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 0598現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/25(日) 10:36:05.31ID: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],”
http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc_ver6.pdf 山下剛サーベイ A PROOF OF ABC CONJECTURE AFTER MOCHIZUKI GO YAMASHITA Date: August 31, 2017. P25 where qE,v = e^2πiτv and τv is in the upper half plane. P94 Lemma 7.4. ([EtTh, Proposition 1.4]) Put where q := e^2πiτ , and U¨ := e^πiz) 0600現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/25(日) 10:58:06.19ID:eIdDsFH8>>591 ”https://ja.wikipedia.org/wiki/%E3%82%AC%E3%82%A6%E3%82%B9%E5%92%8C ガウス和 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。”
http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations.pdf THE ETALE THETA FUNCTION AND ´ ITS FROBENIOID-THEORETIC MANIFESTATIONS Shinichi Mochizuki December 2008 P20 Proposition 1.4. (Relation to the Classical Theta Function) so Θ¨ extends uniquely to a meromorphic function on Y¨ [cf., e.g., [Mumf ], pp.306-307].
[Mumf] D. Mumford, An Analytic Construction of Degenerating Abelian Varieties over Complete Rings, Appendix to [FC]. 0602現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/25(日) 17:08:18.11ID:eIdDsFH8>>601 >q-parameters
モジュラー曲線上の函数としての扱い C の格子 Λ は C 上の楕円曲線 C/Λ を決定する。上で格子の集合上の函数とみなせることを説明したが、同じように楕円曲線の集合の上の函数ともみなすことができる。このようにして、モジュラー形式はモジュラー曲線の上の直線束の切断と考えることができる。たとえば、楕円曲線の j-不変量はモジュラー曲線の有理関数体の生成元である。 直線束の切断としての解釈は次のように説明できる。ベクトル空間 V にたいし射影空間 P(V) 上の函数を考える。V 上の函数 F で V の元 v ≠ 0 の成分の多項式であって、等式 F(cv) = F(v) を 0 でない任意のスカラー c についてみたすようなものを考えると、そのようなものは定数函数しか存在しない。条件をゆるめて多項式の代わりに分母をつけて有理函数を考えれば、F として同じ次数のふたつの斉次多項式の比とすることができる。あるいは F は多項式のままにしておいて、定数 c に関する条件を F(cv) = ckF(v) と緩めれば、そのような函数は k 次の斉次多項式である。斉次多項式の全体は実際には P(V) 上の函数ではないのだから、P(V) の函数が記述する幾何学的な内容を、本当に斉次多項式が記述できるのかと考えるのは自然である。これは代数幾何学において層（この場合は直線束）の切断を考える事に相当する。これは、モジュラー形式についての状況とちょうど対応する話になっている。
予想に谷山の名前が付いているのは，1955年に日光で行われた代数的整数論の国際シンポジウムにおいて谷山豊が楕円曲線と保型形式（automorphic form）との関連について問題の形で言明したことによります．ただし，谷山自身はモジュラー関数だけでは不十分だろうと思っていたようです（[5], pp. 188-189, [11], pp. 248-251）．
数学者サージ・ラングが，この予想に関するヴェイユの発言を徹底的に調べ上げ，その調査結果を「ラング・ファイル」あるいは「谷山・志村ファイル」と呼ばれる文書にまとめたという話は有名です（[5]，pp. 188-191, [8], pp. 137-157）．彼は，ヴェイユが当初予想が成り立つことを信じてはおらず，この予想の成立にはなんの貢献もしていなかったと断定しました．
ご参考 http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kokai-koza/yasuda.pdf 平成19年度(第29回)数学入門公開講座テキスト(京都大学数理解析研究所) R = T 定理の仕組みとその応用 安田 正大
この講座では, Fermat 予想の証明のために Wiles, Taylor-Wiles が確立した R = T 定理に関する最近の発展と応用についてお話します.
