DOCSLIB.ORG

Sir Andrew J. Wiles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sir Andrew J. Wiles ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society March 2017 Volume 64, Number 3 Women's History Month Ad Honorem Sir Andrew J. Wiles page 197 2018 Leroy P. Steele Prize: Call for Nominations page 195 Interview with New AMS President Kenneth A. Ribet page 229 New York Meeting page 291 Sir Andrew J. Wiles, 2016 Abel Laureate. “The definition of a good mathematical problem is the mathematics it generates rather Notices than the problem itself.” of the American Mathematical Society March 2017 FEATURES 197 239229 26239 Ad Honorem Sir Andrew J. Interview with New The Graduate Student Wiles AMS President Kenneth Section Interview with Abel Laureate Sir A. Ribet Interview with Ryan Haskett Andrew J. Wiles by Martin Raussen and by Alexander Diaz-Lopez Allyn Jackson Christian Skau WHAT IS...an Elliptic Curve? Andrew Wiles's Marvelous Proof by by Harris B. Daniels and Álvaro Henri Darmon Lozano-Robledo The Mathematical Works of Andrew Wiles by Christopher Skinner In this issue we honor Sir Andrew J. Wiles, prover of Fermat's Last Theorem, recipient of the 2016 Abel Prize, and star of the NOVA video The Proof. We've got the official interview, reprinted from the newsletter of our friends in the European Mathematical Society; "Andrew Wiles's Marvelous Proof" by Henri Darmon; and a collection of articles on "The Mathematical Works of Andrew Wiles" assembled by guest editor Christopher Skinner. We welcome the new AMS president, Ken Ribet (another star of The Proof). Marcelo Viana, Director of IMPA in Rio, describes "Math in Brazil" on the eve of the upcoming IMO and ICM. For Women's History Month we've got the story of Joan Clarke, an under-recognized English code-breaker during World War II, who like Alan Turing worked on the Enigma Project. To fend off spring fever, read the Notices, submit an article or an item for the BackPage, participate in the commen- tary on the webpage www.ams.org/notices, and send in some Letters to the Editor. —Frank Morgan, Editor-in-Chief FROM THE SECRETARY 252 100 Years Ago: Joan Clarke Allyn Jackson 195 2018 Leroy P. Steele Prizes: Call for Nominations 256 Commutative Algebra Provides a Big Surprise for 268 2017 Class of Fellows of the AMS Craig Huneke's Birthday Irena Swanson 260 Math PhD Careers: New Opportunities ALSO IN THIS ISSUE Emerging Amidst Crisis Yuliy Baryshnikov, Lee DeVille, Richard Laugesen 234 Math in Brazil: Sowing the Seeds Marcelo Viana 265 Book Review A Doubter's Almanac Sheldon Axler 245 Opinion The Common Vision Project: Four Reactions IN EVERY ISSUE Notices AMERICAN MATHEMATICAL SOCIETY of the American Mathematical Society 272 Mathematics People 277 Mathematics Opportunities 278 New Publications Offered by the AMS 283 Mathematics Calendar Game Theory, Alive 286 Classified Advertising Anna R. Karlin, University of Washington, Seattle, and Yuval Peres, Microsoft Research, Redmond, WA 287 Meetings and Conferences of the AMS This beautifully written book introduces the reader to the rich and flourishing subject of game theory. From an exhaus- 304 The Back Page tive account of basics of the classical theory to a wide range of modern themes—including auctions, matching markets, sequential decision making—the book presents a great variety of topics in a rigorous yet entertaining manner. The authors guide the reader through the intricacies of game theory with an exquisite mathematical elegance. For anyone interested in the topic, experts and beginners alike, this book is a must. —Gábor Lugosi, Pompeu Fabra University, Barcelona, Spain 2017; approximately 396 pages; Hardcover; ISBN: 978-1-4704-1982-0; List US$75; AMS members US$60; Order code MBK/101 Also Pre-order your copy at available bookstore.ams.org as an eBook! (800)321-4267 (US & Canada) (401)455-4000 (Worldwide) A MERIC A N M A THEM A TIC A L S OCIETY Leroy P. Steele Prizes The selection committee for these prizes requests nominations for con- sideration for the 2018 awards. Further information about the prizes can be found in the November 2015 Notices pp.1249–1255 (also available at www.ams.org/profession/prizes-awards/ams-prizes/steele-prize). Three Leroy P. Steele Prizes are awarded each year in the following catego- ries: (1) the Steele Prize for Lifetime Achievement: for the cumulative influ- ence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through PhD students; (2) the Steele Prize for Mathematical Exposition: for a book or substantial survey or expos- itory-research paper; and (3) the Steele Prize for Seminal Contribution to Research: for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research. In 2018 the prize for Seminal