Modular Forms and Diophantine Questions
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Report for the Academic Year 1999
l'gEgasag^a3;•*a^oggMaBgaBK>ry^vg^.g^._--r^J3^JBgig^^gqt«a»J^:^^^^^ Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1998-99 PRINCETON • NEW JERSEY HISTORICAL STUDIES^SOCIAl SC^JCE LIBRARY INSTITUTE FOR ADVANCED STUDY PRINCETON, NEW JERSEY 08540 Institute /or ADVANCED STUDY REPORT FOR THE ACADEMIC YEAR 1 998 - 99 OLDEN LANE PRINCETON • NEW JERSEY • 08540-0631 609-734-8000 609-924-8399 (Fax) http://www.ias.edu Extract from the letter addressed by the Institute's Founders, Louis Bamberger and Mrs. FeUx Fuld, to the Board of Trustees, dated June 4, 1930. Newark, New Jersey. It is fundamental m our purpose, and our express desire, that in the appointments to the staff and faculty, as well as in the admission of workers and students, no account shall be taken, directly or indirectly, of race, religion, or sex. We feel strongly that the spirit characteristic of America at its noblest, above all the pursuit of higher learning, cannot admit of any conditions as to personnel other than those designed to promote the objects for which this institution is established, and particularly with no regard whatever to accidents of race, creed, or sex. ni' TABLE OF CONTENTS 4 • BACKGROUND AND PURPOSE 7 • FOUNDERS, TRUSTEES AND OFFICERS OF THE BOARD AND OF THE CORPORATION 10 • ADMINISTRATION 12 • PRESENT AND PAST DIRECTORS AND FACULTY 15 REPORT OF THE CHAIRMAN 18 • REPORT OF THE DIRECTOR 22 • OFFICE OF THE DIRECTOR - RECORD OF EVENTS 27 ACKNOWLEDGMENTS 41 • REPORT OF THE SCHOOL OF HISTORICAL STUDIES FACULTY ACADEMIC ACTIVITIES MEMBERS, VISITORS, -
Sir Andrew Wiles Awarded Abel Prize
Sir Andrew J. Wiles Awarded Abel Prize Elaine Kehoe with The Norwegian Academy of Science and Letters official Press Release ©Abelprisen/DNVA/Calle Huth. Courtesy of the Abel Prize Photo Archive. ©Alain Goriely, University of Oxford. Courtesy the Abel Prize Photo Archive. Sir Andrew Wiles received the 2016 Abel Prize at the Oslo award ceremony on May 24. The Norwegian Academy of Science and Letters has carries a cash award of 6,000,000 Norwegian krone (ap- awarded the 2016 Abel Prize to Sir Andrew J. Wiles of the proximately US$700,000). University of Oxford “for his stunning proof of Fermat’s Citation Last Theorem by way of the modularity conjecture for Number theory, an old and beautiful branch of mathemat- semistable elliptic curves, opening a new era in number ics, is concerned with the study of arithmetic properties of theory.” The Abel Prize is awarded by the Norwegian Acad- the integers. In its modern form the subject is fundamen- tally connected to complex analysis, algebraic geometry, emy of Science and Letters. It recognizes contributions of and representation theory. Number theoretic results play extraordinary depth and influence to the mathematical an important role in our everyday lives through encryption sciences and has been awarded annually since 2003. It algorithms for communications, financial transactions, For permission to reprint this article, please contact: and digital security. [email protected]. Fermat’s Last Theorem, first formulated by Pierre de DOI: http://dx.doi.org/10.1090/noti1386 Fermat in the seventeenth century, is the assertion that 608 NOTICES OF THE AMS VOLUME 63, NUMBER 6 the equation xn+yn=zn has no solutions in positive integers tophe Breuil, Brian Conrad, Fred Diamond, and Richard for n>2. -
2001 Newsletter
ContinuUM Newsletter of the Department of Mathematics at the University of Michigan Summer 2001 On July 31, 2001, Al Taylor stepped down as department Chair after six successful years in the postion. Trevor Wooley has been selected as the new Chair. Due to Trevor’s scheduled sabbatical leave for the 2001-02 academic year, Alejandro Uribe, the current Associate Chair for Academic Affairs, will serve as the interim chair for the coming year. You should hear about Alejandro’s experience as Chair in the next ContinuUM. In the two “Notes from the Chair” columns, Al reflects on his chairmanship, and Trevor lays some plans for the future of the department. Reflections on Six The Future of the Years as Chair Mathematics Department This is a piece I’ve looked forward to The Mathematics Department at writing for some seven years now, my fi- Michigan has experienced extensive nal one for the “Notes from the Chair” change in virtually all aspects of its mis- column. At the end of July, I’ll have fin- sion over the past several years, and we ished my two terms as Chair and am ex- are fortunate indeed that, through Al cited about returning to my normal career Taylor’s commitment and balanced lead- of teaching and research. Alejandro ership through six of the last seven years, Uribe will assume the chairmanship next almost all of these developments have year, filling in for Trevor Wooley, who been for the good. As Al begins some has agreed to serve a 3-year term as Chair, well-deserved relief from administrative the 2002-2005 academic years, after he duties, it seems an opportune moment to returns from a year’s leave in Cambridge highlight some of the developments and and Bonn. -
