>>490 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
上記より下記引用 ・Gaussian integral ∫ ∞ -∞ e-x2 dx = √π ・[archimedean and nonarchimedean] valuations ・Changes of universe as arithmetic changes of coordinates 関連
P6 § 1. Review of the computation of the Gaussian integral § 1.1. Inter-universal Teichm¨uller theory via the Gaussian integral The goal of the present paper is to pave the road, for the reader, from a state of complete ignorance of inter-universal Teichm¨uller theory to a state of general appreciation of the “game plan” of inter-universal Teichm¨uller theory by reconsidering the well-known computation of the Gaussian integral ∫ ∞ -∞ e-x2 dx = √π via polar coordinates from the point of view of a hypothetical high-school student who has studied one-variable calculus and polar coordinates, but has not yet had any exposure to multi-variable calculus.
P7 § 1.3. Introduction of identical but mutually alien copies
P12 § 2. Changes of universe as arithmetic changes of coordinates § 2.1. The issue of bounding heights: the ABC and 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,
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.
P28 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 ex
528 名前：ample 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”! (引用終り)
>>495 追加 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
＜”宇宙”について＞ これ、望月氏の 宇宙 ”relationships between universes”の説明が、下記にあるけど 結構独特で、世間的には、ちょっとズレている気がする。「複数の宇宙の使用は、1960年代の数学」（下記）とかね 一方、（後述の）ちょうど1960年代に、数学基礎論で強制法が考えられて、「強制法は集合論の宇宙 V をより大きい宇宙 V* に拡大する」（下記）みたいな話がある だから、数学基礎論の強制法を知っている人（あるいは、いまどき ”universe”の数学的意味を検索した人）は、IUTの”Inter-universal”という語法に違和感を感じる気がする 代数系なり代数幾何にしろ、集合論や圏論としても、せいぜい集合と類までで収まるはず。（圏論でも、”局所的に小さい (locally small) ”で収まるはず）”宇宙”は、普通は出てこない
＜下記に対訳を作ってみた＞ ＜原文＞ P27 § 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.
＜原文＞ 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.
＜原文＞ 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”!
＜原文＞ 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”.
（>>527） >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
581 名前：コース 原書名 Basic Category Theory 著者名 斎藤 恭司 監修 土岡 俊介 訳 丸善出版 2017年01月 ＜arxiv公開＞ https://arxiv.org/abs/1612.09375 Basic Category Theory Tom Leinster [v1] Fri, 30 Dec 2016 03:02:01 UTC (210 KB) Journal reference: Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, 2014 Download:PDF https://arxiv.org/pdf/1612.09375
P2 （圏論の‘universal’の説明で、the universe of sets と使っている） Properties such as this are called ‘universal’ because they state how the object being described (in this case, the set 1) relates to the entire universe in which it lives (in this case, the universe of sets). The property begins with the words ‘for all sets X’, and therefore says something about the relationship between 1 and every set X: namely, that there is a unique map from X to 1.
P168 (toposの説明で、‘universe of sets’と使っている) For instance, a topos can be regarded as a ‘universe of sets’: Set is the most basic example of a topos, and every topos shares enough features with Set that one can reason with its objects as if they were sets of some exotic kind. On the other hand, a topos can be regarded as a generalized topological space: every space gives rise to a topos (namely, the category of sheaves on it), and topological properties of the space can be reinterpreted in a useful way as categorical properties of its associated topos. (引用終り)
英文で、universeの箇所を引用したが、‘universe of sets’とかで、 ”relates to the entire universe in which it lives (in this case, the universe of sets).”とされている 望月IUTの‘universe’は、明らかに、Leinster氏の書いている意味とは違う気がする もっとも、Leinster氏も‘universe’の厳密な定義を、書いていない（多分、‘universe’の厳密な定義を必要としないからでしょう （P2とP168との間でuniversalは使うが、‘universe’は使わない））
>>527 追加 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
Mutually Alien Copies に関連しそうなところを、下記に引用すると 1）N ・ h “=～” h N be a fixed natural number > 1 2）qN “=～” q 3）“alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”. とか、そのまま読むと、望月ワールド全開で、NHKスペシャル見ているから「同じものを別と見て、かつ同一視する」でしたか、ああこのことかと思いました 普通に読むと、読めないでしょうね ” Gaussian integral に繋げないんだろう”と好意的に読むと、気持ちは分かりますがね（これ数学として成り立つ？ｗ） ここ、説明の一つの山でしょね
(引用開始) P3 Introduction Let N be a fixed natural number > 1. Then the issue of bounding a given nonnegative real number h ∈ R?0 may be understood as the issue of showing that N ・ h is roughly equal to h, i.e., N ・ h “=～” h [cf. §2.3, §2.4]. When h is the height of an elliptic curve over a number field, this issue may be understood as the issue of showing that the height of the [in fact, in most cases, fictional!] “elliptic curve” whose q-parameters are the N-th powers “qN ” of the q-parameters “q” of the given elliptic curve is roughly equal to the height of the given elliptic curve, i.e., that, at least from the point of view of [global] heights, qN “=～” q [cf. §2.3, §2.4].
In order to verify the approximate relation qN “=～” q, one begins by introducing two distinct - i.e., two “mutually alien” - copies of the conventional scheme theory surrounding the given initial Θ-data. Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”.
These two mutually alien copies of conventional scheme theory are glued together - by considering relatively weak underlying structures of the respective conventional scheme theories such as multiplicative monoids and profinite groups - in such a way that the “qN ” in one copy of scheme theory is identified with the “q” in the other copy of scheme theory. This gluing is referred to as the Θ-link. Thus, the “qN ” on the left-hand side of the Θ-link is glued to the “q” on the right-hand side of the Θ-link, i.e., qNLHS “=” qRHS [cf. §3.3, (vii), for more details]. Here, “N” is in fact taken not to be a fixed natural number, but rather a sort of symmetrized average over the values j2, where j = 1,...,l*, and we write l* def = (l ? 1)/2. Thus, the left-hand side of the above display {qj2LHS}j bears a striking formal resemblance to the Gaussian distribution. One then verifies the desired approximate relation qN “=～” q by computing {qj2LHS}j - not in terms of qLHS [which is immediate from the definitions!], but rather - in terms of [the scheme theory surrounding] qRHS [which is a highly nontrivial matter!]. (引用終り) 以上
因みに、” the familiar Galois module “Z＾(1)””とか合ったので下記を引用しておきます
(引用開始) P17 § 2.6. Positive characteristic model for mono-anabelian transport In this example, Galois groups, or ´etale fundamental groups, in some sense play the role that is played by tangent bundles in the classical theory - a situation that is reminiscent of the approach of the [scheme-theoretic] Hodge-Arakelov theory of [HASurI], [HASurII], which is briefly reviewed in §2.14 below. 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. (引用終り) 以上
>>558 追加 > 3）“alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”. >とか、そのまま読むと、望月ワールド全開で、NHKスペシャル見ているから「同じものを別と見て、かつ同一視する」でしたか、ああこのことかと思いました >普通に読むと、読めないでしょうね
下記 フェセンコサーベイ （星の遠アーベル幾何学の進展 数学 vol74-No1 に紹介されている 文献の[6]） を読んでいる ”such gluing isomorphisms by applying various tautological Galois-equivariance properties of such gluing isomorphisms ” (google訳 そのような接着同型の様々なトートロジー的ガロア同変特性を適用することによるそのような接着同型 ) とか 出てくるんだよね（下記） 知らない人には、「え～」てなものでしょうね まして、ショルツェ氏のように、直接IUTの論文を読むと、あまりの奇想天外の発想についていけず 自分なりの独自解釈をしてしまいそうですねｗ
(参考) https://ivanfesenko.org/?page_id=126 Research ? Ivan Fesenko https://ivanfesenko.org/wp-content/uploads/2021/10/notesoniut.pdf [L1] Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Europ. J. Math. (2015) 1:405?440 P15 Monoid-theoretic structures are of essential importance in IUT, since they allow one to construct various gluing isomorphisms. The use of Galois and arithmetic fundamental groups gives rise to canonical splittings objects arising from such gluing isomorphisms by applying various tautological Galois-equivariance properties of such gluing isomorphisms. The computation of the theta-link can be viewed as a sort of passage from monoid-theoretic data to such canonical splittings involving arithmetic fundamental groups, by applying generalised Kummer theory, together with various multiradial algorithms which make essential use of mono-anabelian geometry.
Buzzardが基調講演で取り上げるってことは 既に幾度も書かれていますよ。 今の数学界におけるIUT理論の評価はまさしく、 「一般には受け入れられていない」です。 arXiv:2112.11598v2 (18 Apr 2022) >A great example is Mochizuki’s claimed proof of the ABC >conjecture [Moc21]. This proof has now been published >in a serious research journal, however it is clear that it is >not accepted by the mathematical community in general.
Theorem 3.11 in Part III is somehow reinterpreted in Corollary 3.12 of the same paper in a way that relates to the kind of diophantine inequalities one wishes to prove. One constructs certain arithmetic line bundles of interest within each theatre, a theta version and a q-version (which at the places of bad reduction arises essentially from the q-parameter of the corresponding Tate curve), which give rise to certain theta and q-objects in certain (products of) Frobenioids: the theta and q-pilots. By construction the theta pilot maps to the q-pilot via the horizontal link in the log-theta lattice. One can then proceed and compare the log-volumes of the images of these two objects in the relevant objects constructed via the multiradial algorithm in Theorem 3.11.
>>527 追加 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf P6 Acknowledgements: The author wishes to express his appreciation for the stimulating comments that he has received from numerous mathematicians concerning the theory exposed in the present paper and, especially, his deep gratitude to Fumiharu Kato, Akio Tamagawa, Go Yamashita, Mohamed Sa¨?di, Yuichiro Hoshi, Ivan Fesenko, Fucheng Tan, Emmanuel Lepage, Arata Minamide, and Wojciech Porowski for the very active and devoted role that they played both in discussing this theory with the author and in disseminating it to others. []
>>575 >Theorem 3.11 in Part III is somehow reinterpreted in Corollary 3.12 of the same paper in a way that relates to the kind of diophantine inequalities one wishes to prove. One constructs certain arithmetic line bundles of interest within each theatre, a theta version and a q-version (which at the places of bad reduction arises essentially from the q-parameter of the corresponding Tate curve), which give rise to certain theta and q-objects in certain (products of) Frobenioids: the theta and q-pilots. By construction the theta pilot maps to the q-pilot via the horizontal link in the log-theta lattice. One can then proceed and compare the log-volumes of the images of these two objects in the relevant objects constructed via the multiradial algorithm in Theorem 3.11.