Contribution to Research will be awarded for a paper in Discrete Mathematics/Logic. Further information and instructions for submitting a nomination can be found at the Leroy P. Steele Prizes website: www.ams.org/profession/prizes-awards/ams-prizes/steele-prize Nominations for the Steele Prizes for Lifetime Achievement and for Mathematical Exposition will remain active and receive consideration for three consecutive years. For questions contact the AMS Secretary at [email protected]. The nomination period is February 1, 2017 through March 31, 2017. Call for Nominations for Call Notices of the American Mathematical Society EDITOR-IN-CHIEF CONTACTING THE NOTICES Frank Morgan SUBSCRIPTION INFORMATION ASSOCIATE EDITORS Subscription prices for Volume 64 (2017) are US$637 list; Benjamin Braun, US$509.60 institutional member; US$382.20 individual mem- Alexander Diaz-Lopez, Thomas Garrity, Joel Hass, ber; US$573.30 corporate member. (The subscription price for Stephen Kennedy, Florian Luca, Steven J. Miller, members is included in the annual dues.) A late charge of 10% of Harriet Pollatsek, Carla Savage (ex officio), the subscription price will be imposed upon orders received from Cesar E. Silva, Christina Sormani, Daniel J. Velleman non-members after January 1 of the subscription year. Add for CONSULTANTS postage: Surface delivery outside the United States and India— John Baez, Hélène Barcelo, Ricardo Cortez, Jerry Folland, US$27; in India—US$40; expedited delivery to destinations in Tara Holm, Kenneth A. Ribet North America—US$35; elsewhere—US$120. Subscriptions and orders for AMS publications should be addressed to the ASSISTANT to the EDITOR-IN-CHIEF American Mathematical Society, P.O. Box 845904, Boston, MA Katharine Merow 02284-5904 USA. All orders must be prepaid. SENIOR WRITER/DEPUTY EDITOR MANAGING EDITOR Allyn Jackson Rachel L. Rossi ADVERTISING Notices publishes situations wanted and classified advertising, ADVERTISING COORDINATOR and display advertising for publishers and academic or scientific Anne Newcomb organizations. Advertising requests, materials, and/or questions REPRINT PERMISSIONS should be sent to: Erin M. Buck [email protected] (classified ads) CONTRIBUTING WRITER [email protected] (display ads) Elaine Kehoe PERMISSIONS COMPOSITION, DESIGN and EDITING All requests to reprint Notices articles should be sent to: Kyle Antonevich, John Brady, Anna Hattoy, [email protected] Mary Medeiros, Lori Nero, Karen Ouellette, Courtney Rose, Deborah Smith, Mike Southern, Peter Sykes SUBMISSIONS The editor-in-chief should be contacted about articles for consideration after potential authors have reviewed the “For Supported by the AMS membership, most of this publication, Authors” page at www.ams.org/publications/journals/notices/ including the opportunity to post comments, is freely available noticesauthors. electronically through the AMS website, the Society’s resource for delivering electronic products and services. Use the URL The managing editor should be contacted for additions to our www.ams.org/notices/ to access the Notices on the website. The news sections and for any questions or corrections. Contact the online version of the Notices is the version of record, so it may managing editor at: [email protected] occasionally differ slightly from the print version. Letters to the editor should be sent to: The print version is a privilege of Membership. Graduate students [email protected] at member institutions can opt to receive the print magazine by updating their individual member profiles at https://www.ams. Additions to the Math Calendar should be submitted at: org/cml/update-ams. For questions regarding updating your bit.ly/1SDf5kF profile, please call 800-321-4267. To make suggestions for additions to other sections, and for For back issues see www.ams.org/backvols. Note: Single issues full contact information, see www.ams.org/publications/ of the Notices are not available after one calendar year. journals/notices/noticescontact [Notices of the American Mathematical Society (ISSN 0002-9920) is published monthly except bimonthly in June/July by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA, GST No. 12189 2046 RT****. Periodicals postage paid at Providence, RI, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P.O. Box 6248, Providence, RI 02904-6248 USA.] Publication here of the Society’s street address and the other bracketed information is a technical requirement of the U.S. Postal Service. © Copyright 2017 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society. AD HONOREM SIR ANDREW J. WILES Wiles received the Abel Prize from Crown Prince Haakon of Norway. Wiles ultimately established in a proof both surprising and profound.