Mathematics 2016
Mathematics 2016 press.princeton.edu Contents General Interest 1 New in Paperback 11 Biography 12 History & Philosophy of Science 13 Graduate & Undergraduate Textbooks 16 Mathematical Sciences 20 Princeton Series in Applied Mathematics 21 Annals of Mathematics Studies 22 Princeton Mathematical Series 24 Mathematical Notes 24 Cover image © Cameron Strathdee. Courtesy of iStock. Forthcoming Fashion, Faith, and Fantasy in the New Physics of the Universe Roger Penrose “This gem of a book is vintage Roger Penrose: eloquently argued and deeply original on every page. His perspective on the present crisis and future promise of physics and cosmology provides an important corrective to fashionable thinking at this crucial moment in science. This book deserves the widest possible hearing among specialists and the public alike.” —Lee Smolin, author of Time Reborn: From the Crisis in Physics to the Future of the Universe What can fashionable ideas, blind faith, or pure fantasy possibly have to do with the scienti c quest to understand the universe? Surely, theoreti- cal physicists are immune to mere trends, dogmatic beliefs, or ights of fancy? In fact, acclaimed physicist and bestselling author Roger Penrose argues that researchers working at the extreme frontiers of physics are just as susceptible to these forces as anyone else. In this provocative book, he argues that fashion, faith, and fantasy, while sometimes pro- ductive and even essential in physics, may be leading today’s researchers astray in three of the eld’s most important areas—string theory, quan- tum mechanics, and cosmology. July 2016. 424 pages. 211 line illus. Cl: 978-0-691-11979-3 $29.95 | £19.95 New The Princeton Companion to Applied Mathematics Edited by Nicholas J. -
Katherine E. Stange and Xin Zhang Compositio Mathematica, 155:6 (2019), 1118–1170
KATHERINEE.STANGE personal information email [email protected] website math.colorado.edu/∼kstange office Math. Bldg. 308 · +1 (303) 492 3346 address University of Colorado Boulder Department of Mathematics Campus Box 395 Boulder, CO, USA 80309-0395 research areas Algebraic and algorithmic number theory and arithmetic geometry, including Diophantine approximation, Kleinian groups, elliptic curves and abelian varieties, integer sequences, and cryptography, including elliptic curve, lattice-based, and post-quantum cryptography. education Doctor and Master 2001-2008 Brown University of Mathematics Ph.D. Dissertation: Elliptic nets and elliptic curves Advisor: Joseph H. Silverman Bachelor of 1997-2001 University of Waterloo Mathematics Pure Mathematics With Distinction, Dean’s Honours List history Current Position 2018-present The University of Colorado, Boulder Associate Professor 2012-2018 The University of Colorado, Boulder Assistant Professor Postdoctoral 2011-2012 Stanford University Experience NSF Postdoctoral Fellow Advisor: Brian Conrad 2009-2011 Simon Fraser University, Pacific Institute for the Mathematical Sciences, and the University of British Columbia NSERC/PIMS/NSF Postdoctoral Fellow Advisor: Nils Bruin 2008-2009 Harvard University NSF Postdoctoral Fellow and Junior Lecturer Advisor: Noam Elkies Graduate Fall 2007 Microsoft Research Experience Research Intern, Cryptograph Group Advisor: Kristin Lauter 2 Summer/Fall Volunteer Work 2005 Volunteer, English Teacher, School #27, Izhevsk, Russia Volunteer, Community Projects, -
Fermat's Last Theorem If X, Y, Z and N Are Integers Satisfying Xn + Yn = Zn, Then Either N ≤ 2 Or Xyz = 0
