DOCSLIB.ORG

Inter-Universal Teichmüller Theory II: Hodge-Arakelov-Theoretic Evaluation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Inter-Universal Teichmüller Theory II: Hodge-Arakelov-Theoretic Evaluation INTER-UNIVERSAL TEICHMULLER¨ THEORY II: HODGE-ARAKELOV-THEORETIC EVALUATION Shinichi Mochizuki December 2020 Abstract. In the present paper, which is the second in a series of four pa- pers, we study the Kummer theory surrounding the Hodge-Arakelov-theoretic eval- uation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the l- th root of the] theta function at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point], for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime num- ber l ≥ 5. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th ±ell root of the] theta function at primes of bad reduction of EF in one Θ NF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in an- other Θ±ellNF-Hodge theater. The theory developed in the present paper allows one ×μ to construct certain new versions of this “Θ-link”. One such new version is the Θgau- link, which is similar to the Θ-link, but involves the theta values at l-torsion points, rather than the theta function itself. One important aspect of the constructions ×μ that underlie the Θgau-link is the study of multiradiality properties, i.e., properties of the “arithmetic holomorphic structure” — or, more concretely, the ring/scheme structure — arising from one Θ±ellNF-Hodge theater that may be formulated in such a way as to make sense from the point of the arithmetic holomorphic structure of another Θ±ellNF-Hodge theater which is related to the original Θ±ellNF-Hodge ×μ theater by means of the [non-scheme-theoretic!] Θgau-link. For instance, certain of the various rigidity properties of the ´etale theta function studied in an earlier paper by the author may be intepreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions ×μ that underlie the Θgau-link is the study of “conjugate synchronization” via the ± ±ell Fl -symmetry of a Θ NF-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms — which are free of any conjugacy indeterminacies! — between copies of local absolute Galois groups at the various l-torsion points at which the theta function is evaluated. Conjugate synchronization plays an impor- tant role in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study of coricity properties of [i.e., the study of ×μ objects left invariant by] the Θgau-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia groups. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI Contents: Introduction §1. Multiradial Mono-theta Environments §2. Galois-theoretic Theta Evaluation §3. Tempered Gaussian Frobenioids §4. Global Gaussian Frobenioids Introduction In the following discussion, we shall continue to use the notation of the In- troduction to the first paper of the present series of papers [cf. [IUTchI], §I1]. In particular, we assume that are given an elliptic curve EF over a number field F , together with a prime number l ≥ 5. In the present paper, which forms the sec- ond paper of the series, we study the Kummer theory surrounding the Hodge- Arakelov-theoretic evaluation [cf. Fig. I.1 below] — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory of [HASurI], [HASurII] — of the reciprocal of the l-th root of the theta function 1 √ 2 −1 2 − def 1 (m+ 1 ) n 1 (n+ 1 ) n+ 1 l − · 2 2 · − · 2 2 · 2 Θv = 1 qv ( 1) qv Uv m∈Z n∈Z [cf. [EtTh], Proposition 1.4; [IUTchI], Example 3.2, (ii)] at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point] in the context of the theory of Θ±ellNF-Hodge theaters developed in [IUTchI]. Here, relative to the notation bad of [IUTchI], §I1, v ∈ V ; qv denotes the q-parameter at v of the given elliptic curve EF over a number field F; Uv denotes the standard multiplicative coordinate on the Tate curve obtained by localizing EF at v.Letq be a 2l-th root of qv. v Then these “theta values at l-torsion points” will, up to a factor given by a 2l-th root of unity, turn out to be of the form [cf. Remark 2.5.1, (i)] j2 q v [Frobenius-like!] Kummer [´etale-like!] Frobenioid-theoretic Galois-theoretic ´etale ......... theta function theta function evalu- ⇓ ation evalu- ⇓ ation [Frobenius-like!] Kummer [´etale-like!] Frobenioid-theoretic Galois-theoretic ......... theta values theta values Fig. I.1: The Kummer theory surrounding Hodge-Arakelov-theoretic evaluation INTER-UNIVERSAL TEICHMULLER¨ THEORY II 3 —wherej ∈{0, 1,...,l def=(l − 1)/2},soj is uniquely determined by its image def