Load more

Recommended publications
  • 2006 Annual Report
    Contents Clay Mathematics Institute 2006 James A. Carlson Letter from the President 2 Recognizing Achievement Fields Medal Winner Terence Tao 3 Persi Diaconis Mathematics & Magic Tricks 4 Annual Meeting Clay Lectures at Cambridge University 6 Researchers, Workshops & Conferences Summary of 2006 Research Activities 8 Profile Interview with Research Fellow Ben Green 10 Davar Khoshnevisan Normal Numbers are Normal 15 Feature Article CMI—Göttingen Library Project: 16 Eugene Chislenko The Felix Klein Protocols Digitized The Klein Protokolle 18 Summer School Arithmetic Geometry at the Mathematisches Institut, Göttingen, Germany 22 Program Overview The Ross Program at Ohio State University 24 PROMYS at Boston University Institute News Awards & Honors 26 Deadlines Nominations, Proposals and Applications 32 Publications Selected Articles by Research Fellows 33 Books & Videos Activities 2007 Institute Calendar 36 2006 Another major change this year concerns the editorial board for the Clay Mathematics Institute Monograph Series, published jointly with the American Mathematical Society. Simon Donaldson and Andrew Wiles will serve as editors-in-chief, while I will serve as managing editor. Associate editors are Brian Conrad, Ingrid Daubechies, Charles Fefferman, János Kollár, Andrei Okounkov, David Morrison, Cliff Taubes, Peter Ozsváth, and Karen Smith. The Monograph Series publishes Letter from the president selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. The next volume in the series will be Ricci Flow and the Poincaré Conjecture, by John Morgan and Gang Tian. Their book will appear in the summer of 2007. In related publishing news, the Institute has had the complete record of the Göttingen seminars of Felix Klein, 1872–1912, digitized and made available on James Carlson.
    [Show full text]
  • A MATHEMATICIAN's SURVIVAL GUIDE 1. an Algebra Teacher I
    A MATHEMATICIAN’S SURVIVAL GUIDE PETER G. CASAZZA 1. An Algebra Teacher I could Understand Emmy award-winning journalist and bestselling author Cokie Roberts once said: As long as algebra is taught in school, there will be prayer in school. 1.1. An Object of Pride. Mathematician’s relationship with the general public most closely resembles “bipolar” disorder - at the same time they admire us and hate us. Almost everyone has had at least one bad experience with mathematics during some part of their education. Get into any taxi and tell the driver you are a mathematician and the response is predictable. First, there is silence while the driver relives his greatest nightmare - taking algebra. Next, you will hear the immortal words: “I was never any good at mathematics.” My response is: “I was never any good at being a taxi driver so I went into mathematics.” You can learn a lot from taxi drivers if you just don’t tell them you are a mathematician. Why get started on the wrong foot? The mathematician David Mumford put it: “I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate.” 1.2. A Balancing Act. The other most common response we get from the public is: “I can’t even balance my checkbook.” This reflects the fact that the public thinks that mathematics is basically just adding numbers. They have no idea what we really do. Because of the textbooks they studied, they think that all needed mathematics has already been discovered.