ここで考えている反例 a^l + b^l = c^l において, 条件 a, b, c の最大公約数が 1 であり, さらに a + 1 が 4 の倍数で b が偶数であると仮定しても一般性を失わないのでそう仮定することにします. このとき楕円曲 線 Ea,b が存在するとすると, 非常におかしなことが起こるということに Frey は気づきました. 一般に有理 数体上の楕円曲線 E が与えられると, E の極小判別式と呼ばれる整数 ?E と E の導手と呼ばれる正の整 数 NE とが定まります. E の導手のほうが E の極小判別式の絶対値よりも小さいのですが, E = Ea,b に 関しては NE が ?E と比べて極端に小さくなります. ところが Szpiro の予想1という予想があって, E の 導手が E の極小判別式と比べて極端に小さくなることはないと思われているので Ea,b が存在するとする とおかしなことになります.
16. R = T Mazur は R を考えるアイデアを創始し, いろんなアプローチによる R の研究方法を提唱しました. その うちの一つとして, 上の設定とは少し異なるモジュライ問題の下で, 写像 R → T を考え, それが同型である ことを肥田の変形というものを用いて示しました. Wiles [W] と Taylor-Wiles [TW] は, 上に設定したよう な状況の下での同型 R → T の証明の基本戦略を開発し, それを用いて特別な場合の谷山-志村予想を解決し ました. (引用終り) 以上 0634１３２人目の素数さん2020/10/30(金) 05:21:46.84ID:iuPqYV+w>>632 >ほいよ
https://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions Weierstrass's elliptic functions (抜粋) In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass. This class of functions are also referred to as p-functions and generally written using the symbol p (a calligraphic lowercase p). The p functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points. They can be used to parametrize elliptic curves over the complex numbers, thus establishing an equivalence to complex tori. Genus one solutions of differential equations can be written in terms of Weierstrass elliptic functions. Notably, the simplest periodic solutions of the Korteweg?de Vries equation are often written in terms of Weierstrass p-functions. ＜google英訳＞ ワイエルシュトラスの楕円関数である楕円関数特に単純な形をとります。それらはカールワイエルシュトラスにちなんで名付けられました。このクラスの関数はp関数とも呼ばれ、一般に記号p（カリグラフィの小文字のp）を使用して記述されます。p関数は、トーラスによるリーマン球の分岐した二重被覆を構成し、4点で分岐します。これらを使用して、複素数の楕円曲線をパラメーター化し、複素トーラスとの同等性を確立できます。の属1ソリューション微分方程式は、ワイエルシュトラスの楕円関数で書くことができます。特に、Korteweg?de Vries方程式の最も単純な周期解は、Weierstrassのp関数で記述されることがよくあります。 0637現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/10/30(金) 17:22:53.69ID:ANa+nMVb>>629 追加 > ６．つまりは、p > 5で a^p+b^p=c^p→ 楕円曲線 y2=x(x-a^p)(x+b^p) →谷山・志村予想（モジュラリティ定理（ｑ展開））＋ε予想→フェルマーの最終定理解決 > という流れだったのです
（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 Teichmuller Theory. PDF NEW !! (2020-04-04) P3 “elliptic curve” whose q-parameters are the N-th powers “q^N ” 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, q^N “≒” q [cf. §2.3, §2.4].
http://www.kurims.kyoto-u.ac.jp/~gokun/DOCUMENTS/abc2019Jul5.pdf 山下剛サーベイA proof of the abc conjecture after Mochizuki.preprint. Go Yamashita last updated on 8/July/2019 P27 (4) l is a prime number l ≧ 5 such that l is prime 略 the q-parameters of EF P39 where E has bad reduction with q-parameter qE,v 略 where qE,v = e^2πiτv and τv is in the upper half plane. (引用終り) 以上 0639１３２人目の素数さん2020/10/30(金) 19:53:06.70ID:iuPqYV+w>>636 ＞複素数の楕円曲線をパラメーター化し、複素トーラスとの同等性を確立できます まったく理解できてないでしょ
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
P23 LIST OF PARTICIPANTS (36). (xxxi) Seidai Yasuda, Osaka University, Japan;