j ∈|Fl| = Fl/{±1} = {0} Fl [cf. the notation of [IUTchI], §I1]. In order to understand the significance of Kummer theory in the context of Hodge-Arakelov-theoretic evaluation, it is important to recall the notions of “Frobenius-like” and “´etale-like” mathematical structures [cf. the discussion of [IUTchI], §I1]. In the present series of papers, the Frobenius-like structures consti- tuted by [the monoidal portions of] Frobenioids — i.e., more concretely, by various monoids — play the important role of allowing one to construct gluing isomor- phisms such as the Θ-link which lie outside the framework of conventional scheme/ring theory [cf. the discussion of [IUTchI], §I2]. Such gluing isomor- phisms give rise to Frobenius-pictures [cf. the discussion of [IUTchI], §I1]. On the other hand, the ´etale-like structures constituted by various Galois and arith- metic fundamental groups give rise to the canonical splittings of such Frobenius- pictures furnished by corresponding ´etale-pictures [cf. the discussion of [IUTchI], §I1]. In [IUTchIII], absolute anabelian geometry will be applied to these Galois and arithmetic fundamental groups to obtain descriptions of alien arithmetic holomorphic structures, i.e., arithmetic holomorphic structures that lie on the opposite side of a Θ-link from a given arithmetic holomorphic structure [cf. the discussion of [IUTchI], §I3]. Thus, in light of the equally crucial but substantially different roles played by Frobenius-like and ´etale-like structures in the present series of papers, it is of crucial importance to be able to relate corresponding Frobenius-like and ´etale-like versions of vari- ous objects to one another. This is the role played by Kummer theory. In particular, in the present paper, we shall study in detail the Kummer theory that relates Frobenius-like and ´etale- like versions of the theta function and its theta values at l-torsion points to one another [cf. Fig. I.1]. One important notion in the theory of the present paper is the notion of mul- tiradiality. To understand this notion, let us recall the ´etale-picture discussed in [IUTchI], §I1 [cf. [IUTchI], Fig. I1.6]. In the context of the present paper, we shall be especially interested in the ´etale-like version of the theta function and its ±ell theta values constructed in each D-Θ±ellNF-Hodge theater (−)HT D-Θ NF;thus, one can think of the ´etale-picture under consideration as consisting of the diagram given in Fig. I.2 below. As discussed earlier, we shall ultimately be interested in applying various absolute anabelian reconstruction algorithms to the various arith- metic fundamental groups that [implicitly] appear in such ´etale-pictures in order to obtain descriptions of alien holomorphic structures, i.e., descriptions of objects that arise on one “spoke” [i.e., “arrow emanating from the core”] that make sense from the point of view of another spoke. In this context, it is natural to classify the various algorithms applied to the arithmetic fundamental groups lying in a given spoke as follows [cf. Example 1.7]: · We shall refer to an algorithm as coric if it in fact only depends on input data arising from the mono-analytic core of the ´etale-picture, i.e., the data that is common to all spokes. 4 SHINICHI MOCHIZUKI · We shall refer to an algorithm as uniradial if it expresses the objects constructed from the given spoke in terms that only make sense within the given spoke. · We shall refer to an algorithm as multiradial if it expresses the objects constructed from the given spoke in terms of corically constructed objects, i.e., objects that make sense from the point of view of other spokes. Thus, multiradial algorithms are compatible with simultaneous execution at multiple spokes [cf. Example 1.7, (v); Remark 1.9.1], while uniradial algorithms may only be consistently executed at a single spoke. Ultimately, in the present series of papers, we shall be interested — relative to the goal of obtaining “descriptions of alien holomorphic structures” — in the establishment of multiradial algorithms for constructing the objects of interest, e.g., [in the context of the present paper] the ´etale-like versions of the theta functions and the corresponding theta values discussed above. Typically, in order to obtain such multiradial algorithms, i.e., algorithms that make sense from the point of view of other spokes, it is necessary to allow for some sort of “indeterminacy” in the descriptions that appear in the algorithms of the objects constructed from the given spoke.
