下記より ” and Hoshi-Mochizuki-Minamide, the construction of arithmetic operads ” とありますな、ワッハッハｗｗｗ
（参考） https://www.mfo.de/occasion/2110a/www_view The Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics) Homotopic and Geometric Galois Theory 7 Mar - 13 Mar 2021 ID: 2110a Organizers Benjamin Collas, Bayreuth Pierre Dèbes, Villeneuve d'Ascq Hiroaki Nakamura, Osaka Jakob Stix, Frankfurt Public Abstract https://www.mfo.de/document/2110a/Public-Abstract-2010a
46 名前：.pdf
Abstract A fundamental idea in studying the absolute Galois group of a field is to make it act on geometric objects such as Galois covers, étale cohomology groups and fundamental groups. The following research topics emphasize this seminal idea: (a) Galois covers, G-torsors and their parametrizing families, (b) motivic Galois representations, (c) anabelian towers of fundamental groups. Striking advances have recently shed new light on the whole topic:
(c) in Anabelian Geometry: the successful introduction of methods from étale homotopy theory (Schmidt-Stix) and from motivic A1-homotopy theory for moduli stacks of curves (Collas), the import of operads (Fresse-Horel) which echo the Galois techniques of Pop and Hoshi-Mochizuki-Minamide, the construction of arithmetic operads for Hurwitz moduli spaces (Westerland-Wickelgren). []
>>45 >The following research topics emphasize this seminal idea: (a) Galois covers, G-torsors and their parametrizing families, (b) motivic Galois representations, (c) anabelian towers of fundamental groups. >Striking advances have recently shed new light on the whole topic: >(c) in Anabelian Geometry: the successful introduction of methods from étale homotopy theory >(Schmidt-Stix) and from motivic A1-homotopy theory for moduli stacks of curves (Collas), the import >of operads (Fresse-Horel) which echo the Galois techniques of Pop and , >the construction of arithmetic operads for Hurwitz moduli spaces (Westerland-Wickelgren).
”Striking advances have recently shed new light on the whole topic:” ですね
ここで、”Striking advances have recently shed new light on the whole topic:”に該当するのは、”November 2020”の１）”Explicit estimates in inter-universal Teichmüller theory”ですね ”March 2017”は、ちょっと古いですね たぶんね
>>52 追加下記ご参考 2018年からの流れですね （参考） www.kurims.kyoto-u.ac.jp/~bcollas/documents/MFO-owr1816a_report_Introduction.pdf Mathematisches Forschungsinstitut Oberwolfach Report No. 17/2018 DOI: 10.4171/OWR/2018/17 Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups Organised by Benjamin Collas, Bayreuth Pierre Dèbes, Villeneuve d’Ascq Michael D. Fried, Billings 15 April – 21 April 2018
Abstract. The geometric study of the absolute Galois group of the rational numbers has been a highly active research topic since the first milestones: Hilbert’s Irreducibility Theorem, Noether’s program, Riemann’s Existence Theorem. It gained special interest in the last decades with Grothendieck’s “Esquisse d’un programme”, his “Letter to Faltings” and Fried’s introduction of Hurwitz spaces. It grew on and thrived on a wide range of areas, e.g. formal algebraic geometry, Diophantine geometry, group theory. The recent years have seen the development and integration in algebraic geometry and Galois theory of new advanced techniques from algebraic stacks, p-adic representations and homotopy theories. It was the goal of this mini-workshop, to bring together an international panel of young and senior experts to draw bridges towards these fields of research and to incorporate new methods, techniques and structures in the development of geometric Galois theory
P5 3. Galois Anabelian and Homotopical Geometry
Schmidt and Stix presented their joint work: they showed how to use étale homotopic methods and Mochizuki’s work to deduce the existence of anabelian Zariski-neighbourhoods in smooth variety of any dimension. Schmidt first explained the necessary requirements and difficulties in Artin-Mazur-Friedlander pointed-unpointed étale homotopy theory, then Stix presented the proof based on Tamagawa’s idea of Jacobian approximation of rational points via the existence of a certain retract.
