IUTを読むための用語集資料集スレ

[過去ﾛｸﾞ] IUTを読むための用語集資料集スレ (1002ﾚｽ)
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179(2): 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 09:47:14.02 ID:2Y0qBKwb(1/9) AAS
>>173 補足
新しい動きはもう始まっている

例えば下記
"In this paper I consider a different approach to this problem of understanding the fluidity of ring structures and in particular to the problem of quantifying the fluidity of the additive structures on the set OΔK ∪ {0} for a p-adic field K.
I began thinking of this problem in Kyoto (Spring 2018) and my preoccupation with it became more or less permanent on my return from Kyoto."
とかあるよね。Kirti Joshi氏は、自分なりに理解しようとしているんだ！

"Mochizuki’s anabelian reconstruction yoga (see [22] and its references) provides"とかある
yoga＝ヨガかｗ

”This paper would not be possible without Shinichi Mochizuki’s bold, audacious, deep and profoundly original ideas which continue to be a source of inspiration for me?in particular his truly remarkable and astounding discovery that there are arithmetic properties of non-zero elements of a fixed p-adic field which are independent of the ring structure of this field. ”
Mochizuki氏に対する最大限の賛辞ですね（＾＾

https://ejje.weblio.jp/content/yoga
yogaとは weblio
瑜伽(ゆが)、ヨガ、ヨガの行(ぎよう)、(身心の健康のために行なう)ヨガ
1【ヒンズー教】瑜伽(ゆが), ヨガ; ヨガの行(ぎよう) 《五感の作用を制して精神統一を旨とする瞑想(めいそう)的修行法》.

https://arxiv.org/pdf/1906.06840.pdf
Mochizuki’s anabelian variation of ring structures and formal groups
Kirti Joshi December 11, 2019
(抜粋)
1 A prelude
Shinichi Mochizuki has shown that a p-adic field K (the term p-adic field in this paper will mean a finite extension of Qp for some prime number p) can be recovered from its absolute Galois group GK (as a topological group) equipped with its filtration by inertia subgroups
(“the upper numbering filtration”)

つづく

180(1): 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 09:47:59.66 ID:2Y0qBKwb(2/9) AAS
>>179
つづき

; later he has refined this result and shown that K may be recovered from the topological group GK and one Lubin-Tate character of GK (see [14] and [20]).

On the other hand, given a padic field K1, the Jarden-Ritter Theorem (see [8]) provides a characterization of all p-adic fields
K2 such that one has a topological isomorphism GK2 ' GK1 of their absolute Galois groups and
it is well-known that for every prime p, pairs of fields with this property always exists.
Mochizuki’s anabelian reconstruction yoga (see [22] and its references) provides, starting with
the topological group G ' GK, the reconstruction amphora of G (see [6, 5, 7, 20, 21, 22] and
its references for the contents of the reconstruction amphora; in [9] I introduced this term as a
convenient short-form and memory-aid) which contains all quantities related to K which are reconstructed from the topological group G such as the prime p,
the topological monoids O^*K (the group of units of the ring of integers OK of K) and OΔK the multiplicative monoid of non-zero elements of OK (this notation is due to Mochizuki).

However the ring OK is not contained in the reconstruction amphora of G.
Moreover Mochizuki’s Reconstruction yoga also asserts that if one has an isomorphism of
topological groups
GK1 〜＝ GK2
then an isomorphism of the topological monoids
OΔK1 〜＝ OΔK2
may also be reconstructed from it (see [6, Section 2]).

つづく

181(1): 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 09:48:56.07 ID:2Y0qBKwb(3/9) AAS
>>180

つづき

In this paper I consider a different approach to this problem of understanding the fluidity of ring structures and in particular to the problem of quantifying the fluidity of the additive structures on the set OΔK ∪ {0} for a p-adic field K.
I began thinking of this problem in Kyoto (Spring 2018) and my preoccupation with it became more or less permanent on my return from Kyoto.

The idea, which I elaborate here, occurred to me in a recent lecture by Michael Hopkins at the Arizona Winter School (2019).
In one of his lectures, Hopkins narrated an anecdote about Daniel Quillen’s discovery of the role of formal groups in topological cohomology theories:
in particular Quillen’s assertion (to Hopkins) that “as addition rule for Chern classes fails to hold,
it must therefore fail in worst possible way?namely by means of a formal group”
(I am paraphrasing both Hopkins and Quillen here).

つづく

182: 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 09:49:20.20 ID:2Y0qBKwb(4/9) AAS
>>181
つづき

As was also pointed out to me by Taylor Dupuy, Mochizuki recognized a long time ago (see for
instance [15, Section 4]) that arithmetic applications of anabelian geometry lead naturally to the
deep and difficult problem of understanding the line bundles and (Arakelov) degrees (or Arakelov
Chern classes) in the presence of anabelian variation of ring structures and he resolved this problem
by means of his theory of Frobenioids and realified Frobenioids [18] and Arakelov-Hodge theoretic
evaluation methods culminating in [10, 11, 12, 13].