Load more

Recommended publications
  • An Introduction to P-Adic Teichmüller Theory Astérisque, Tome 278 (2002), P
    Astérisque SHINICHI MOCHIZUKI An introduction to p-adic Teichmüller theory Astérisque, tome 278 (2002), p. 1-49 <http://www.numdam.org/item?id=AST_2002__278__1_0> © Société mathématique de France, 2002, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Astérisque 278, 2002, p. 1-49 AN INTRODUCTION TO p-ADIC TEICHMULLER THEORY by Shinichi Mochizuki Abstract. — In this article, we survey a theory, developed by the author, concerning the uniformization of p-adic hyperbolic curves and their moduli. On the one hand, this theory generalizes the Fuchsian and Bers uniformizations of complex hyperbolic curves and their moduli to nonarchimedean places. It is for this reason that we shall often refer to this theory as p-adic Teichmiiller theory, for short. On the other hand, this theory may be regarded as a fairly precise hyperbolic analogue of the Serre-Tate theory of ordinary abelian varieties and their moduli. The central object of p-adic Teichmiiller theory is the moduli stack of nilcurves. This moduli stack forms a finite flat covering of the moduli stack of hyperbolic curves in positive characteristic. It parametrizes hyperbolic curves equipped with auxiliary "uniformization data in positive characteristic." The geometry of this moduli stack may be analyzed combinatorially locally near infinity.
    [Show full text]
  • Remarks on Aspects of Modern Pioneering Mathematical Research
    REMARKS ON ASPECTS OF MODERN PIONEERING MATHEMATICAL RESEARCH IVAN FESENKO Inter-universal Teichmüller (IUT) theory of Shinichi Mochizuki1 was made public at the end of August of 2012. Shinichi Mochizuki had been persistently working on IUT for the previous 20 years. He was supported by Research Institute for Mathematical Sciences (RIMS), part of Kyoto University.2 The IUT theory studies cardinal properties of integer numbers. The simplicity of the definition of numbers and of statements of key distinguished problems about them hides an underlying immense compexity and profound depth. One can perform two standard operations with numbers: add and multiply. Prime numbers are ‘atoms’ with respect to multiplication. Several key problems in mathematics ask hard (and we do not know how hard until we see a solution) questions about relations between prime numbers and the second operation of addition. More generally, the issue of hidden relations between multiplication and addition for integer numbers is of most fundamental nature. The problems include abc-type inequalities, the Szpiro conjecture for elliptic curves over number fields, the Frey conjecture for elliptic curves over number fields and the Vojta conjecture for hyperbolic curves over number fields. They are all proved as an application of IUT. But IUT is more than a tool to solve famous conjectures. It is a new fundamental theory that might profoundly influence number theory and mathematics. It restores in number theory the place, role and value of topological groups, as opposite to the use of their linear representations only. IUT is the study of deformation of arithmetic objects by going outside conventional arithmetic geometry, working with their groups of symmetries, using categorical geometry structures and applying deep results of mono-anabelian geometry.