    [Show full text]
  • Manjul Bhargava
    The Work of Manjul Bhargava Manjul Bhargava's work in number theory has had a profound influence on the field. A mathematician of extraordinary creativity, he has a taste for simple problems of timeless beauty, which he has solved by developing elegant and powerful new methods that offer deep insights. When he was a graduate student, Bhargava read the monumental Disqui- sitiones Arithmeticae, a book about number theory by Carl Friedrich Gauss (1777-1855). All mathematicians know of the Disquisitiones, but few have actually read it, as its notation and computational nature make it difficult for modern readers to follow. Bhargava nevertheless found the book to be a wellspring of inspiration. Gauss was interested in binary quadratic forms, which are polynomials ax2 +bxy +cy2, where a, b, and c are integers. In the Disquisitiones, Gauss developed his ingenious composition law, which gives a method for composing two binary quadratic forms to obtain a third one. This law became, and remains, a central tool in algebraic number theory. After wading through the 20 pages of Gauss's calculations culminating in the composition law, Bhargava knew there had to be a better way. Then one day, while playing with a Rubik's cube, he found it. Bhargava thought about labeling each corner of a cube with a number and then slic- ing the cube to obtain 2 sets of 4 numbers. Each 4-number set naturally forms a matrix. A simple calculation with these matrices resulted in a bi- nary quadratic form. From the three ways of slicing the cube, three binary quadratic forms emerged.
    [Show full text]
  • MY UNFORGETTABLE EARLY YEARS at the INSTITUTE Enstitüde Unutulmaz Erken Yıllarım
    MY UNFORGETTABLE EARLY YEARS AT THE INSTITUTE Enstitüde Unutulmaz Erken Yıllarım Dinakar Ramakrishnan `And what was it like,’ I asked him, `meeting Eliot?’ `When he looked at you,’ he said, `it was like standing on a quay, watching the prow of the Queen Mary come towards you, very slowly.’ – from `Stern’ by Seamus Heaney in memory of Ted Hughes, about the time he met T.S.Eliot It was a fortunate stroke of serendipity for me to have been at the Institute for Advanced Study in Princeton, twice during the nineteen eighties, first as a Post-doctoral member in 1982-83, and later as a Sloan Fellow in the Fall of 1986. I had the privilege of getting to know Robert Langlands at that time, and, needless to say, he has had a larger than life influence on me. It wasn’t like two ships passing in the night, but more like a rowboat feeling the waves of an oncoming ship. Langlands and I did not have many conversations, but each time we did, he would make a Zen like remark which took me a long time, at times months (or even years), to comprehend. Once or twice it even looked like he was commenting not on the question I posed, but on a tangential one; however, after much reflection, it became apparent that what he had said had an interesting bearing on what I had been wondering about, and it always provided a new take, at least to me, on the matter. Most importantly, to a beginner in the field like I was then, he was generous to a fault, always willing, whenever asked, to explain the subtle aspects of his own work.
    [Show full text]
  • Curriculum Vitae
    CURRICULUM VITAE JANNIS A. ANTONIADIS Department of Mathematics University of Crete 71409 Iraklio,Crete Greece PERSONAL Born on 5 of September 1951 in Dryovouno Kozani, Greece. Married with Sigrid Arnz since 1983, three children : Antonios (26), Katerina (23), Karolos (21). EDUCATION-EMPLOYMENT 1969-1973: B.S. in Mathematics at the University of Thessaloniki, Greece. 1973-1976: Military Service. 1976-1979: Assistant at the University of Thessaloniki, Greece. 1979-1981: Graduate student at the University of Cologne, Germany. 1981 Ph.D. in Mathematics at the University of Cologne. Thesis advisor: Prof. Dr. Curt Meyer. 1982-1984: Lecturer at the University of Thessaloniki, Greece. 