Inter-universal geometry と ABC予想 (応援スレ) 55

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818: １３２人目の素数さん [sage] 2021/06/25(金) 00:09:14 ID:EpKfSbds

>>817
つづき

（参考）（キーワード：perfectoid shimura variety で検索した結果のトップ3）
Perfectoid Shimura varieties | SpringerLinkhttps:
P Scholze 著 ・ 2016 ・ 被引用数: 3 ? The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit a new kind of period map, called the Hodge?Tate period map, towards the flag variety.

Perfectoid Shimura varieties - CiNii 論文
Japanese journal of mathematics. 3rd series. Japanese journal of mathematics. 3rd series 11(1), 15-32, 2016. Springer Japan. Tweet. キーワード. Shimura varieties ・ Galois representations ・ perfectoid spaces. 各種コード. NII論文ID(NAID?) ...

Perfectoid Shimura varieties
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P Scholze 著 ・ 2016 ・ 被引用数: 3 ? in the cohomology of locally symmetric spaces such as arithmetic hyperbolic 3-?manifolds. The key technical tool was a theory of Shimura varieties with infinite level at p: As p-adic analytic spaces, they are perfectoid, and admit ...

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。
志村多様体ははじめ志村五郎により虚数乗法論の一般化の中で導入された。志村は解析的に定義されたその多様体が数論的な対象であることを示した。すなわち、志村多様体は反射体とよばれる数体の上定義される。
1970年代に、ピエール・ドリーニュ(Pierre Deligne)は、志村の仕事の公理的なフレームワークを作り出した。同時期にロバート・ラングランズ(Robert Langlands)は、ラングランズ・プログラムの文脈において、モチーフ的L-函数（英語版）(Motivic L-function)と保型形式のL-函数の対応のある例を志村多様体が作り上げることに注目した。志村多様体のコホモロジーの中に現れる保型形式は、一般的な保型形式よりも研究しやすい。たとえば、保型形式に対応するガロア表現を構成することができる。

つづく

http://rio2016.5ch.net/test/read.cgi/math/1623558298/818

819: １３２人目の素数さん [sage] 2021/06/25(金) 00:09:58 ID:EpKfSbds

>>818
つづき

https://en.wikipedia.org/wiki/Shimura_variety
Shimura variety
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.
On the basis of their results, Robert Langlands made a prediction that the Hasse-Weil zeta function of any algebraic variety W defined over a number field would be a product of positive and negative powers of automorphic L-functions, i.e. it should arise from a collection of automorphic representations.[1] However philosophically natural it may be to expect such a description, statements of this type have only been proved when W is a Shimura variety.[4]
In the words of Langlands:
To show that all L-functions associated to Shimura varieties ? thus to any motive defined by a Shimura variety ? can be expressed in terms of the automorphic L-functions of [his paper of 1970] is weaker, even very much weaker, than to show that all motivic L-functions are equal to such L-functions. Moreover, although the stronger statement is expected to be valid, there is, so far as I know, no very compelling reason to expect that all motivic L-functions will be attached to Shimura varieties.[5]
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1623558298/819

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