    [Show full text]
  • Report on Discussions, Held During the Period March 15 – 20, 2018, Concerning Inter-Universal Teichm¨Uller Theory (Iutch)
    REPORT ON DISCUSSIONS, HELD DURING THE PERIOD MARCH 15 – 20, 2018, CONCERNING INTER-UNIVERSAL TEICHMULLER¨ THEORY (IUTCH) Shinichi Mochizuki February 2019 §1. The present document is a report on discussions held during the period March 15 – 20, 2018, concerning inter-universal Teichm¨uller theory (IUTch). These discussions were held in a seminar room on the fifth floor of Maskawa Hall, Kyoto University, according to the following schedule: · March 15 (Thurs.): 2PM ∼ between 5PM and 6PM, · March 16 (Fri.): 10AM ∼ between 5PM and 6PM, · March 17 (Sat.): 10AM ∼ between 5PM and 6PM, · March 19 (Mon.): 10AM ∼ between 5PM and 6PM, · March 20 (Tues.): 10AM ∼ between 5PM and 6PM. (On the days when the discussions began at 10AM, there was a lunch break for one and a half to two hours.) Participation in these discussions was restricted to the following mathematicians (listed in order of age): Peter Scholze, Yuichiro Hoshi, Jakob Stix,andShinichi Mochizuki. All four mathematicians participated in all of the sessions listed above (except for Hoshi, who was absent on March 16). The existence of these discussions was kept confidential until the conclusion of the final session. From an organizational point of view, the discussions took the form of “negotations” between two “teams”: one team (HM), consisting of Hoshi and Mochizuki, played the role of explaining various aspects of IUTch; the other team (SS), consisting of Scholze and Stix, played the role of challenging various aspects of the explanations of HM. Most of the sessions were conducted in the following format: Mochizuki would stand and explain various aspects of IUTch, often supplementing oral explanations by writing on whiteboards using markers in various colors; the other participants remained seated, for the most part, but would, at times, make questions or comments or briefly stand to write on the whiteboards.
    [Show full text]
  • The Paradox of the Proof | Project Wordsworth
    The Paradox of the Proof | Project Wordsworth Home | About the Authors If you enjoy this story, we ask that you consider paying for it. Please see the payment section below. The Paradox of the Proof By Caroline Chen MAY 9, 2013 n August 31, 2012, Japanese mathematician Shinichi Mochizuki posted four O papers on the Internet. The titles were inscrutable. The volume was daunting: 512 pages in total. The claim was audacious: he said he had proved the ABC Conjecture, a famed, beguilingly simple number theory problem that had stumped mathematicians for decades. Then Mochizuki walked away. He did not send his work to the Annals of Mathematics. Nor did he leave a message on any of the online forums frequented by mathematicians around the world. He just posted the papers, and waited. Two days later, Jordan Ellenberg, a math professor at the University of Wisconsin- Madison, received an email alert from Google Scholar, a service which scans the Internet looking for articles on topics he has specified. On September 2, Google Scholar sent him Mochizuki’s papers: You might be interested in this. http://projectwordsworth.com/the-paradox-of-the-proof/[10/03/2014 12:29:17] The Paradox of the Proof | Project Wordsworth “I was like, ‘Yes, Google, I am kind of interested in that!’” Ellenberg recalls. “I posted it on Facebook and on my blog, saying, ‘By the way, it seems like Mochizuki solved the ABC Conjecture.’” The Internet exploded. Within days, even the mainstream media had picked up on the story. “World’s Most Complex Mathematical Theory Cracked,” announced the Telegraph.
    [Show full text]
  • Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms
    J. Math. Sci. Univ. Tokyo 22 (2015), 939–1156. Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms By Shinichi Mochizuki Abstract. In the present paper, which forms the third part of a three-part series on an algorithmic approach to absolute anabelian geometry, we apply the absolute anabelian technique of Belyi cuspi- dalization developed in the second part, together with certain ideas contained in an earlier paper of the author concerning the category- theoretic representation of holomorphic structures via either the topo- logical group SL2(R) or the use of “parallelograms, rectangles, and squares”, to develop a certain global formalism for certain hyperbolic orbicurves related to a once-punctured elliptic curve over a number field. This formalism allows one to construct certain canonical rigid integral structures, which we refer to as log-shells, that are obtained by applying the logarithm at various primes of a number field. Moreover, although each of these local logarithms is “far from being an isomor- phism”both in the sense that it fails to respect the ring structures involved and in the sense (cf. Frobenius morphisms in positive char- acteristic!) that it has the effect of exhibiting the “mass”represented by its domain as a “somewhat smaller collection of mass” than the “mass”represented by its codomain, this global formalism allows one to treat the logarithm operation as a global operation on a number field which satisfies the property of being an “isomomorphism up to an appropriate renormalization operation”, in a fashion that is rem- iniscent of the isomorphism induced on differentials by a Frobenius lifting, once one divides by p.
