つづき （参考） https://en.wikipedia.org/wiki/Axiom_schema Axiom schema https://plato.stanford.edu/entries/schema/ Stanford Encyclopedia of Philosophy Schema (抜粋) First published Fri May 28, 2004; substantive revision Tue Aug 2, 2016 A schema (plural: schemata, or schemas), also known as a scheme (plural: schemes), is a linguistic “template”, “frame”, or “pattern” together with a rule for using it to specify a potentially infinite multitude of phrases, sentences, or arguments, which are called instances of the schema. Schemas are used in logic to specify rules of inference, in mathematics to describe theories with infinitely many axioms, and in semantics to give adequacy conditions for definitions of truth.
複 数 の universe がある , とする 視 点 を 提 案 している 人 [ Ham12 ] もいる 。 この Hamkins の 論 文 は , n -Category Cafe や Math Over?ow ( ここ や ここ や ここ ) などで 話 題 にな っ ている 。 References [Low] Zhen Lin Low. Universes for category theory, arXiv:1304.5227 . https://arxiv.org/abs/1304.5227 [Submitted on 18 Apr 2013 (v1), last revised 28 Nov 2014 (this version, v2)] Universes for category theory Zhen Lin Low The Grothendieck universe axiom asserts that every set is a member of some set-theoretic universe U that is itself a set. One can then work with entities like the category of all U-
721 名前：sets or even the category of all locally U-small categories, where U is an "arbitrary but fixed" universe, all without worrying about which set-theoretic operations one may legitimately apply to these entities. Unfortunately, as soon as one allows the possibility of changing U, one also has to face the fact that universal constructions such as limits or adjoints or Kan extensions could, in principle, depend on the parameter U. We will prove this is not the case for adjoints of accessible functors between locally presentable categories (and hence, limits and Kan extensions), making explicit the idea that "bounded" constructions do not depend on the choice of U. (引用終り) 以上 []
11.^ この議論の発端は、MathOverflowの記事 Philosophy behind Mochizuki’s work on the ABC conjecture である https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture Philosophy behind Mochizuki's work on the ABC conjecture edited Jun 28 '13 at 1:13 community wiki James D. Taylor
Answers edited May 4 '18 at 5:21 community wiki Marty
edited Sep 9 '12 at 15:59 community wiki Minhyong Kim
edited Oct 20 '12 at 17:29 community wiki Vesselin Dimitrov
https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/244783/1/B76-02.pdf 宇宙際Teichmuller理論入門(On the examination and further development of inter-universal Teichmuller theory) 星 裕一郎 Aug-2019 数理解析研究所講究録別冊 B76 (抜粋) P83 § 1. 円分物 この §1 では, その対象の輸送の遂行の際に重要な役割を果たす 円分 物 (cyclotome) という概念についての解説を行います. 円分物とは何でしょうか. それは Tate 捻り “Zb(1)” のことです. 広義には, Zb(1) の 商や, あるいは, “(Q/Z)(1)” という可除な変種も円分物と呼ばれます. 遠アーベル幾何学 において, この円分物の “管理” は非常に重要です. この点について, もう少し説明しましょう. (引用終り)
冒頭からワカランｗ（＾＾； Tate 捻り “Zb(1)”？ 下記かな？ https://en.wikipedia.org/wiki/Tate_twist Tate twist (抜粋) In number theory and algebraic geometry, the Tate twist,[1] named after John Tate, is an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V) is a representation of GK on a finite-dimensional vector space V over the field Qp of p-adic numbers, then the Tate twist of V, denoted V(1), is the representation on the tensor product V?Qp(1), where Qp(1) is the p-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure Ks of K). More generally, if m is a positive integer, the mth Tate twist of V, denoted V(m), is the tensor product of V with the m-fold tensor product of Qp(1). Denoting by Qp(?1) the dual representation of Qp(1), the -mth Tate twist of V can be defined as V ◯X Q_p(-1)^{◯X m}. References 'The Tate Twist', in Lecture Notes in Mathematics', Vol 1604, 1995, Springer, Berlin p.98-102
>>685 追加 佐藤 周友先生ね（＾＾ https://ncatlab.org/nlab/show/Tate+twist Tate twist Last revised on February 9, 2018 Contents 1. Idea 2. Definition (抜粋) 1. Idea Cohomology theories often have two aspects, which one might refer to as geometric and arithmetic. A prototypical example is the l-adic cohomology Hi(X,Zl) of a (sufficiently nice) scheme X over a field k of characteristic p, with l coprime to p. This is not only an abelian group (the geometric aspect), but a representation of the absolute Galois group of k (the arithmetic aspect). In the case of the singular cohomology of a complex manifold, the ‘arithmetic’ aspect arises a
731 名前：s the Hodge structure on the cohomology groups. It has been speculated (for example by Manin?) that there should be some kind of ‘Galois group’ whose representations are Hodge structures, (and similarly for mixed Hodge modules vs perverse sheaves, and so on) but this remains mysterious; it may be that a good theory of algebraic geometry over F1 (the would-be “field with one element”) would provide an explanation. Tate twists play an important role in cohomology theories with this dual geometric and arithmetic aspect, allowing one to express Poincare duality canonically, that is, without choosing an orientation of one’s geometric object (scheme, complex manifold, …). 3. References https://arxiv.org/abs/math/0610426 [Submitted on 13 Oct 2006] p-adic etale Tate twists and arithmetic duality Kanetomo Sato Graduate School of Mathematics Nagoya University In this paper, we define, for arithmetic schemes with semistable reduction, p-adic objects playing the roles of Tate twists in etale topology, and establish their fundamental properties. Comments: 66 papges. to appear in Ann. Sci. Ec. Norm. Sup. (4) https://arxiv.org/pdf/math/0610426.pdf https://researchers.chuo-u.ac.jp/Profiles/3/0000248/profile.html?lang=ja 教授 サトウ カネトモ 佐藤 周友 []
>>685 追加 「the familiar Galois module “Z(1)”, i.e., the “Tate twist”」か（＾＾； （参考） www.kurims.kyoto-u.ac.jp/~motizuki/alien.pdf The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichm¨uller Theory By Shinichi Mochizuki Received xxxx xx, 2016. Revised xxxx xx, 2020. (抜粋) P17 2.6. Positive characteristic model for mono-anabelian transport
One notion of central importance in this example ? and indeed throughout inter-universal Teichm¨uller theory! ? is the notion of a cyclotome, a term which is used to refer to an isomorphic copy of some quotient [by a closed submodule] of the familiar Galois module “Z(1)”, i.e., the “Tate twist” of the trivial Galois module “Z”, or, alternatively, the rank one free Z-module equipped with the action determined by the cyclotomic character. Also, if p is a prime number, then we shall write Z=p for the quotient Z/Zp. Example 2.6.1. Mono-anabelian transport via the Frobenius morphism in positive characteristic.
参考 https://en.wikipedia.org/wiki/Belyi%27s_theorem Belyi's theorem This is a result of G. V. Belyi from 1979.
759 名前：At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.
Quotients of the upper half-plane It follows that the Riemann surface in question can be taken to be H/Γ with H the upper half-plane and Γ of finite index in the modular group, compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular curve.
Belyi functions A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C) ramified only over three points, which after a Mobius transformation may be taken to be {\displaystyle \{0,1,\infty \}}\{0,1,\infty \}. Belyi functions may be described combinatorially by dessins d'enfants.
Belyi functions and dessins d'enfants ? but not Belyi's theorem ? date at least to the work of Felix Klein; he used them in his article (Klein 1879) to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).[1]
Applications Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.
References 1 le Bruyn, Lieven (2008), Klein's dessins d'enfant and the buckyball. http://www.neverendingbooks.org/index.php/kleins-dessins-denfant-and-the-buckyball
https://en.wikipedia.org/wiki/G._V._Belyi G. V. Belyi Belyi won a prize of the Moscow Mathematical Society in 1981, and was an invited speaker at the International Congress of Mathematicians in 1986.[1] []