例えば （>>5より） www.kurims.kyoto-u.ac.jp/~yuichiro/papers.html 星裕一郎の論文 (抜粋) １）Explicit estimates in inter-universal Teichmüller theory (with Shinichi Mochizuki, Ivan Fesenko, Arata Minamide, and Wojciech Porowski) RIMS Preprint 1933 (November 2020): (PDF). www.kurims.kyoto-u.ac.jp/~yuichiro/rims1933.pdf Abstract. We also obtain an explicit estimate concerning “Fermat’s Last Theorem” (FLT) - i.e., to the effect that FLT holds for prime exponents > 1.615 ・ 10^14 - which is sufficient to give an alternative
58 名前： proof of the first case of Fermat’s Last Theorem. (引用終り)
5^ ABC予想が K = 1 かつ ε = 1 で正しければ、互いに素な自然数 A, B, C が A + B = C を満たすとき C < (rad ABC)2 が成り立つ。互いに素な自然数 a, b, c が an + bn = cn を満たすと仮定すると、an, bn, cn は互いに素より、A = an, B = bn, C = cn を代入して c^n< rad a^nb^nc^n)^2 が成り立つ。一般に rad x^n= rad x=< x であるから、 rad (a^nb^nc^n)^2=< (abc)^2<(c^3)^2=c^6 となる。ゆえに cn < c6, c > 1 より n < 6。n = 3, 4, 5 については古典的な証明があるので定理が証明される (山崎 2010, p. 11)。
出典 27^ a b SHINICHI MOCHIZUKI; IVAN FESENKO, YUICHIRO HOSHI,ARATA MINAMIDE, AND WOJCIECH POROWSKI (2020-11-30). Explicit Estimates in Inter-universal Teichm¨uller Theory (Report). 京都大学数理解析研究所 []
（>>6より） https://arxiv.org/pdf/2004.13108.pdf PROBABILISTIC SZPIRO, BABY SZPIRO, AND EXPLICIT SZPIRO FROM MOCHIZUKI'S COROLLARY 3.12 TAYLOR DUPUY AND ANTON HILADO Date: April 30, 2020.
Abstract. In particular, for an elliptic curve in initial theta data we show how to derive uniform Szpiro (with explicit numerical constants). The inequalities we get will be strictly weaker than [Moc15b, Theorem 1.10] but the proofs are more transparent, modifiable, and user friendly. All of these inequalities are derived from an probabilistic version of [Moc15a, Corollary 3.12] formulated in [DH20b] based on the notion of random measurable sets.
1991 Peter Kronheimer 1993 Jörg Brüdern and Jens Franke 1996 Gero Friesecke and Stefan Sauter 1998 Alice Guionnet 2000 Luca Trevisan 2003 Paul Biran 2007 ゴ・バオ・チャウ 2010 Nicola Gigli and László Székelyhidi Jr.
なお、参考(>>5より) http://www.kurims.kyoto-u.ac.jp/~motizuki/Explicit%20estimates%20in%20IUTeich.pdf Explicit Estimates in Inter-universal Teichmuller Theory. PDF NEW!! (2020-11-30) いわゆる南出論文 (抜粋)
1．<“µ6-version”の発端> P7 One fundamental observation - due to Porowski - that underlies the theory of the present paper is the following: n satisfies the conditions (1), (2) if and only if n = 6
2．<“µ6-version”の詳細> P1 Contents 3. µ6-Theory for [EtTh] 22 4. µ6-Theory for [IUTchI-III] 25 5. µ6-Theory for [IUTchIV] 32 P32 Theorem 5.1. (Log-volume estimates for the “µ6-version” of Θpilot objects) in the situation of the “µ6-version” of [IUTchIII], Corollary 3.12 [cf. Remark 4.2.6]
（まとめ） 1．南出論文はCor3.12を前提したものではありません。 2．Abstractより” In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results, we first establish a version of the theory of ´etale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime “2”. ” とあります 3．[EtTh]からIUTを全部見直しています 4．南出論文には、IUTのエッセンスが凝集されていると見ました。わずか50ページの南出論文を読んでから、それと対比しながら必要に応じてIUTを読めば良いと思います。
＞2. Abstractより ”In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results, we first establish a version of the theory of ´etale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime “2”. ” とあります
下記ですね ”[EtTh] S. Mochizuki, The Etale Theta Function and its Frobenioid-theoretic Manifestations, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349.”です
145 名前：!! (2008-12-12) Comments NEW !! (2016-07-12) Related expositions NEW!! (2015-04-26) Responses to Questions on Frobenioids (cf., especially, Questions 3, 8, 23, 24) NEW!! (2015-12-06)
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
Abstract. We develop the theory of the tempered anabelian and Frobenioid-theoretic aspects of the “´etale theta function”, i.e., the Kummer class of the classical formal algebraic theta function associated to a Tate curve over a nonarchimedean mixedcharacteristic local field. In particular, we consider a certain natural “environment” for the study of the ´etale theta function, which we refer to as a “mono-theta environment” - essentially a Kummer-theoretic version of the classical theta trivialization - and show that this mono-theta environment satisfies certain remarkable rigidity properties involving cyclotomes, discreteness, and constant multiples, all in a fashion that is compatible with the topology of the tempered fundamental group and the extension structure of the associated tempered Frobenioid.
Contents: §1. The Tempered Anabelian Rigidity of the Etale Theta Function ´ §2. The Theory of Theta Environments §3. Tempered Frobenioids §4. General Bi-Kummer Theory §5. The Etale Theta Function via Tempered Frobenioid []
