補足資料下記 熟読下さい（＾＾ https://en.wikipedia.org/wiki/Infinitesimal Infinitesimal (抜粋) Infinitesimals in teaching Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.[14][15]
14. Ely, Robert (2010). "Nonstandard student conceptions about infinitesimals" (PDF). Journal for Research in Mathematics Education. 41 (2): 117?146. JSTOR 20720128. Archived (PDF) from the original on 2019-05-06. 15. Katz, Karin Usadi; Katz, Mikhail G. (2010). "When is .999... less than1?" (PDF). The Montana Mathematics Enthusiast. 7 (1): 3?30. arXiv:1007.3018. ISSN 1551-3440. Archived from the original (PDF) on 2012-12-07. Retrieved 2012-12-07.
Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Consequently present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it.[4]（訳: 今日では、解析学の授業において無限小量について述べることはあまり一般的ではない。その結果、当世の学生はこの言葉づかいに全く習熟していない。にも拘らず、未だにそれを扱うことが必要である）
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf Research Institute for Mathematical Sciences - Kyoto University, Japan PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元 Online Seminar - Algebraic & Arithmetic Geometry Laboratoire Paul Painleve - Universite de Lille, France Version 1 - ε? - 09/10/2020
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/IUT-references.html Promenade in Inter-Universal Teichmuller Theory Org.: Collas (RIMS); Debes, Fresse (Lille).
The Programme of the seminar contains a selection of ~30 references with respect to (1) Diophantine Geometry, (2) IUT Geometry, and (3) Anabelian Geometry. We indicate some links towards the key opuses as well as some complementary notes and proceedings. 0536現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/17(土) 10:25:45.84ID:02Kfs2KS メモ https://afst.centre-mersenne.org/item/?id=AFST_2009_6_18_S2_5_0 https://afst.centre-mersenne.org/article/AFST_2009_6_18_S2_5_0.pdf The Way to the Proof of Fermat’s Last Theorem Gerhard Frey Annales de la Faculte des sciences de Toulouse : Mathematiques, Serie 6, Tome 18 (2009) no. S2, pp. 5-23.
http://backup.itsoc.org/review/05pl1.pdf The Way to the Proof of Fermat ’s Last Theorem Gerhard Frey 1This paper is based on a talk at the ISIT meeting 1997. The author wants to thank the organizers for the invitation and the warm hospitality 0537ぷっちゃん2020/10/17(土) 12:01:32.25ID:QjI40yYH ＞よくわからない定義に出くわしたとき，
幕府クン（＝慶喜クン） 「わけわからない発言で煙に巻いて誤魔化す」
数学に興味ないくせに、わかった風な顔をしたがるペテン師の態度ですねｗ 0538現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/17(土) 16:31:00.47ID:02Kfs2KS 下記、Goldfeld, Modular forms, elliptic curves, and the ABC-conjecture が、なかなか良いね
§1. The ABC-Conjecture. The ABC-conjecture was first formulated by David Masser and Joseph Osterl´e (see [Ost]) in 1985. Curiously, although this conjecture could have been formulated in the last century, its discovery was based on modern research in the theory of function fields and elliptic curves, which suggests that it is a statement about ramification in arithmetic algebraic geometry. The ABC-conjecture seems connected with many diverse and well known problems in number theory and always seems to lie on the boundary of what is known and what is unknown. We hope to elucidate the beautiful connections between elliptic curves, modular forms and the ABC-conjecture. Conjecture (ABC). Let A, B, C be non-zero, pairwise relatively prime, rational integers satisfying A + B + C = 0. Define N = Πp|ABC p to be the squarefree part of ABC. Then for every ε > 0, there exists κ(ε) > 0 such that max(|A|, |B|, |C|) < κ(ε)N1+ε. A weaker version of the ABC-conjecture (with the same notation as above) may be given as follows. Conjecture (ABC) (weak). For every ε > 0, there exists κ(ε) > 0 such that |ABC| 1/3 < κ(ε)N1+ε.