    [Show full text]
  • Arxiv:2106.07269V2 [Math.NT] 20 Jun 2021
    Some instructive mathematical errors Richard P. Brent∗ September 23, 2021 Abstract We describe various errors in the mathematical literature, and con- sider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their pub- lication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Fi- nally, we mention some instructive errors that have been detected in the author’s own published papers. 1 Introduction Those that fail to learn from history are doomed to repeat it. Winston Churchill [24]. Since mathematics is a human endeavour, errors can and do occur. It is worth studying past errors and learning from them to reduce the likelihood arXiv:2106.07269v3 [math.NT] 22 Sep 2021 of making similar errors in the future. We are concerned with nontrivial errors made by working mathemati- cians, not with the class of errors that school-children often make in their homework. ∗Mathematical Sciences Institute, Australian National University, Canberra, Australia 1 The errors that we consider can be grouped into three broad categories. 1. Well-known errors made by prominent mathematicians (see §2). 2. Errors discovered by the author in other mathematicians’ work (§3). 3.
    [Show full text]
  • Promenade in Inter-Universal Teichmüller Theory
    Research Institute for Mathematical Sciences - Kyoto University, Japan PROMENADE IN INTER-UNIVERSAL TEICHMÜLLER THEORY - 復元 Online Seminar - Algebraic & Arithmetic Geometry Laboratoire Paul Painlevé - Université de Lille, France INTRODUCTION Mochizuki’s Inter-Universal Teichmüller theory (IUT) has been developed with the goal of providing +휀 a deeper understanding of the abc-conjecture– for every 휀 > 0, the relation rad(푎푏푐)1 < 푐 has only finitely many solutions for coprime integers (푎, 푏, 푐) ∈ N>0 such that 푎 + 푏 = 푐 (abc). This conjecture, that is deeply rooted in arithmetic geometry, is to be understood as a certain rigidity property on the intertwining of the multiplicative and additive monoid structures of (N>0, ⊞, ⊠), whose estimate is hidden within the many isomorphic identifications used in arithmetic and Diophantine geometry. The seminal achievement of Mochizuki’s IUT is to provide a new geometry that brings an estimate of the (abc) rigidity property within reach. IUT theory is a geometry of the moduli stack of elliptic curves ℳ1,1 whose 퐾-points are endowed with certain rigid Diophantine arithmetic line bundle invariants with place-wise compatible arithmetic and geometric symmetries, and that are embedded in various types of non-rigid anabelian étale containers. The consideration of rings/schemes and fields in terms of mono-anabelian geometry and up-to some indeterminacies, for example at nonarchimedean places of 퐾s, as abstract ⊞⇑⊠-monoids, allows the decoupling of their ⊞⇑⊠-monoids structures. This deconstruction- reconstruction process – or Fukugen 復元 , relies on the non (mono) anabelianity of sub-푝-adic fields and allows to track the isomorphic identifications of ring structures, which in turns provides an abc-estimate at the level of heights of line bundles.
