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A Look at the ABC Conjecture Via Elliptic Curves
A Look at the ABC Conjecture via Elliptic Curves Nicole Cleary Brittany DiPietro Alexander Hill Gerard D.Koffi Beihua Yan Abstract We study the connection between elliptic curves and ABC triples. Two important results are proved. The first gives a method for finding new ABC triples. The second result states conditions under which the power of the new ABC triple increases or decreases. Finally, we present two algorithms stemming from these two results. 1 Introduction The ABC conjecture is a central open problem in number theory. It was formu- lated in 1985 by Joseph Oesterl´eand David Masser, who worked separately but eventually proposed equivalent conjectures. Like many other problems in number theory, the ABC conjecture can be stated in relatively simple, understandable terms. However, there are several profound implications of the ABC conjecture. Fermat's Last Theorem is one such implication which we will explore in the second section of this paper. 1.1 Statement of The ABC Conjecture Before stating the ABC conjecture, we define a few terms that are used frequently throughout this paper. Definition 1.1.1 The radical of a positive integer n, denoted rad(n), is defined as the product of the distinct prime factors of n. That is, rad(n) = Q p where p is prime. pjn 1 Example 1.1.2 Let n = 72 = 23 · 32: Then rad(n) = rad(72) = rad(23 · 32) = 2 · 3 = 6: Definition 1.1.3 Let A; B; C 2 Z. A triple (A; B; C) is called an ABC triple if A + B = C and gcd(A; B; C) = 1. -
Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
Remembrance of Alan Baker
Hardy-Ramanujan Journal 42 (2019), 4-11Hardy-Ramanujan 4-11 Remembrance of Alan Baker Gisbert W¨ustholz REMEMBRANCE OF ALAN BAKER It was at the conference on Diophantische Approximationen organized by Theodor Schneider by which took place at Oberwolfach Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter SchlickeweiGISBERT a graduate WUSTHOLZ¨ student of Schneider at the Albert Ludwigs University at Freiburg i. Br.. We both were invited to the conference to help Schneider organize the conference in practical matters. In particular we were determined to make the speakers write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for the future. Indeed if you open the webpage of the Mathematisches Forschungsinstitut Oberwolfach you find a link to the Oberwolfach Digital Archive where you can find all these Documents It was at the conference on Diophantische Approximationen organized by Theodor Schneider which took place at fromOberwofach 1944-2008. Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter Schlickewei a graduateAt this student conference of Schneider I was at verythe Albert-Ludwig-University much impressed and at Freiburg touched i. Br.. by We seeing both were the invitedFields to Medallist the conference Baker whoseto help work Schneider I had to studied organize the in conference the past in year practical to matters.do the Inp-adic particular analogue we were determined of Baker's to make recent the versions speakers A write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for sharpeningthe future. -
The Abc Conjecture
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2002 The abc conjecture Jeffrey Paul Wheeler University of Tennessee Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Recommended Citation Wheeler, Jeffrey Paul, "The abc conjecture. " Master's Thesis, University of Tennessee, 2002. https://trace.tennessee.edu/utk_gradthes/6013 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Jeffrey Paul Wheeler entitled "The abc conjecture." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. Pavlos Tzermias, Major Professor We have read this thesis and recommend its acceptance: Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Jeffrey Paul Wheeler entitled "The abc Conjecture." I have examined the finalpaper copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. Pavlas Tzermias, Major Professor We have read this thesis and recommend its acceptance: 6.<&, ML 7 Acceptance for the Council: The abc Conjecture A Thesis Presented for the Master of Science Degree The University of Tenne�ee, Knoxville Jeffrey Paul Wheeler August 2002 '"' 1he ,) \ � �00� . -
Elliptic Curves and the Abc Conjecture
Elliptic Curves and the abc Conjecture Anton Hilado University of Vermont October 16, 2018 Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 1 / 37 Overview 1 The abc conjecture 2 Elliptic Curves 3 Reduction of Elliptic Curves and Important Quantities Associated to Elliptic Curves 4 Szpiro's Conjecture Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 2 / 37 The Radical Definition The radical rad(N) of an integer N is the product of all distinct primes dividing N Y rad(N) = p: pjN Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 3 / 37 The Radical - An Example rad(100) = rad(22 · 52) = 2 · 5 = 10 Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 4 / 37 The abc Conjecture Conjecture (Oesterle-Masser) Let > 0 be a positive real number. Then there is a constant C() such that, for any triple a; b; c of coprime positive integers with a + b = c, the inequality c ≤ C() rad(abc)1+ holds. Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 5 / 37 The abc Conjecture - An Example 210 + 310 = 13 · 4621 Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 6 / 37 Fermat's Last Theorem There are no integers satisfying xn + y n = zn and xyz 6= 0 for n > 2. Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 7 / 37 Fermat's Last Theorem - History n = 4 by Fermat (1670) n = 3 by Euler (1770 - gap in the proof), Kausler (1802), Legendre (1823) n = 5 by Dirichlet (1825) Full proof proceeded in several stages: Taniyama-Shimura-Weil (1955) Hellegouarch (1976) Frey (1984) Serre (1987) Ribet (1986/1990) Wiles (1994) Wiles-Taylor (1995) Anton Hilado (UVM) Elliptic Curves and the abc Conjecture October 16, 2018 8 / 37 The abc Conjecture Implies (Asymptotic) Fermat's Last Theorem Assume the abc conjecture is true and suppose x, y, and z are three coprime positive integers satisfying xn + y n = zn: Let a = xn, b = y n, c = zn, and take = 1. -