THEOREM OF THE DAY Fermat's Last Theorem If x, y, z and n are integers satisfying xn + yn = zn, then either n ≤ 2 or xyz = 0. It is easy to see that we can assume that all the integers in the theorem are positive. So a legitimate, but totally different, way of asserting the theorem, invented by the engineer Edward C. Molina, is the following: we take a ball at random from Urn A; then replace it and take a 2nd ball 5 × 5 at random. Do the same for Urn B. The probability that both A balls are blue, for the urns shown here, is 7 7 . The probability that both B 4 2 3 2 2 2 2 balls are the same colour (both blue or both red) is ( 7 ) + ( 7 ) . Now the Pythagorean triple 5 = 3 + 4 tells us that the probabilities are equal: 25 9 16 > 49 = 49 + 49 . What if we choose n 2 balls with replacement? Can we again fill each of the urns with N balls, red and blue, so that taking n with replacement will give equal probabilities? Fermat’s Last Theorem says: only in the trivial case where all the balls in Urn A are blue (which includes, vacuously, the possibility that N = 0). Another, much more profound restatement: if an + bn, for n > 2 and positive integers a and b, is again an n-th power of an integer then the elliptic curve y2 = x(x − an)(x + bn), known as the Frey curve, cannot be modular (is not a rational map of a modular curve). -
Sir Andrew J. Wiles
The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to Sir Andrew J. Wiles University of Oxford “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” Number theory, an old and beautiful branch of the upper half of the complex plane, and naturally factor mathematics, is concerned with the study of arithmetic through shapes known as modular curves. An elliptic properties of the integers. In its modern form the subject curve is said to be modular if it can be parametrized by is fundamentally connected to complex analysis, algebraic a map from one of these modular curves. The modularity geometry, and representation theory. Number theoretic conjecture, proposed by Goro Shimura, Yutaka Taniyama, results play an important role in our everyday lives through and André Weil in the 1950s and 60s, claims that every encryption algorithms for communications, financial elliptic curve defined over the rational numbers is modular. transactions, and digital security. In 1984, Gerhard Frey associated a semistable elliptic Fermat’s Last Theorem, first formulated by Pierre de curve to any hypothetical counterexample to Fermat’s Fermat in the 17th century, is the assertion that the Last Theorem, and strongly suspected that this elliptic equation xn + yn = zn has no solutions in positive integers curve would not be modular. Frey’s non-modularity was for n>2. Fermat proved his claim for n=4, Leonhard Euler proven via Jean-Pierre Serre’s epsilon conjecture by found a proof for n=3, and Sophie Germain proved the Kenneth Ribet in 1986. -
Summary of 2005 Research Activities
Summary of 2005 Research Activities The activities of CMI researchers and research programs are sketched below. Researchers and programs are selected by the Scientific Advisory Board (see inside back cover). Clay Research Fellows CMI researchers Bo’az Klartag graduated from Tel Aviv University. His three-year appointment began in September 2005. He is based at the Institute for Advanced Study in Princeton, New Jersey. David Speyer graduated from UC Berkeley and is working at the University of Michigan. He has a five-year appointment. Barry Mazur’s Clay Public Lecture “Are there unsolved problems about numbers?” Klartag and Speyer joined current research fellows March 31, 2005. MSRI program on Probability, Manjul Bhargava (Princeton University), Daniel Algorithms and Statistical Physics. Biss (University of Chicago), Alexei Borodin (Caltech), Maria Chudnovsky (Princeton), Sergei Simon Levin (Princeton), June 26–July 16, 2005. Gukov (Caltech), Elon Lindenstrauss (Princeton), PCMI program on Mathematical Biology. Ciprian Manolescu (Columbia University), Charles Peskin (Courant Institute, NYU), June 26– Maryam Mirzakhani (Princeton University), Ben July 16, 2005. PCMI program on Mathematical Green (MIT), András Vasy (Stanford) and Akshay Biology. Venkatesh (Princeton). David Morrison (Duke), August 1–December 16, Book Fellows 2005. KITP program on Mathematical Structures in String Theory. Appointed in 2005 were Steven Finch (Boston University), who worked on a second volume of Robert Bartnik (Canberra), August 8–December 23, “Mathematical Constants,” Grigory Mikhalkin 2005. Newton Institute program on Global Problems (Toronto & Utah), who is writing a book on “Tropical in Mathematical Relativity. Geometry and Amoebas,” and John Morgan (Columbia University) and Gang Tian (Princeton Nigel Hitchin (Oxford), September 1–30, 2005. -
Brian Conrad, Assistant Professor of Mathematics, College of Literature