Conjecture. (Szpiro, 1981) Let E be an elliptic curve over Q which is a global minimal model with discriminant Δ and conductor N. Then for every ε > 0, there exists κ(ε) > 0 such that Δ < κ(ε)N6+ε. We show that Szpiro’s conjecture above is equivalent to the weak ABC-conjecture. Let A, B, C be coprime integers satisfying A + B + C = 0 and ABC 6= 0. Set N = Πp|ABCp. Consider the Frey-Hellegouarch curve EA,B : y2 = x(x - A)(x + B). A minimal model for EA,B has discriminant (ABC)2・ 2-s and conductor N ・ 2-t for certain absolutely bounded integers s, t, (see Frey [F1]). Plugging this data into Szpiro’s conjecture immediately shows the equivalence.
[F1] FREY, G., Links between stable elliptic curves and certain diophantine equations, Annales Universiatis Saraviensis, Vol 1, No. 1 (1986), 1-39. [F2] FREY, G., Links between elliptic curves and solutions of A-B=C, Journal of the Indian Math. Soc. 51 (1987), 117-145. (引用終り) 以上 0540ぷっちゃん2020/10/17(土) 17:23:37.39ID:QjI40yYH>>538 モジュラー形式も楕円曲線も理解できないシロウトには無縁だね
江戸総攻撃の前に行なわれた勝と新政府軍参謀西郷隆盛との交渉により、 江戸城は4月11日に新政府軍に明け渡された。 彰義隊や旧幕臣の暴発を恐れた慶喜は 4月11日午前3時に寛永寺大慈院を出て水戸へ向かった。 水戸では弘道館の至善堂にて引き続き謹慎した後、 7月に徳川家が駿府に移封されると、慶喜も駿河の宝台院に移って謹慎した。 これにより、徳川家による政権は幕を閉じた。 0546１３２人目の素数さん2020/10/18(日) 14:26:59.10ID:ufbJ1e15 フロべニオイドって自然な定義なのか？ 0547現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日) 19:38:34.42ID:ZLSkSSTT メモ貼る https://stacks.math.columbia.edu/bibliography The Stacks project Table of contentsBibliography (抜粋) Grothendieck, A., Standard conjectures on algebraic cycles Grothendieck, Alexander, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA 2) Grothendieck, Alexander, Fondements de la geometrie algebrique Grothendieck, Alexander, La theorie des classes de Chern Grothendieck, Alexander, Revetements etales et groupe fondamental (SGA 1) Grothendieck, Alexander, Sur quelques points d'algebre homologique Grothendieck, Alexander, Technique de descente et theoremes d'existence en geometrie algebrique. I. Generalites. Descente par morphismes fidelement plats Grothendieck, Alexander, Technique de descente et theoremes d'existence en geometrie algebrique. II. Le theoreme d'existence en theorie formelle des modules Grothendieck, Alexander, Techniques de construction et theoremes d'existence en geometrie algebrique. III. Preschemas quotients Grothendieck, Alexander, Techniques de construction et theoremes d'existence en geometrie algebrique. IV. Les schemas de Hilbert Grothendieck, Alexander and Dieudonne, Jean, Elements de geometrie algebrique I Grothendieck, Alexander and Dieudonne, Jean, Elements de geometrie algebrique I Grothendieck, Alexander and Dieudonne, Jean, Elements de geometrie algebrique II Grothendieck, Alexander and Murre, Jacob P., The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme Grothendieck, Alexander and Raynaud, Michel and Rim, Dock Sang, Groupes de monodromie en geometrie algebrique. I Grothendieck, Alexandre, Seminaire de geometrie algebrique du Bois-Marie 1965-66, Cohomologie l-adique et fonctions L, SGA5 0548現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/18(日) 19:51:52.82ID:ZLSkSSTT>>546 >フロべニオイドって自然な定義なのか？
P3 Modus Operandi & Leitfaden. As a new geometry, the essence of Mochizuki’s IUT is to introduce a new semiotic system - formalism, terminology, and their interactions - that can be unsettling at first. This programme proposes a 3 layers approach with precise references, examples, and analogies. Because IUT discovery also benefits from a non-linear and spiralling approach, we provide further indications for an independent wandering: Mochizuki recommends to start with the introductory [Alien] - young arithmetic-geometers can also consult [Fes15] for a shorter overview. We also recommend to begin with §Intro - §3.6-7 ibid. for a direct encounter with IUT’s semiotic, then to follow one’s own topics of interest according to Fig. 1, which also indicates some topic-wise references as entry-points - [EtTh], [GenEll], etc. Within the “canon” [IUTChI]-[IUTChIV], our recommendation is to start with [IUTChIII] §Introduction. Intuition of the reader can further rely on the strongly consistent terminology of IUT - e.g. Frobenioid, mono-anabelian transport, arithmetic analytic.