1984-1990: Associate Professor at the University of Crete, Greece. 1990-now: Professor at the University of Crete, Greece. 2003-2007 Chairman of the Department VISITING POSITIONS - University of Cologne Germany, during the period from December 1981 until Ferbruary 1983 as researcher of the German Research Council (DFG). - MPI-for Mathematics Bonn Germany, during the periods: from May until September of the year 1985, from May until September of the year 1986, from July until January of the year 1988 and from July until September of the year 1988. - University of Heidelberg Germany, during the period: from September 1993 until January 1994, as visiting Professor. University of Cyprus, during the period from January 2008 until May 2008, as visiting Professor. Again for the Summer Semester of 2009 and of the Winter Semester 2009-2010. LONG TERM VISITS: - University
    [Show full text]
  • A Mathematical Tonic
    NATURE|Vol 443|14 September 2006 BOOKS & ARTS reader nonplussed. For instance, he discusses a problem in which a mouse runs at constant A mathematical tonic speed round a circle, and a cat, starting at the centre of the circle, runs at constant speed Dr Euler’s Fabulous Formula: Cures Many mathematician. This is mostly a good thing: directly towards the mouse at all times. Does Mathematical Ills not many mathematicians know much about the cat catch the mouse? Yes, if, and only if, it by Paul Nahin signal processing or the importance of complex runs faster than the mouse. In the middle of Princeton University Press: 2006. 404 pp. numbers to engineers, and it is fascinating to the discussion Nahin suddenly abandons the $29.95, £18.95 find out about these subjects. Furthermore, it mathematics and puts a discrete approximation is likely that a mathematician would have got into his computer to generate some diagrams. Timothy Gowers hung up on details that Nahin cheerfully, and His justification is that an analytic formula for One of the effects of the remarkable success rightly, ignores. However, his background has the cat’s position is hard to obtain, but actu- of Stephen Hawking’s A Brief History Of Time the occasional disadvantage as well, such as ally the problem can be neatly solved without (Bantam, 1988) was a burgeoning of the mar- when he discusses the matrix formulation of such a formula (although it is good to have ket for popular science writing. It did not take de Moivre’s theorem, the fundamental result his diagrams as well).
    [Show full text]
  • Interviewed by T. Christine Stevens)
    KENNETH A. ROSS JANUARY 8, 2011 AND JANUARY 5, 2012 (Interviewed by T. Christine Stevens) How did you get involved in the MAA? As a good citizen of the mathematical community, I was a member of MAA from the beginning of my career. But I worked in an “AMS culture,” so I wasn’t actively involved in the MAA. As of January, 1983, I had never served on an MAA committee. But I had been Associate Secretary of the AMS from 1971 to 1981, and thus Len Gillman (who was MAA Treasurer at the time) asked me to be MAA Secretary. There was a strong contrast between the cultures of the AMS and the MAA, and my first two years were very hard. Did you receive mentoring in the MAA at the early stages of your career? From whom? As a graduate student at the University of Washington, I hadn’t even been aware that the department chairman, Carl Allendoerfer, was serving at the time as MAA President. My first mentor in the MAA was Len Gillman, who got me involved with the MAA. Being Secretary and Treasurer, respectively, we consulted a lot, and he was the one who helped me learn the MAA culture. One confession: At that time, approvals for new unbudgeted expenses under $500 were handled by the Secretary, the Treasurer and the Executive Director, Al Wilcox. The requests usually came to me first. Since Len was consistently tough, and Al was a push-over, I would first ask the one whose answer would agree with mine, and then with a 2-0 vote, I didn’t have to even bother the other one.