SUMMARY OF PERSONNEL ACTIONS REGENTS AGENDA July 2020 ANN ARBOR CAMPUS – Recommendations for approval 1. New appointments and promotions for regular associate and full professor ranks, with tenure. (1) Das, Aileen, promotion to associate professor of classical studies, with tenure, and associate professor of Middle East studies, without tenure, College of Literature, Science, and the Arts, effective September 1, 2020 (currently assistant professor of classical studies, and assistant professor of Middle East studies). 2. Reappointments of regular instructional staff and selected academic and administrative staff. (1) Armstrong, William F., M.D., Franklin Davis Johnston Collegiate Professor of Cardiovascular Medicine, Medical School, effective September 1, 2020 through August 31, 2025 ( also professor of internal medicine, with tenure). (2) Bach, David S., M.D., Park W. Willis III Collegiate Professor of Cardiovascular Medicine, Medical School, effective September 1, 2020 through August 31, 2025 (also clinical professor, Department of Internal Medicine). (3) Bayraktar, Erhan, Susan Meredith Smith Professor of Actuarial Sciences, College of Literature, Science, and the Arts, effective September 1, 2020 through August 31, 2025 (also professor of mathematics, with tenure). (4) Behar, Ruth, Victor Haim Perera Collegiate Professor of Anthropology, College of Literature, Science, and the Arts, effective September 1, 2020 through August 31, 2025 (also professor of anthropology, with tenure). (5) Bloch, Anthony M., Alexander Ziwet Collegiate Professor of Mathematics, College of Literature, Science, and the Arts, effective September 1, 2020 through August 31, 2025 (also professor of mathematics, with tenure). (6) Bound, John, George E. Johnson Collegiate Professor of Economics, College of Literature, Science, and the Arts, effective September 1, 2020 through August 31, 2025 (also professor of economics, with tenure). -
On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises
On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Breuil, Christophe, Brian Conrad, Fred Diamond, and Richard L. Taylor. 2001. On the modularity of elliptic curves over Q: Wild 3- adic exercises. Journal of the American Mathematical Society 14(4): 843-939. Published Version doi:10.1090/S0894-0347-01-00370-8 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3626807 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume Numb er Xxxx XXXX Pages S XX ON THE MODULARITY OF ELLIPTIC CURVES OVER Q WILD ADIC EXERCISES CHRISTOPHE BREUIL BRIAN CONRAD FRED DIAMOND AND RICHARD TAYLOR Introduction In this pap er building on work of Wiles Wi and of Wiles and one of us RT TW we will prove the following two theorems see x Theorem A If E is an el liptic curve then E is modular Q Theorem B If GalQQ GL F is an irreducible continuous representation with cyclotomic determinant then is modular We will rst remind the reader of the content of these results and then briey outline the metho d of pro of Z If N is a p ositive integer then we let N denote the subgroup of SL consisting of matrices that mo dulo N are of the form The quotient -
Elliptic Curves, Modular Forms, and Their L-Functions
STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 58 Elliptic Curves, Modular Forms, and Their L-functions Álvaro Lozano-Robledo American Mathematical Society Institute for Advanced Study stml-58-lozano-cov.indd 1 1/5/11 1:17 PM http://dx.doi.org/10.1090/stml/058 Elliptic Curves, Modular Forms, and Their L-functions STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 58 Elliptic Curves, Modular Forms, and Their L-functions Álvaro Lozano-Robledo American Mathematical Society, Providence, Rhode Island Institute for Advanced Study, Princeton, New Jersey Editorial Board of the Student Mathematical Library Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell Series Editor for the Park City Mathematics Institute John Polking Cover art courtesy of Karl Rubin, using MegaPOV, which is based on POV-Ray, both of which are open source, freely available software. 2000 Mathematics Subject Classification. Primary 14H52, 11G05; Secondary 11F03, 11G40. For additional information and updates on this book, visit www.ams.org/bookpages/stml-58 Library of Congress Cataloging-in-Publication Data Lozano-Robledo, Alvaro,´ 1978– Elliptic curves, modular forms, and their L-functions / Alvaro´ Lozano-Robledo. p. cm. — (Student mathematical library ; v. 58. IAS/Park City mathemat- ical subseries) Includes bibliographical references and index. ISBN 978-0-8218-5242-2 (alk. paper) 1. Curves, Elliptic. 2. Forms, Modular. 3. L-functions. 4. Number theory. I. Title. QA567.2.E44L69 2010 516.352—dc22 2010038952 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. -
Algebra & Number Theory
Algebra & Number Theory Volume 4 2010 No. 4 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale