Inter-universal Teichmüller theory. I: Construction of Hodge theaters
type: Item
instance of: scholarly article
title: Inter-universal Teichmüller theory. I: Construction of Hodge theaters
zbMATH Open document ID: 1465.14002
DOI: 10.4171/PRIMS/57-1-1
author: Shinichi Mochizuki
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2021-03-06T00:00:00Z
review text: In this series of papers on Inter-Universal Teichmüller Theory [this paper; ibid. 57, No.~1--2, 209--401 (2021; Zbl 1465.14003); ibid. 57, No.~1--2, 403--626 (2021; Zbl 1465.14004); ibid. 57, No.~1--2, 627--723 (2021; Zbl 1465.14005)], the author aims to prove the ABC conjecture of Masser and Oesterlé, in close to effective form. Recall that in its simplest form, the ABC conjecture states that for all $\epsilon>0$ there is some constant $C_\epsilon$ such that for all coprime positive integers $a,b,c$ satisfying $a+b=c$, one has $c\leq C_\epsilon (\prod_{p|abc} p)^{1+\epsilon}$. Here the product runs over all primes $p$ dividing one of $a$, $b$ and $c$, but crucially not counting multiplicity. It is arguably the most central open Diophantine problem. For example, applied to $a=x^n$, $b=y^n$ and $c=z^n$ it formally implies that there are at most finitely many counterexamples to Fermat's Last Theorem; if $C_\epsilon$ can be made explicit, it thus in principle reduces it to a finite computation. A proof of the ABC conjecture would also lead to a new proof of Mordell's conjecture [\textit{N.~D. Elkies}, Int. Math. Res. Not. 1991, No.~7, 99--109 (1991; Zbl 0763.11016)]. A closely related conjecture is the Szpiro conjecture, which relates conductor and discriminant of elliptic curves; the translation is given by passing to the Frey-Hellegouarch curve $y^2=x(x-a)(x+b)$. The ABC conjecture is known to imply the Szpiro conjecture, but the converse fails, essentially because the discriminant of elliptic curves does not contain an ``Archimedean factor''. The author observed in [Math. J. Okayama Univ. 52, 1--28 (2010; Zbl 1221.14024)] that if one formulates suitably uniform versions of both conjectures for number fields, they become equivalent. (Additionally, one can arrange for auxiliary congruence conditions at finitely many primes, which is used $2$-adically in the present paper.) With this in mind, the author starts in these papers with the datum of an elliptic curve $E$ ove…
reviewed by: Peter Scholze
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G32
zbMATH DE Number: 7317908
zbMATH Keywords: inter-universal Teichmüller theory
zbMATH Keywords: Hodge theater
zbMATH Keywords: global multiplicative subspace
zbMATH Keywords: canonical generator
zbMATH Keywords: punctured elliptic curve
zbMATH Keywords: theta-link
zbMATH Keywords: étale theta function
zbMATH Keywords: absolute anabelian geometry
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7317908