    [Show full text]
  • I. Overview of Activities, April, 2005-March, 2006 …
    MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 …......……………………. 2 Innovations ………………………………………………………..... 2 Scientific Highlights …..…………………………………………… 4 MSRI Experiences ….……………………………………………… 6 II. Programs …………………………………………………………………….. 13 III. Workshops ……………………………………………………………………. 17 IV. Postdoctoral Fellows …………………………………………………………. 19 Papers by Postdoctoral Fellows …………………………………… 21 V. Mathematics Education and Awareness …...………………………………. 23 VI. Industrial Participation ...…………………………………………………… 26 VII. Future Programs …………………………………………………………….. 28 VIII. Collaborations ………………………………………………………………… 30 IX. Papers Reported by Members ………………………………………………. 35 X. Appendix - Final Reports ……………………………………………………. 45 Programs Workshops Summer Graduate Workshops MSRI Network Conferences MATHEMATICAL SCIENCES RESEARCH INSTITUTE ANNUAL REPORT FOR 2005-2006 I. Overview of Activities, April, 2005-March, 2006 This annual report covers MSRI projects and activities that have been concluded since the submission of the last report in May, 2005. This includes the Spring, 2005 semester programs, the 2005 summer graduate workshops, the Fall, 2005 programs and the January and February workshops of Spring, 2006. This report does not contain fiscal or demographic data. Those data will be submitted in the Fall, 2006 final report covering the completed fiscal 2006 year, based on audited financial reports. This report begins with a discussion of MSRI innovations undertaken this year, followed by highlights
    [Show full text]
  • 2012-13 Annual Report of Private Giving
    MAKING THE EXTRAORDINARY POSSIBLE 2012–13 ANNUAL REPORT OF PRIVATE GIVING 2 0 1 2–13 ANNUAL REPORT OF PRIVATE GIVING “Whether you’ve been a donor to UMaine for years or CONTENTS have just made your first gift, I thank you for your Letter from President Paul Ferguson 2 Fundraising Partners 4 thoughtfulness and invite you to join us in a journey Letter from Jeffery Mills and Eric Rolfson 4 that promises ‘Blue Skies ahead.’ ” President Paul W. Ferguson M A K I N G T H E Campaign Maine at a Glance 6 EXTRAORDINARY 2013 Endowments/Holdings 8 Ways of Giving 38 POSSIBLE Giving Societies 40 2013 Donors 42 BLUE SKIES AHEAD SINCE GRACE, JENNY AND I a common theme: making life better student access, it is donors like you arrived at UMaine just over two years for others — specifically for our who hold the real keys to the ago, we have truly enjoyed our students and the state we serve. While University of Maine’s future level interactions with many alumni and I’ve enjoyed many high points in my of excellence. friends who genuinely care about this personal and professional life, nothing remarkable university. Events like the surpasses the sense of reward and Unrestricted gifts that provide us the Stillwater Society dinner and the accomplishment that accompanies maximum flexibility to move forward Charles F. Allen Legacy Society assisting others to fulfill their are one of these keys. We also are luncheon have allowed us to meet and potential. counting on benefactors to champion thank hundreds of donors.
    [Show full text]
  • Robert P. Langlands Receives the Abel Prize
    Robert P. Langlands receives the Abel Prize The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for Advanced Study, Princeton, USA “for his visionary program connecting representation theory to number theory.” Robert P. Langlands has been awarded the Abel Prize project in modern mathematics has as wide a scope, has for his work dating back to January 1967. He was then produced so many deep results, and has so many people a 30-year-old associate professor at Princeton, working working on it. Its depth and breadth have grown and during the Christmas break. He wrote a 17-page letter the Langlands program is now frequently described as a to the great French mathematician André Weil, aged 60, grand unified theory of mathematics. outlining some of his new mathematical insights. The President of the Norwegian Academy of Science and “If you are willing to read it as pure speculation I would Letters, Ole M. Sejersted, announced the winner of the appreciate that,” he wrote. “If not – I am sure you have a 2018 Abel Prize at the Academy in Oslo today, 20 March. waste basket handy.” Biography Fortunately, the letter did not end up in a waste basket. His letter introduced a theory that created a completely Robert P. Langlands was born in New Westminster, new way of thinking about mathematics: it suggested British Columbia, in 1936. He graduated from the deep links between two areas, number theory and University of British Columbia with an undergraduate harmonic analysis, which had previously been considered degree in 1957 and an MSc in 1958, and from Yale as unrelated.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:1407.1093V1 [Math.NT]
    MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Abstract. We show that the cyclotomic Iwasawa–Greenberg Main Conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg Main Conjecture was established in [18], in com- bination with work of Kato [13], for a large class of newforms f ∈ Sk(Γ0(N)) that are ordinary at an odd prime p ∤ N, subject to k ≡ 2 (mod p − 1) and certain conditions on the mod p Galois representation associated with f. The purpose of this note is to extend this result to the case where p | N (in which case k is necessarily equal to 2). ∞ n Recall that the coefficients an of the q-expansion f = n=1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f) are algebraic integers that generate a finite extension Q(f) ⊂ C of Q. Let p be an odd primeP and let L be a finite extension of the completion of Q(f) at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q(f) ֒→ Qp). Suppose that f is ordinary at p with respect to L in the sense that ap is a unit in the ring of integers O of L.
    [Show full text]