※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.
※ Hodge-Arakelov and p-adic Teichmuller theories stand as important models for IUT, which also relies on key categorical constructions - e.g. Frobenioids and anabelioids. These aspects are not included in this programme - we refer to [Alien] and the canon for references - they can be the object of additional talks by specialists. 以上 0554現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/20(火) 08:06:28.04ID:V6fYxSC9https://rio2016.5ch.net/test/read.cgi/math/1600350445/541 IUT と ABC予想 (応援スレ) 49
>※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.
http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元 Online Seminar - Algebraic & Arithmetic Geometry Laboratoire Paul Painleve - Universite de Lille, France (抜粋) P2 ※ In order to keep the length of this guide (incl. 〜 25 tables, figures, and diagrams) strictly shorter than the IUT corpus - 〜 1200 pages with a piece of anabelian geometry, 〜 675 pages for the canon, and 〜 170 pages for the introductory [Alien] - some details have been omitted, some approximations were made; they should be negligible for our goal. Content will be updated according to the progress of the seminar, see version and date.
P3 Within the “canon” [IUTChI]-[IUTChIV], our recommendation is to start with [IUTChIII] §Introduction. Intuition of the reader can further rely on the strongly consistent terminology of IUT - e.g. Frobenioid, mono-anabelian transport, arithmetic analytic.
※ Hodge-Arakelov and p-adic Teichmuller theories stand as important models for IUT, which also relies on key categorical constructions - e.g. Frobenioids and anabelioids. These aspects are not included in this programme - we refer to [Alien] and the canon for references - they can be the object of additional talks by specialists.
つづく 0555現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/20(火) 08:07:05.29ID:V6fYxSC9>>554 つづき >※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.
そうか、この[Alien]っていうのが、重要な論文なんだね〜（＾＾ P4 [Alien]： [Alien] S. Mochizuki, “The mathematics of mutually alien copies: From Gaussian integrals to Inter-universal Teichmuller theory,” RIMS Preprint no. 1854, 169p. Jul. 2016, Eprint available on-line.
http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf [7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-04-04)
Abstract Inter-universal Teichm¨uller theory may be described as a construction of certain canonical deformations of the ring structure of a number field equipped with certain auxiliary data, which includes an elliptic curve over the number field and a prime number ? 5. In the present paper, we survey this theory by focusing on the rich analogies between this theory and the classical computation of the Gaussian integral. The main common features that underlie these analogies may be summarized as follows: ・ the introduction of two mutually alien copies of the object of interest; ・ the computation of the effect -i.e., on the two mutually alien copies of the object of interest -of two-dimensional changes of coordinates by considering the effect on infinitesimals;
・ the passage from planar cartesian to polar coordinates and the resulting splitting, or decoupling, into radial -i.e., in more abstract valuation-theoretic terminology, “value group” -and angular -i.e., in more abstract valuation-theoretic terminology, “unit group” -portions; ・ the straightforward evaluation of the radial portion by applying the quadraticity of the exponent of the Gaussian distribution; ・ the straightforward evaluation of the angular portion by considering the metric geometry of the group of units determined by a suitable version of the natural logarithm function.
[Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”.] After reviewing the classical computation of the Gaussian integral, we give a detailed survey of inter-universal Teichm¨uller theory by concentrating on the common features listed above. The paper concludes with a discussion of various historical aspects of the mathematics that appears in inter-universal Teichm¨uller theory. (引用終り) 以上 0557１３２人目の素数さん2020/10/20(火) 16:18:17.10ID:8nlx/Wj4>>552 ＞頭に入れておくのが良い
https://en.wikipedia.org/wiki/Brian_Conrad Brian Conrad Brian Conrad (born November 20, 1970), is an American mathematician and number theorist, working at Stanford University. Previously, he taught at the University of Michigan and at Columbia University. Conrad and others proved the modularity theorem, also known as the Taniyama-Shimura Conjecture. He proved this in 1999 with Christophe Breuil, Fred Diamond and Richard Taylor, while holding a joint postdoctoral position at Harvard University and the Institute for Advanced Study in Princeton, New Jersey.
https://en.wikipedia.org/wiki/Kiran_Kedlaya Kiran Sridhara Kedlaya (/?k?r?n ??ri?d?r k?d?l??j?/;[2] born July 1974) is an Indian American mathematician. He currently is a Professor of Mathematics and the Stefan E. Warschawski Chair in Mathematics[3] at the University of California, San Diego. 0559現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/20(火) 17:34:00.99ID:lsCoo7pb>>558 誤変換タイポ訂正
例 基礎体を有理数体とすると、クロネッカー・ウェーバーの定理は、代数体 K が Q のアーベル拡大であることと、ある円分体 Q(ζn) の部分体であることが同値であることを言っている[15]。従って、K の導手はそのようなものの中で最も小さな n である。
局所導手や分岐との関係 大域導手は局所導手の積である。[17]
結局、有限素点が L/K で分岐していることと、それが f(L/K) を割ることは同値である。[18] 無限素点 v は導手の中にあらわれることと、v が実素点で、L で複素素点となることとが同値である。 0589現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/24(土) 20:32:01.25ID:i6I9Q5ne ＜転載＞ 純粋・応用数学(含むガロア理論)5 https://rio2016.5ch.net/test/read.cgi/math/1602034234/149 http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元
のP3で、Fig. 1. IUT, Topics & References as potential entry points. があるよね その図で、一番外のリングで灰色部分が、[Alien]： http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf [7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF
”http://www.kurims.kyoto-u.ac.jp/~bcollas/IUT/documents/RIMS-Lille%20-%20Promenade%20in%20Inter-Universal%20Teichm%C3%BCller%20Theory.pdf PROMENADE IN INTER-UNIVERSAL TEICHMULLER THEORY - 復元 のP3で、Fig. 1. IUT, Topics & References as potential entry points. があるよね その図で、一番外のリングで灰色部分が、[Alien]： http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf [7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF”
補足します（＾＾ ・Fig. 1（IUT曼荼羅）で、同心円 一番外が[Alien]、以下中心に向けて、IUT1〜4があり、IUT4が一番内側 ・外周は、ほぼ6等分され、頂点から右回りの各ゾーンで、1)IUT Geometry、2)Diophantine [GenEII]、3)Anabelian [AbTopIII]、4)Geometrical [IUTChII]、5)Category [Fr]-[An]、6)Meta-Abelian Theta [EtTh] と記されている ・そして、各ゾーンで白抜きで、プランクの箇所がところどころある。この部分、”無し”ってこと。 例えば、IUT4が関連するのは２つのゾーン、IUT GeometryゾーンとDiophantineゾーンのみ ・で、一番外が[Alien]のさらに外が、従来の数学界ってことなのでしょうね〜ｗ ・”※ We have also found the synthetic and selfcontent [Yam17] to be particularly helpful as a bridge between [Alien] and the “canon”.” とあるから、 [Alien] 読むのが良さそうってこと
歴史 このガウス和の別の表現は、次のようなものである： Σ{r} e^{2πir^2}/p} 二次ガウス和は、テータ関数の理論と密接に関連している。 0592現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/25(日) 09:22:48.97ID:eIdDsFH8>>590 メモ貼る （参考） http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.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) Contents § 2. Changes of universe as arithmetic changes of coordinates
§ 2.1. The issue of bounding heights: the ABC and Szpiro Conjectures A brief exposition of various conjectures related to this issue of bounding heights of rational points may be found in [Fsk], §1.3. In this context, the case where the algebraic curve under consideration is the projective line minus three points corresponds most directly to the so-called ABC and − by thinking of this projective line as the “λ-line” that appears in discussions of the Legendre form of the Weierstrass equation for an elliptic curve − 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], Proposition 3.4]. Here, it is also useful to recall [cf. [GenEll], Theorem 2.1] that, in the situation of the ABC or Szpiro Conjectures, one may assume, without loss of generality, that, for any given finite set Σ of [archimedean and nonarchimedean] valuations of the rational number field Q, the rational points under consideration lie, at each valuation of Σ, inside some compact subset [i.e., of the set of rational points of the projective line minus three points over some finite extension of the completion of Q at this valuation] satisfying certain properties.
つづく 0593現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/25(日) 09:23:08.65ID:eIdDsFH8>>592 つづき In particular, when one computes the height of a rational point of the projective line minus three points as a suitable weighted sum of the valuations of the q-parameters of the corresponding elliptic curve, one may ignore, up to bounded discrepancies, contributions to the height that arise, say, from the archimedean valuations or from the nonarchimedean valuations that lie over some “exceptional” prime number such as 2.
§ 2.2. Arithmetic degrees as global integrals
§ 2.7. The apparatus and terminology of mono-anabelian transport Example 2.6.1 is exceptionally rich in structural similarities to inter-universal Teichm¨uller theory, which we proceed to explain in detail as follows. One way to understand these structural similarities is by considering the quite substantial portion of terminology of inter-universal Teichm¨uller theory that was, in essence, inspired by Example 2.6.1: (i) Links between “mutually alien” copies of scheme theory: One central aspect of inter-universal Teichm¨uller theory is the study of certain “walls”, or “filters” − which are often referred to as “links” − that separate two “mutually alien” copies of conventional scheme theory [cf. the discussions of [IUTchII], Remark 3.6.2; [IUTchIV], Remark 3.6.1]. The main example of such a link in inter-universal Teichm¨uller theory is constituted by [various versions of] the Θ-link. The log-link also plays an important role in inter-universal Teichm¨uller theory. The main motivating example for these links which play a central role in inter-universal Teichm¨uller theory is the Frobenius morphism ΦηX of Example 2.6.1. From the point of view of the discussion of §1.4, §1.5, §2.2, §2.3, §2.4, and §2.5, such a link corresponds to a change of coordinates.
§ 2.10. Inter-universality: changes of universe as changes of coordinates One fundamental aspect of the links [cf. the discussion of §2.7, (i)] − namely, the Θ-link and log-link − that occur in inter-universal Teichm¨uller theory is their incompatibility with the ring structures of the rings and schemes that appear in their domains and codomains. In particular, when one considers the result of transporting an ´etale-like structure such as a Galois group [or ´etale fundamental group] across such a link [cf. the discussion of §2.7, (iii)], one must abandon the interpretation of such a Galois group as a group of automorphisms of some ring [or field] structure [cf. [AbsTopIII], Remark 3.7.7, (i); [IUTchIV], Remarks 3.6.2, 3.6.3], i.e., one must regard such a Galois group as an abstract topological group that is not equipped with any of the “labelling structures” that arise from the relationship between the Galois group and various scheme-theoretic objects. It is precisely this state of affairs that results in the quite central role played in inter-universal Teichm¨uller theory by results in [mono-]anabelian geometry, i.e., by results concerned with reconstructing various scheme-theoretic structures from an abstract topological group that “just happens” to arise from scheme theory as a Galois group/´etale fundamental group.
