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

>>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)

http://rio2016.5ch.net/test/read.cgi/math/1592654877/187

435: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/10(月) 08:11:17.31 ID:gEQArxFG

メモ
https://carmonamateo.github.io/letters/LMC17.pdf
Dear Carmona, 13.11.2017
There is a very substantive mathematical difference between the theory of Galois categories/topoi as developed in SGA1/SGA4 and the theory of anabelioids as developed in my paper "The Geometry of Anabelioids":
Namely, the notion of slimness allows one to work with 1-categories of (slim) anabelioids, whereas the theory of Galois categories/topoi as developed in SGA1/SGA4 gives rise to 2-categories of Galois categories/topoi. In particular, "Galois groups"
(i.e., in the classical sense) arise naturally as groups of 1-morphisms in 1-categories of slim anabelioids, which is a very substantive mathematical difference from the way in which they arise in 2-categories of Galois categories/topoi, i.e., as groups of 2-morphisms in 2-categories.
This difference between 1- vs. 2-categories or 1- vs. 2-morphisms plays a fundamental role in the theory of anabelioids (as developed both in my paper "The
Geometry of Anabelioids", as well as in subsequent papers, e.g., papers on combinatorial anabelian geometry). Put another way, this difference may be understood
as being analogous to the difference between
Algebraic spaces (which form a 1-category)
and
(Deligne-Mumford) algebraic stacks (which form a 2-category).

Of course, algebraic spaces and (Deligne-Mumford) algebraic stacks are closely
related, in the sense that both arise by considering gluing operations in the etale
topology of schemes. On the other hand, the substantive difference between 1-
and 2-categories gives rise to many substantive mathematical differences in various
geometric arguments. In particular, this substantive difference between 1- and 2-
categories is sufficiently significant as to render extremely strange and unnatural
any attempt to use the same terminology for both algebraic spaces and algebraic
stacks.
Sincerely, Shinichi Mochizuki

http://rio2016.5ch.net/test/read.cgi/math/1592654877/435

598: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/25(日) 10:36:05.31 ID:eIdDsFH8

>>592
"Szpiro Conjectures. In this case, the height of a rational point may
be thought of as a suitable weighted sum of the valuations of the q-parameters of
the elliptic curve determined by the rational point at the nonarchimedean primes of potentially multiplicative reduction [cf. the discussion at the end of [Fsk], §2.2; [GenEll],”

”q-parameter”：多分下記の楕円テータ関数 「q = e^2πiτ」だろうね（＾＾；
https://member.ipmu.jp/yuji.tachikawa/
Yuji Tachikawa 立川裕二
https://member.ipmu.jp/yuji.tachikawa/lectures/
List of lectures
(抜粋)
・2016年10月 場の量子論の数学と二次元四次元対応 (第67回「数学との遭遇」中央大) [詳細]
・2012年5月 数学者のための場の理論 (駒場) [講義ノート]
・2012年10月 数学者のための超対称場の理論 (京都大) [講義のページ]
https://member.ipmu.jp/yuji.tachikawa/lectures/2014-butsurisuugaku2/
https://member.ipmu.jp/yuji.tachikawa/lectures/2014-butsurisuugaku2/notes.pdf
物理数学II (2014)講義ノート
(抜粋)
P14
楕円テータ関数
昔は q = e^πiτ
最近は (すくなくとも純粋数学および弦理論では)
q = e^2πiτ。
Mathematica はまだ前者の定義。

https://ja.wikipedia.org/wiki/%E3%83%86%E3%83%BC%E3%82%BF%E9%96%A2%E6%95%B0
テータ関数
楕円テータ関数の定義
楕円テータ関数（だえんテータかんすう、英: elliptic theta function）は、以下のように定義された関数である[10][9]。 ただし、Im τ >0, q:=e^πiτ である。
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1592654877/598

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