universal K(X)?-formal group (here K(X)? is the monoid of non-zero meromorphic functions on a smooth curve X/K over a p-adic field K, which binds together all the additive structures on K(X)?∪{0} compatibly with the universal additive structure on K?∪{0} and hence a non-zero meromorphic function on X is given by a universal additive expression which is independent of the ring structure of K(X)?∪{0} 0122１３２人目の素数さん2019/11/23(土) 16:17:43.30ID:Wq0d9HjX o類昌俊 0123１３２人目の素数さん2019/11/23(土) 16:18:13.83ID:Wq0d9HjX o類昌俊 0124現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/23(土) 18:28:50.78ID:iKDSmfWl>>64 ＞そもそもホッジ理論とアラケロフ理論ってのは元々解析的な観点では離れた理論ではないんであって、 ＞望月さんは遠アーベル幾何学的な代数寄りのアプローチをしたのが当時新しかったわけで
IUTからみで、前半2回のworkshopは リストアップされている しかし、後半2回のworkshopは、リストにないね（＾＾； ３） Invitation to inter-universal Teichmuller Theory (IUT) RIMS workshop, September 1 - 4 2020 ４） Inter-universal Teichmuller Theory (IUT) Summit 2020 RIMS workshop, September 8 - 11 2020
https://kskedlaya.org/ Kiran Sridhara Kedlaya (抜粋) Professor of Mathematics Stefan E. Warschawski Chair in Mathematics Department of Mathematics, Room 7202 University of California, San Diego https://kskedlaya.org/confs.cgi Conferences in arithmetic geometry 2020 ・Foundations and Perspectives of Anabelian Geometry, May 18-22, Kyoto, Japan ・Combinatorial Anabelian Geometry and Related Topics, June 29-July 3, Kyoto, Japan 0131現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/23(土) 22:01:07.08ID:iKDSmfWl>>56 >Jakob Stix (Frankfurt Univ., Germany),
”Riemann”でファイル内検索かけると http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF) (抜粋) P34 Finally, in the context of the normalized determinants that appear in (a), it is interesting to note the role played by the prime number theorem ? i.e., in essence, the Riemann zeta function [cf. Proposition 1.6 and its proof] ? in the computation of “inter-universal analytic torsion” given in the proof of Theorem 1.10.
P48 In this context, it is of interest to observe that the form of the “ term” δ1/2 ・ log(δ) is strongly reminiscent of well-known intepretations of the Riemann hypothesis in terms of the asymptotic behavior of the function defined by considering the number of prime numbers less than a given natural number. Indeed, from the point of view of weights [cf. also the discussion of Remark 2.2.2 below], it is natural to regard the [logarithmic] height of a line bundle as an object that has the same weight as a single Tate twist, or, from a more classical point of view, “2πi” raised to the power 1. On the other hand, again from the point of view of weights, the variable “s” of the Riemann zeta function ζ(s) may be thought of as corresponding precisely to the number of Tate twists under consideration, so a single Tate twist corresponds to “s = 1”. 0143現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/24(日) 15:19:50.24ID:GGJQySam あれま～！ このスレが4位だよｗ（＾＾；
http://www.kurims.kyoto-u.ac.jp/~motizuki/research-japanese.html 望月新一の過去と現在の研究 南出新氏による、IUTeichにおける明示的な不等式に関する講演のスライドを掲載 http://www.kurims.kyoto-u.ac.jp/~motizuki/Minamide%20---%20Explicit%20estimates%20in%20inter-universal%20Teichmuller%20theory%20(in%20progress).pdf Explicit estimates in inter-universal Teichm¨uller theory (in progress) (joint work w/ I. Fesenko, Y. Hoshi, S. Mochizuki, and W. Porowski) Arata Minamide RIMS, Kyoto University November 2, 2018 (引用終り) 以上 0158現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/25(月) 20:57:10.67ID:1A25DpO+>>156 因みに、Ivan Fesenko 氏 （東工大はIUT派か） http://www.math.titech.ac.jp/~somekawa/AGS/AGSeminarTIT.html 東工大 数論・幾何学セミナー 10月24日（水） 16:00～17:00 東工大本館2階 234セミナー室 （いつもと曜日と場所が異なりますので御注意下さい！） Ivan Fesenko 氏（University of Nottingham） 「Two 2d adelic structures on elliptic surfaces and the BSD conjecture」 要旨： Two-dimensional local non-archimedean local fields arising from two-dimensional arithmetic geometry, e.g. formal power series over p-adic numbers, have two distinct integral structures: of rank 1 and of rank 2. Correspondingly, there are two distinct two-dimensional adelic structures on elliptic surfaces. Interestingly, they have a number of similarities with two symmetries of IUT. My talk will explain how an interaction between the two adelic structures on proper models of elliptic curves over global fields helps us to understand the meaning of the classical BSD conjecture and produce its equivalent reformulation in purely adelic terms. Part of this work is joint work with W. Czerniawska and P. Dolce. （google訳） 2次元算術幾何学から生じる2次元局所非アルキメデス局所場、例えば p進数上の正式なべき級数には、ランク1とランク2の2つの異なる積分構造があります。 これに対応して、楕円面には2つの異なる2次元のアデリック構造があります。 興味深いことに、IUTの2つの対称性と多くの類似点があります。 私の講演では、グローバルフィールド上の楕円曲線の適切なモデル上の2つのアデル構造間の相互作用が、古典的なBSD予想の意味を理解し、純粋なアデル用語で同等の再定式化を生成する方法を説明します。 この作業の一部は、W。チェルニアウスカおよびP.ドルチェとの共同作業です。 0159現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2019/11/25(月) 21:02:06.32ID:1A25DpO+>>157 補足
"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
