Redalyc.Solution for Fermat's Last Theorem
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Arxiv:2003.01675V1 [Math.NT] 3 Mar 2020 of Their Locations
DIVISORS OF MODULAR PARAMETRIZATIONS OF ELLIPTIC CURVES MICHAEL GRIFFIN AND JONATHAN HALES Abstract. The modularity theorem implies that for every elliptic curve E=Q there exist rational maps from the modular curve X0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y (z) where X(z) and Y (z) satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the q - expansions of these parametrizations at any cusp. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X0(N). We also examine a connection between the al- gebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects. 1. Introduction and statement of results The modularity theorem [2, 12] guarantees that for every elliptic curve E of con- ductor N there exists a weight 2 newform fE of level N with Fourier coefficients in Z. The Eichler integral of fE (see (3)) and the Weierstrass }-function together give a rational map from the modular curve X0(N) to the coordinates of some model of E: This parametrization has singularities wherever the value of the Eichler integral is in the period lattice. Kodgis [6] showed computationally that many of the zeros of the Eichler integral occur at CM points. -
Arxiv:Math/0701554V2 [Math.HO] 22 May 2007 Pythagorean Triples and a New Pythagorean Theorem
Pythagorean Triples and A New Pythagorean Theorem H. Lee Price and Frank Bernhart January 1, 2007 Abstract Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational (integer) right triangles from a given one. 1. Harmonic and Symphonic Squares Consider an arbitrary triangle with altitude α corresponding to base β (see Figure 1a). Assuming that the base angles are acute, suppose that a square of side η is inscribed as shown in Figure 1b. arXiv:math/0701554v2 [math.HO] 22 May 2007 Figure 1: Triangle and inscribed square Then α,β,η form a harmonic sum, i.e. satisfy (1). The equivalent formula αβ η = α+β is also convenient, and is found in some geometry books. 1 1 1 + = (1) α β η 1 Figure 2: Three congruent harmonic squares Equation (1) remains valid if a base angle is a right angle, or is obtuse, save that in the last case the triangle base must be extended, and the square is not strictly inscribed ( Figures 2a, 2c ). Starting with the right angle case (Figure 2a) it is easy to see the inscribed square uniquely exists (bisect the right angle), and from similar triangles we have proportion (β η) : β = η : α − which leads to (1). The horizontal dashed lines are parallel, hence the three triangles of Figure 2 have the same base and altitude; and the three squares are congruent and unique. Clearly (1) applies to all cases. -
Supersingular Zeros of Divisor Polynomials of Elliptic Curves of Prime Conductor
Kazalicki and Kohen Res Math Sci(2017) 4:10 DOI 10.1186/s40687-017-0099-8 R E S E A R C H Open Access Supersingular zeros of divisor polynomials of elliptic curves of prime conductor Matija Kazalicki1* and Daniel Kohen2,3 *Correspondence: [email protected] Abstract 1Department of Mathematics, p p University of Zagreb, Bijeniˇcka For a prime number , we study the zeros modulo of divisor polynomials of rational cesta 30, 10000 Zagreb, Croatia elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, Full list of author information is 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of available at the end of the article supersingular elliptic curves over Fp. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form vE .This allows us to prove that if the root number of E is −1 then all supersingular j-invariants of elliptic curves defined over Fp are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in Fp seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of vE corresponding to supersingular elliptic curves defined over Fp are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest. -
A Glimpse of the Laureate's Work