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. -
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. -
Publications of Shou-Wu Zhang
Publications of Shou-Wu Zhang 1. Gerd Faltings, Lectures on the arithmetic Riemann-Roch theorem. Notes taken by Shouwu Zhang. Annals of Mathematics Studies, 127. (1992). x+100 pp. amazon 2. Positive line bundles on arithmetic surfaces. Ann. of Math. (2) 136 (1992), no. 3, pp 569{587. pdf 3. Admissible pairing on a curve. Invent. Math. 112 (1993), no. 1, pp 171{193. pdf 4. Geometric reductivity at Archimedean places. Internat. Math. Res. No- tices 1994, no. 10, 425 ff., approx. 9 pp. pdf 5. Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8 (1995), no. 1, pp 187{221. pdf 6. Small points and adelic metrics. J. Algebraic Geom. 4 (1995), no. 2, 28100 pdf 7. Heights and reductions of semi-stable varieties. Compositio Math. 104 (1996), no. 1, pp 77{105. pdf 8. (with Lucien Szpiro, Emmanuel Ullmo) Equir´epartition´ des petits points. Invent. Math. 127 (1997), no. 2, pp 337{347. pdf 9. Heights of Heegner cycles and derivatives of L-series. Invent. Math. 130 (1997), no. 1, pp 99{152. pdf 10. Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147 (1998), no. 1, pp 159{165. pdf 11. Small points and Arakelov theory. Proc. of ICM, Doc. Math. 1998, Extra Vol. II, pp 217{225 (electronic). pdf 12. (with Dorian Goldfeld) The holomorphic kernel of the Rankin-Selberg con- volution. Asian J. Math. 3 (1999), no. 4, pp 729{747. pdf 13. Distribution of almost division points. Duke Math. J. 103 (2000), no. 1, pp 39{46. pdf 14. -
Fermat's Last Theorem
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions.[1] The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death.[2] After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[3] It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.[4] Contents The 1670 edition of Diophantus's Arithmetica includes Fermat's Overview commentary, referred to as his "Last Pythagorean origins Theorem" (Observatio Domini Petri Subsequent developments and solution de Fermat), posthumously published Equivalent statements of the theorem by his son. -
NÉRON MODELS 1. Introduction Néron Models Were Introduced by André Néron (1922-1985) in His Seminar at the IHES in 1961. He
NERON´ MODELS DINO LORENZINI Abstract. N´eronmodels were introduced by Andr´eN´eronin his seminar at the IHES in 1961. This article gives a brief survey of past and recent results in this very useful theory. KEYWORDS N´eronmodel, weak N´eronmodel, abelian variety, group scheme, elliptic curve, semi-stable reduction, Jacobian, group of components. MATHEMATICS SUBJECT CLASSIFICATION: 14K15, 11G10, 14L15 1. Introduction N´eronmodels were introduced by Andr´eN´eron(1922-1985) in his seminar at the IHES in 1961. He gave a summary of his results in a 1961 Bourbaki seminar [81], and a longer summary was published in 1962 in Crelle's journal [82]. N´eron'sown full account of his theory is found in [83]. Unfortunately, this article1 is not completely written in the modern language of algebraic geometry. N´eron'sinitial motivation for studying models of abelian varieties was his study of heights (see [84], and [56], chapters 5 and 10-12, for a modern account). In [83], N´eronmentions his work on heights2 by referring to a talk that he gave at the International Congress in Edinburgh in 1958. I have been unable to locate a written version of this talk. A result attributed to N´eronin [54], Corollary 3, written by Serge Lang in 1963, might be a result that N´erondiscussed in 1958. See also page 438 of [55]. There is a vast literature on the theory of N´eronmodels. The first mention of N´eron models completely in the language of schemes may be in an article of Jean-Pierre Serre at the International Congress in Stockholm in 1962 ([94], page 195). -
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. -
Dynamics on Berkovich Spaces in Low Dimensions
Dynamics on Berkovich Spaces in Low Dimensions Mattias Jonsson Contents 1 Introduction..................................................................................... 205 2 Tree Structures ................................................................................. 220 3 The Berkovich Affine and Projective Lines ................................................... 236 4 Action by Polynomial and Rational Maps .................................................... 253 5 Dynamics of Rational Maps in One Variable ................................................. 270 6 The Berkovich Affine Plane Over a Trivially Valued Field .................................. 283 7 The Valuative Tree at a Point .................................................................. 297 8 Local Plane Polynomial Dynamics ............................................................ 328 9 The Valuative Tree at Infinity.................................................................. 335 10 Plane Polynomial Dynamics at Infinity ....................................................... 352 References .......................................................................................... 361 1 Introduction The goal of these notes is twofold. First, I’d like to describe how Berkovich spaces enter naturally in certain instances of discrete dynamical systems. In particular, I will try to show how my own work with Charles Favre [FJ07, FJ11]onvaluative dynamics relates to the dynamics of rational maps on the Berkovich projective line as initiated by Juan Rivera-Letelier