In this context, we remark that it is also this state of affairs that gave rise to the term “inter-universal”: That is to say, the notion of a “universe”, as well as the use of multiple universes within the discussion of a single set-up in arithmetic geometry, already occurs in the mathematics of the 1960’s, i.e., in the mathematics of Galois categories and ´etale topoi associated to schemes. On the other hand, in this mathematics of the Grothendieck school, typically one only considers relationships between universes − i.e., between labelling apparatuses for sets − that are induced by morphisms of schemes, i.e., in essence by ring homomorphisms. The most typical example of this sort of situation is the functor between Galois categories of ´etale coverings induced by a morphism of connected schemes. By contrast, the links that occur in inter-universal Teichm¨uller theory are constructed by partially dismantling the ring structures of the rings in their domains and codomains [cf. the discussion of §2.7, (vii)], hence necessarily result in much more complicated relationships between the universes − i.e., between the labelling apparatuses for sets − that are adopted in the Galois categories that occur in the domains and codomains of these links, i.e., relationships that do not respect the various labelling apparatuses for sets that arise from correspondences between the Galois groups that appear and the respective ring/scheme theories that occur in the domains and codomains of the links.
That is to say, it is precisely this sort of situation that is referred to by the term “inter-universal”. Put another way, a change of universe may be thought of [cf. the discussion of §2.7, (i)] as a sort of abstract/combinatorial/arithmetic version of the classical notion of a “change of coordinates”. In this context, it is perhaps of interest to observe that, from a purely classical point of view, the notion of a [physical] “universe” was typically visualized as a copy of Euclidean three-space. Thus, from this classical point of view, a “change of universe” literally corresponds to a “classical change of the coordinate system − i.e., the labelling apparatus − applied to label points in Euclidean three-space”! Indeed, from an even more elementary point of view, perhaps the simplest example of the essential phenomenon under consideration here is the following purely combinatorial phenomenon: Consider the string of symbols 010 − i.e., where “0” and “1” are to be understood as formal symbols. Then, from the point of view of the length two substring 01 on the left, the digit “1” of this substring may be specified by means of its “coordinate relative to this substring”, namely, as the symbol to the far right of the substring 01. In a similar vein, from the point of view of the length two substring 10 on the right, the digit “1” of this substring may be specified by means of its “coordinate relative to this substring”, namely, as the symbol to the far left of the substring 10. On the other hand, neither of these specifications via “substring-based coordinate systems” is meaningful to the opposite length two substring; that is to say, only the solitary abstract symbol “1” is simultaneously meaningful, as a device for specifying the digit of interest, relative to both of the “substring-based coordinate systems”.
Finally, in passing, we note that this discussion applies, albeit in perhaps a somewhat trivial way, to the isomorphism of Galois groups ΨηX : GK〜→ GK induced by the Frobenius morphism ΦηX in Example 2.6.1, (i): That is to say, from the point of view of classical ring theory, this isomorphism of Galois groups is easily seen to coincide with the identity automorphism of GK. On the other hand, if one takes the point of view that elements of various subquotients of GK are equipped with labels that arise from the isomorphisms ρ or κ of Example 2.6.1, (ii), (iii), i.e., from the reciprocity map of class field theory or Kummer theory, then one must regard such labelling apparatuses as being incompatible with the Frobenius morphism ΦηX . Thus, from this point of view, the isomorphism ΦηX must be regarded as a “mysterious, indeterminate isomorphism” [cf. the discussion of §2.7, (iii)]. (引用終り) 以上