A glimpse of the Laureate’s work Alex Bellos Fermat’s Last Theorem – the problem that captured planets moved along their elliptical paths. By the beginning Andrew Wiles’ imagination as a boy, and that he proved of the nineteenth century, however, they were of interest three decades later – states that: for their own properties, and the subject of work by Niels Henrik Abel among others. There are no whole number solutions to the Modular forms are a much more abstract kind of equation xn + yn = zn when n is greater than 2. mathematical object. They are a certain type of mapping on a certain type of graph that exhibit an extremely high The theorem got its name because the French amateur number of symmetries. mathematician Pierre de Fermat wrote these words in Elliptic curves and modular forms had no apparent the margin of a book around 1637, together with the connection with each other. They were different fields, words: “I have a truly marvelous demonstration of this arising from different questions, studied by different people proposition which this margin is too narrow to contain.” who used different terminology and techniques. Yet in the The tantalizing suggestion of a proof was fantastic bait to 1950s two Japanese mathematicians, Yutaka Taniyama the many generations of mathematicians who tried and and Goro Shimura, had an idea that seemed to come out failed to find one. By the time Wiles was a boy Fermat’s of the blue: that on a deep level the fields were equivalent. Last Theorem had become the most famous unsolved The Japanese suggested that every elliptic curve could be problem in mathematics, and proving it was considered, associated with its own modular form, a claim known as by consensus, well beyond the reaches of available the Taniyama-Shimura conjecture, a surprising and radical conceptual tools. -
The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes By
The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes by Tony Feng1 Oberwolfach April 2-8, 2017 1version of April 11, 2017 Contents Note to the reader 5 Part 1. Day One 7 1. An overview of the Gross-Zagier and Waldspurger formulas (Yunqing Tang) 8 2. The stacks Bunn and Hecke (Timo Richarz) 13 3. (Moduli of Shtukas I) (Doug Ulmer) 17 4. Moduli of Shtukas II (Brian Smithling) 21 Part 2. Day Two 25 5. Automorphic forms over function fields (Ye Tian) 26 6. The work of Drinfeld (Arthur Cesar le Bras) 30 7. Analytic RTF: Geometric Side (Jingwei Xiao) 36 8. Analytic RTF: Spectral Side (Ilya Khayutin) 41 Part 3. Day Three 47 9. Geometric Interpretation of Orbital Integrals (Yihang Zhu) 48 10. Definition and properties of Md (Jochen Heinloth) 52 Part 4. Day Four 57 11. Intersection theory on stacks (Michael Rapoport) 58 12. LTF for Cohomological Correspondences (Davesh Maulik) 64 µ 13. Definition and description of HkM;d: expressing Ir(hD) as a trace (Liang Xiao) 71 3 4 CONTENTS 14. Alternative calculation of Ir(hD) (Yakov Varshavsky) 77 Part 5. Day Five 83 15. Comparison of Md and Nd; the weight factors (Ana Caraiani) 84 16. Horocycles (Lizao Ye) 89 17. Cohomological spectral decomposition and finishing the proof (Chao Li) 93 NOTE TO THE READER 5 Note to the reader This document consists of notes I live-TEXed during the Arbeitsgemeinschaft at Oberwolfach in April 2017. They should not be taken as a faithful transcription of the actual lectures; they represent only my personal perception of the talks. -
[The PROOF of FERMAT's LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] the World's Most Famous Math Problem the World's Most Famous Math Problem
0Eft- [The PROOF of FERMAT'S LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] The World's Most Famous Math Problem The World's Most Famous Math Problem [ THE PROOF OF FERMAT'S LAST THEOREM AND OTHER MATHEMATICAL MYSTERIES I Marilyn vos Savant ST. MARTIN'S PRESS NEW YORK For permission to reprint copyrighted material, grateful acknowledgement is made to the following sources: The American Association for the Advancement of Science: Excerpts from Science, Volume 261, July 2, 1993, C 1993 by the AAAS. Reprinted by permission. Birkhauser Boston: Excerpts from The Mathematical Experience by Philip J. Davis and Reuben Hersh © 1981 Birkhauser Boston. Reprinted by permission of Birkhau- ser Boston and the authors. The Chronicleof Higher Education: Excerpts from The Chronicle of Higher Education, July 7, 1993, C) 1993 Chronicle of HigherEducation. Reprinted by permission. The New York Times: Excerpts from The New York Times, June 24, 1993, X) 1993 The New York Times. Reprinted by permission. Excerpts from The New York Times, June 29, 1993, © 1993 The New York Times. Reprinted by permission. Cody Pfanstieh/ The poem on the subject of Fermat's last theorem is reprinted cour- tesy of Cody Pfanstiehl. Karl Rubin, Ph.D.: The sketch of Dr. Wiles's proof of Fermat's Last Theorem in- cluded in the Appendix is reprinted courtesy of Karl Rubin, Ph.D. Wesley Salmon, Ph.D.: Excerpts from Zeno's Paradoxes by Wesley Salmon, editor © 1970. Reprinted by permission of the editor. Scientific American: Excerpts from "Turing Machines," by John E. Hopcroft, Scientific American, May 1984, (D 1984 Scientific American, Inc. -