This paper would not be possible without Shinichi Mochizuki’s bold, audacious, deep and profoundly original ideas which continue to be a source of inspiration for me?in particular his truly remarkable and astounding discovery that there are arithmetic properties of non-zero elements of a fixed p-adic field which are independent of the ring structure of this field.
I am deeply indebted to him for many conversations on many topics surrounding his ideas and for his continued support and encouragement.
(引用終り)
以上

183: 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 15:00:43.76 ID:2Y0qBKwb(5/9) AAS
>>179
>新しい動きはもう始まっている
>例えば下記
>"In this paper I consider a different approach to this problem of understanding the fluidity of ring structures and in particular to the problem of quantifying the fluidity of the additive structures on the set OΔK ∪ {0} for a p-adic field K.
I> began thinking of this problem in Kyoto (Spring 2018) and my preoccupation with it became more or less permanent on my return from Kyoto."
>とかあるよね。Kirti Joshi氏は、自分なりに理解しようとしているんだ！

これがあるべき姿だと思う
自分なりに消化しようとしている

加藤文元先生の「まったく新しい考え」は、いいけど
「既存の数学では語れない」とか
それも1つの見方だろうが

じゃあ、「もっと新しい見方で、既存の数学とIUTを統一する数学を作ればいいべ」
というのが、Kirti Joshi 氏のスタンスだよね

これがあるべき姿だと思う
自分なりに消化しようとしている

185(2): 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 17:46:31.28 ID:2Y0qBKwb(6/9) AAS
>>166 追加

”Shimura curves”
http://www.math.columbia.edu/~chaoli/
Chao Li's homepage
http://www.math.columbia.edu/~chaoli/docs/ShimuraCurves.html
Shimura curves

In the 60s, Shimura studied certain algebraic curves as analogues of classical modular curves in order to construct class fields of totally real number fields. These curves were later coined "Shimura curves" and vastly generalized by Deligne. We will take a tour of the rich geometry and arithmetic of Shimura curves. Along the way, we may encounter tessellations of disks, quaternion algebras, abelian surfaces, elliptic curves with CM, Hurwitz curves ... and the answer to life, the universe and everything.

[-] Contents
Review of Modular Curves
Shimura curves
Moduli interpretation and class fields
Hurwitz curves

Briefly speaking, Shimura curves are simply one-dimensional Shimura varieties. I have accomplished my trivial notion task because I have told you a trivial notion. But obviously it does not help much if you do not know what the term Shimura varieties means. It only takes 5 chapters in Milne's notes in order to define them ? not too bad ? but initially Shimura invented them really because they are natural analogues of classical modular curves.

https://math.dartmouth.edu/~jvoight/articles/shimura-clay-proceedings-071707.pdf
Shimura curve computations
John Voight 1991 Mathematics Subject Classification.
Abstract. We introduce Shimura curves first as Riemann surfaces and then
as moduli spaces for certain abelian varieties. We give concrete examples of
these curves and do some explicit computations with them.
1. Introduction: modular curves

つづく

186(1): 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 17:47:17.30 ID:2Y0qBKwb(7/9) AAS
>>185
つづき

https://ja.wikipedia.org/wiki/%E5%BF%97%E6%9D%91%E5%A4%9A%E6%A7%98%E4%BD%93
志村多様体（Shimura variety）とは代数多様体であってモジュラー曲線の高次元化とみなせるような整数論で重要な対象である。

歴史
「志村多様体」と言う命名はピエール・ドリーニュ(Pierre Deligne)が導入し、彼は志村理論の中で独立した抽象的な形をしている部分の研究を推し進めた。ドリーニュの定式化では、志村多様体はホッジ構造のあるタイプのパラメータ空間である。このようにして、彼らは、レベル構造を持つ楕円曲線のモジュライ空間がそうであったように、モジュラ曲線の自然に高次元への一般化を作り出した。

例
d = 1 （例えば、F = Q や D ◯x R ＝〜 M2(R)）のとき、D× の十分小さな算術的部分群（英語版）(arithmetic subgroup)を固定すると、志村曲線を得ることができ、この構成から得られる曲線は既にコンパクトである（すなわち、射影的）。
明らかに方程式が知られている志村曲線の例は、以下の括弧の中の種数のフルヴィッツ曲線（英語版）(Hurwitz curve)である。

つづく

187: 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 17:48:03.31 ID:2Y0qBKwb(8/9) AAS
>>186

つづき

https://en.wikipedia.org/wiki/Shimura_variety
Shimura variety
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties.

History
In Deligne's formulation, Shimura varieties are parameter spaces of certain types of Hodge structures. Thus they form a natural higher-dimensional generalization of modular curves viewed as moduli spaces of elliptic curves with level structure.

Role in the Langlands program
Shimura varieties play an outstanding role in the Langlands program. The prototypical theorem, the Eichler?Shimura congruence relation, implies that the Hasse?Weil zeta function of a modular curve is a product of L-functions associated to explicitly determined modular forms of weight 2. Indeed, it was in the process of generalization of this theorem that Goro Shimura introduced his varieties and proved his reciprocity law. Zeta functions of Shimura varieties associated with the group GL2 over other number fields and its inner forms (i.e. multiplicative groups of quaternion algebras) were studied by Eichler, Shimura, Kuga, Sato, and Ihara.

以上
(引用終り)

188: 現代数学の系譜雑談 ◆yH25M02vWFhP [] 2020/07/19(日) 17:53:16.29 ID:2Y0qBKwb(9/9) AAS
>>185
”Shimura curves”は、志村多様体の１次元版か
でも、複素１次元ぽいな

”ピエール・ドリーニュ(Pierre Deligne)が導入し、・・彼らは、レベル構造を持つ楕円曲線のモジュライ空間がそうであったように、モジュラ曲線の自然に高次元への一般化を作り出した。”
とあるから、楕円曲線を拡張したものかね？（＾＾

” Zeta functions of Shimura varieties associated with the group GL2 over other number fields and its inner forms (i.e. multiplicative groups of quaternion algebras) were studied by Eichler, Shimura, Kuga, Sato, and Ihara.”
Sato＝佐藤幹夫かな？

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