    [Show full text]
  • 2014 Annual Report
    CLAY MATHEMATICS INSTITUTE www.claymath.org ANNUAL REPORT 2014 1 2 CMI ANNUAL REPORT 2014 CLAY MATHEMATICS INSTITUTE LETTER FROM THE PRESIDENT Nicholas Woodhouse, President 2 contents ANNUAL MEETING Clay Research Conference 3 The Schanuel Paradigm 3 Chinese Dragons and Mating Trees 4 Steenrod Squares and Symplectic Fixed Points 4 Higher Order Fourier Analysis and Applications 5 Clay Research Conference Workshops 6 Advances in Probability: Integrability, Universality and Beyond 6 Analytic Number Theory 7 Functional Transcendence around Ax–Schanuel 8 Symplectic Topology 9 RECOGNIZING ACHIEVEMENT Clay Research Award 10 Highlights of Peter Scholze’s contributions by Michael Rapoport 11 PROFILE Interview with Ivan Corwin, Clay Research Fellow 14 PROGRAM OVERVIEW Summary of 2014 Research Activities 16 Clay Research Fellows 17 CMI Workshops 18 Geometry and Fluids 18 Extremal and Probabilistic Combinatorics 19 CMI Summer School 20 Periods and Motives: Feynman Amplitudes in the 21st Century 20 LMS/CMI Research Schools 23 Automorphic Forms and Related Topics 23 An Invitation to Geometry and Topology via G2 24 Algebraic Lie Theory and Representation Theory 24 Bounded Gaps between Primes 25 Enhancement and Partnership 26 PUBLICATIONS Selected Articles by Research Fellows 29 Books 30 Digital Library 35 NOMINATIONS, PROPOSALS AND APPLICATIONS 36 ACTIVITIES 2015 Institute Calendar 38 1 ach year, the CMI appoints two or three Clay Research Fellows. All are recent PhDs, and Emost are selected as they complete their theses. Their fellowships provide a gener- ous stipend, research funds, and the freedom to carry on research for up to five years anywhere in the world and without the distraction of teaching and administrative duties.
    [Show full text]
  • Notices of the American Mathematical Society
    ISSN 0002-9920 of the American Mathematical Society February 2004 Volume 51, Number 2 Geometrization of 3-Manifolds via the Ricci Flow page 184 RIMS, an Institute for Japan and the World page 194 Los Angeles Meeting page 290 Lawrenceville Meeting page 292 Houston Meeting page 295 Heawood's demonstration of Kempe's error (see page 207) Document conversion between Microsoft™ WORD and LaTeX is easier than watching PING-PONG! Paintwork © 2003 Alex Chikrii. a, rd \i Word2TeX and TeX2Word are either registered trademarks or trademarks of Chikrii Softlab in the United States and/ or other countries. Microsoft, Microsoft Word are either registered trademarks or trademarks of Microsoft Corporation in the United States and/ or other countries. Other product and company names mentioned herein may be trademarks of their respective owners. ~ Chikrii Soft lab ~ http:/jwww.chikrii.com International Mathematics Research Notices Editors Website: http://imrn.hindawi.com Morris Weisfeld IMRN provides very fast publication of research articles of high current interest in all Managing Editor areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics. Issues are published as frequently as nec­ Dan Abramovich essary. IMRN is expected to publish 80± issues in volume 2004. The articles of the IMRN are Enrico Arbarello reviewed/indexed in COMPUMATH Citation Index, Current Contents, lSI Alerting Services, Joseph Bernstein Mathematical Reviews, Science Citation Index, SciSearch, and Zentralblatt fUr Mathematik. Enrico Bombieri There are no page charges. Submissions are made by email to [email protected]. Richard E. Borcherds New print subscribers shall receive a free copy of all back volumes, i.e., volumes 1991-2003.