The Theorem of Pythagoras
The Most Famous Theorem The Theorem of Pythagoras feature Understanding History – Who, When and Where Does mathematics have a history? In this article the author studies the tangled and multi-layered past of a famous result through the lens of modern thinking, look- ing at contributions from schools of learning across the world, and connecting the mathematics recorded in archaeological finds with that taught in the classroom. Shashidhar Jagadeeshan magine, in our modern era, a very important theorem being Iattributed to a cult figure, a new age guru, who has collected a band of followers sworn to secrecy. The worldview of this cult includes number mysticism, vegetarianism and the transmigration of souls! One of the main preoccupations of the group is mathematics: however, all new discoveries are ascribed to the guru, and these new results are not to be shared with anyone outside the group. Moreover, they celebrate the discovery of a new result by sacrificing a hundred oxen! I wonder what would be the current scientific community’s reaction to such events. This is the legacy associated with the most ‘famous’ theorem of all times, the Pythagoras Theorem. In this article, we will go into the history of the theorem, explain difficulties historians have with dating and authorship and also speculate as to what might have led to the general statement and proof of the theorem. Vol. 1, No. 1, June 2012 | At Right Angles 5 Making sense of the history Why Pythagoras? Greek scholars seem to be in Often in the history of ideas, especially when there agreement that the first person to clearly state PT has been a discovery which has had a significant in all its generality, and attempt to establish its influence on mankind, there is this struggle to find truth by the use of rigorous logic (what we now call mathematical proof), was perhaps Pythagoras out who discovered it first. -
San Diego Public Library New Additions September 2008
San Diego Public Library New Additions September 2008 Adult Materials 000 - Computer Science and Generalities California Room 100 - Philosophy & Psychology CD-ROMs 200 - Religion Compact Discs 300 - Social Sciences DVD Videos/Videocassettes 400 - Language eAudiobooks & eBooks 500 - Science Fiction 600 - Technology Foreign Languages 700 - Art Genealogy Room 800 - Literature Graphic Novels 900 - Geography & History Large Print Audiocassettes Newspaper Room Audiovisual Materials Biographies Fiction Call # Author Title FIC/ABE Abé, Shana. The dream thief FIC/ABRAHAMS Abrahams, Peter, 1947- Delusion [SCI-FI] FIC/ADAMS Adams, Douglas, 1952- Dirk Gently's holistic detective agency FIC/ADAMSON Adamson, Gil, 1961- The outlander : a novel FIC/ADLER Adler, Elizabeth (Elizabeth A.) Meet me in Venice FIC/AHERN Ahern, Cecelia, 1981- There's no place like here FIC/ALAM Alam, Saher, 1973- The groom to have been FIC/ALEXANDER Alexander, Robert, 1952- The Romanov bride FIC/ALI Ali, Tariq. Shadows of the pomegranate tree FIC/ALLEN Allen, Preston L., 1964- All or nothing [SCI-FI] FIC/ALLSTON Allston, Aaron. Star wars : legacy of the force : betrayal [SCI-FI] FIC/ANDERSON Anderson, Kevin J. Darksaber FIC/ARCHER Archer, Jeffrey, 1940- A prisoner of birth FIC/ARCHER Archer, Jeffrey, 1940- A prisoner of birth FIC/ARCHER Archer, Jeffrey, 1940- Cat o'nine tales and other stories FIC/ASARO Asaro, Catherine. The night bird FIC/AUSTEN Austen, Jane, 1775-1817. Emma FIC/AUSTEN Austen, Jane, 1775-1817. Mansfield Park FIC/AUSTEN Austen, Jane, 1775-1817. Minor works FIC/AUSTEN Austen, Jane, 1775-1817. Northanger Abbey and Persuasion FIC/AUSTEN Austen, Jane, 1775-1817. Sense and sensibility FIC/BAHAL Bahal, Aniruddha, 1967- Bunker 13 FIC/BALDACCI Baldacci, David. -
HISTORY of MATHEMATICS 1 Pythagorean Triples
