https://www.uvm.edu/~tdupuy/papers.html [ Taylor Dupuy's Homepage] 論文集 なお、（メモ）TAYLOR DUPUYは、arxiv投稿で [SS17]を潰した（下記） 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. P14 Remark 3.8.3. (1) The assertion of [SS17, pg 10] is that (3.3) is the only relation between the q-pilot and Θ-pilot degrees. The assertion of [Moc18, C14] is that [SS17, pg 10] is not what occurs in [Moc15a]. The reasoning of [SS17, pg 10] is something like what follows: P15 (2) We would like to point out that the diagram on page 10 of [SS17] is very similar to the diagram on §8.4 part 7, page 76 of the unpublished manuscript [Tan18] which Scholze and Stix were reading while preparing [SS17]. References [SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017. 1, 1, 1e, 2, 7.5.3 ( www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html ) [Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2
なお "[SS17] Peter Scholze and Jakob Stix, Why abc is still a conjecture., 2017."は、2018の気がする "[Tan18] Fucheng Tan, Note on IUT, 2018. 1, 2"が見つからない。"the unpublished manuscript [Tan18]"とはあるのだが（＾＾ 代わりに、ヒットした下記でも、どぞ （2018の何月かが不明だが、2018.3のSS以降かも）
www.kurims.kyoto-u.ac.jp/~motizuki/Tan%20---%20Introduction%20to%20inter-universal%20Teichmuller%20theory%20(slides).pdf Introduction to Inter-universal Teichm¨uller theory Fucheng Tan RIMS, Kyoto University 2018 To my limited experiences, the following seem to be an option for people who wish to get to know IUT without spending too much time on all the details. ・ Regard the anabelian results and the general theory of Frobenioids as blackbox. ・ Proceed to read Sections 1, 2 of [EtTh], which is the basis of IUT. ・ Read [IUT-I] and [IUT-II] (briefly), so as to know the basic definitions. ・ Read [IUT-III] carefully. To make sense of the various definitions/constructions in the second half of [IUT-III], one needs all the previous definitions/results. ・ The results in [IUT-IV] were in fact discovered first. Section 1 of [IUT-IV] allows one to see the construction in [IUT-III] in a rather concrete way, hence can be read together with [IUT-III], or even before. S. Mochizuki, The ´etale theta function and its Frobenioid-theoretic manifestations. S. Mochizuki, Inter-universal Teichm¨uller Theory I, II, III, IV.
www.kurims.kyoto-u.ac.jp/daigakuin/Tan.pdf 教員名: 譚 福成（Tan, Fucheng） P-adic Hodge theory plays an essential role in Mochizuki's proof of Grothendieck's Anabelian Conjecture. Recently, I have been studying anabeian geometry and Mochizuki's Inter-universal Teichmuller theory, which is in certain sense a global simulation of p-adic comparison theorem.
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.
3月にドイツ Oberwolfach で新たな #IUTABC カンファレンス。Stixもオーガナイザーで参加 Homotopic and Geometric Galois Theory 7 Mar - 13 Mar 2021 Organizers Benjamin Collas, Bayreuth Pierre Dèbes, Villeneuve d'Ascq Hiroaki Nakamura, Osaka Jakob Stix, Frankfurt
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.