    [Show full text]
  • Arxiv:2004.13108V2 [Math.NT] 29 Apr 2020 N“Nta Ht Aabitfo H Edo Oui,[O1a Corollary 5.0.1
    PROBABILISTIC SZPIRO, BABY SZPIRO, AND EXPLICIT SZPIRO FROM MOCHIZUKI’S COROLLARY 3.12 TAYLOR DUPUY AND ANTON HILADO Abstract. In [DH20b] we gave some explicit formulas for the “indeterminacies” ind1, ind2, ind3 in Mochizuki’s Inequality as well as a new presentation of initial theta data. In the present paper we use these explicit formulas, together with our probabilistic formulation of [Moc15a, Corollary 3.12] to derive variants of Szpiro’s inequality (in the spirit of [Moc15b]). In par- ticular, 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. Contents 1. Introduction 1 2. Explicit Computations in Tensor Packets 5 3. Conductors, Minimal Discriminants, and Ramification 10 4. Estimates on p-adic Logarithms 16 5. Archimedean Logarithms 18 6. Upper Bounds on Hulls 18 7. Probabilistic Versions of the Mochizuki and Szpiro Inequalities 22 8. Deriving Explicit Constants For Szpiro’s Inequality From Mochizuki’s Inequality 29 References 35 1. Introduction arXiv:2004.13108v2 [math.NT] 29 Apr 2020 In [DH20b] we gave a probabilistic interpretation of Mochizuki’s Inequality (Corollary 3.12 of [Moc15a]). In the present paper we perform some explicit computations using this inequal- ity to derive three inequalities which we will call “Probabilistic Szpiro”, “Baby Szpiro”, and “Explicit Szpiro”.
    [Show full text]
  • The Impenetrable Proof
    NEWS FEATURE 178 | NATURE | VOL 526 | 8 OCTOBER 2015 © 2015 Macmillan Publishers Limited. All rights reserved FEATURE NEWS The impenetrable proof that Mochizuki had been working on the conjecture for years and had been finalizing his work. That same day, Tamagawa e-mailed the news to one of his collaborators, number theorist Ivan Fesenko of the Univer- Shinichi Mochizuki claims sity of Nottingham, UK. Fesenko immediately downloaded the papers and started to read. But he soon became “bewildered”, he says. “It was to have solved one of the impossible to understand them.” Fesenko e-mailed some top experts in Mochizuki’s field of arithmetic most important problems in geometry, and word of the proof quickly spread. Within days, intense chatter began on mathematical blogs and online forums (see Nature mathematics. The trouble is, http://doi.org/725; 2012). But for many researchers, early elation about the proof quickly turned to scepticism. Everyone — even those whose hardly anyone can work out area of expertise was closest to Mochizuki’s — was just as flummoxed by the papers as Fesenko had been. To complete the proof, Mochizuki had whether he’s right. invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, By Davide Castelvecchi wrote on his blog a few days after the paper appeared. Three years on, Mochizuki’s proof remains in mathematical limbo — neither debunked nor accepted by the wider community.
    [Show full text]
  • Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
    Quanta Magazine Titans of Mathematics Clash Over Epic Proof of ABC Conjecture Two mathematicians have found what they say is a hole at the heart of a proof that has convulsed the mathematics community for nearly six years. By Erica Klarreich In a report posted online today, Peter Scholze of the University of Bonn and Jakob Stix of Goethe University Frankfurt describe what Stix calls a “serious, unfixable gap” within a mammoth series of papers by Shinichi Mochizuki, a mathematician at Kyoto University who is renowned for his brilliance. Posted online in 2012, Mochizuki’s papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory. Despite multiple conferences dedicated to explicating Mochizuki’s proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called “a sense of infinite regress.” Between 12 and 18 mathematicians who have studied the proof in depth believe it is correct, wrote Ivan Fesenko of the University of Nottingham in an email. But only mathematicians in “Mochizuki’s orbit” have vouched for the proof’s correctness, Conrad commented in a blog discussion last December. “There is nobody else out there who has been willing to say even off the record that they are confident the proof is complete.” Nevertheless, wrote Frank Calegari of the University of Chicago in a December blog post, “mathematicians are very loath to claim that there is a problem with Mochizuki’s argument because they can’t point to any definitive error.” That has now changed.
    [Show full text]