HISTORY OF MATHEMATICS Summer 2010 Pythagoras and Fermat 1 Pythagorean Triples A Pythagorean triple is a triple of positive integers (a; b; c) such that a2 +b2 = c2. The simplest such triple is (3; 4; 5). Here are some facts about these triples. Since this is a math course, proofs are provided. I'll abbreviate Pythagorean triple to pt. 1. If (a; b; c) is a pt, then so is (ma; mb; mc) for any positive integer m (mul- tiples of pt's are pt's). In fact, (ma)2 + (mb)2 = m2a2 + m2b2 = m2(a2 + b2) = m2c2 = (mc)2: As an example, from the triple (3; 4; 5) we get the triples (6; 8; 10), (9; 12; 15), (12; 16; 20), etc. 2. If (a; b; c) is a pt and if the positive integer d divides a; b, and c, then (a=d; b=d; c=d) is a pt. In fact, ( ) ( ) ( ) a 2 b 2 a2 b2 a2 + b2 c62 c 2 + = + = = = : d d d2 d2 d2 d2 d Because of these two first properties, the basic pt's are those in which a; b; c do not have a common divisor other than 1. We make the following definition: A pt is said to be primitive, or to be a primitive Pythagorean triple if the greatest common divisor of a; b; c is one. Every Pythagorean triple is either primitive or a multiple of a primitive one. Let's continue with the properties 4. If (a; b; c) is a ppt, then c is odd and a; b have opposite parity; that is, one of a; b is odd, the other one even. -
Pythagorean Triples MAS 100 Concepts of Mathematics Sample Paper Jane D
Pythagorean Triples MAS 100 Concepts of Mathematics Sample Paper Jane D. Student September 2013 Everyone is familiar with the Pythagorean Theorem, which states that if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a2 +b2 = c2. Also familiar is the “3-4-5” right triangle. It is an example of three whole numbers that satisfy the equation in the theorem. Other such examples, from less familiar to downright obscure, are right triangles with side lengths 5-12-13, 35-12-37, and 45-28-53. Why are these interesting? Well, if you try to look for right triangles with whole number side lengths by trial and error, you will mostly be frustrated, since most of the time two whole number side lengths go with an irrational third side, like 1-1-√2. Is there a pattern to the whole number triples? If so, what is it? Problem A. Describe all possible triples (called Pythagorean triples) of whole numbers (a,b,c) which can occur as the lengths of the three sides of a right triangle. In other words, find all positive whole num- ber solutions to the equation a2 + b2 = c2. A complete solution was known to the Greeks by about 500 BCE ([3], p.37). Weisstein gives a thorough survey of solutions to this and related problems on his MathWorld website [4]. In this paper we describe a solution which uses a beautiful map called stereographic projection. Similar presentations can be found in [1] and [2]. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Andrew Wiles's Marvelous Proof
Andrew Wiles’s Marvelous Proof Henri Darmon ermat famously claimed to have discovered “a truly marvelous proof” of his Last Theorem, which the margin of his copy of Diophantus’s Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, AndrewF Wiles’s marvelous proof has been public for over two decades and has now earned him the Abel Prize. According to the prize citation, Wiles merits this recogni- tion “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood It is also a centerpiece and appreciated Wiles giving his first lecture in Princeton about his by a novice of the “Langlands approach to proving the Modularity Conjecture in (like the ten- early 1994. year-old Andrew program,” the Wiles browsing of Theorem 2 of Karl Rubin’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “Langlands program,” the his local pub- imposing, ambitious edifice of results and conjectures lic library), yet edifice of results and which has come to dominate the number theorist’s view eluding the con- conjectures which of the world. This program has been described as a “grand certed efforts of unified theory” of mathematics. Taking a Norwegian per- the most brilliant has come to dominate spective, it connects the objects that occur in the works minds for well of Niels Hendrik Abel, such as elliptic curves and their over three cen- the number theorist’s associated Abelian integrals and Galois representations, turies, becoming with (frequently infinite-dimensional) linear representa- over its long his- view of the world.