Kummer Surfaces: 200 Years of Study
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La Controverse De 1874 Entre Camille Jordan Et Leopold Kronecker
La controverse de 1874 entre Camille Jordan et Leopold Kronecker. * Frédéric Brechenmacher ( ). Résumé. Une vive querelle oppose en 1874 Camille Jordan et Leopold Kronecker sur l’organisation de la théorie des formes bilinéaires, considérée comme permettant un traitement « général » et « homogène » de nombreuses questions développées dans des cadres théoriques variés au XIXe siècle et dont le problème principal est reconnu comme susceptible d’être résolu par deux théorèmes énoncés indépendamment par Jordan et Weierstrass. Cette controverse, suscitée par la rencontre de deux théorèmes que nous considèrerions aujourd’hui équivalents, nous permettra de questionner l’identité algébrique de pratiques polynomiales de manipulations de « formes » mises en œuvre sur une période antérieure aux approches structurelles de l’algèbre linéaire qui donneront à ces pratiques l’identité de méthodes de caractérisation des classes de similitudes de matrices. Nous montrerons que les pratiques de réductions canoniques et de calculs d’invariants opposées par Jordan et Kronecker manifestent des identités multiples indissociables d’un contexte social daté et qui dévoilent des savoirs tacites, des modes de pensées locaux mais aussi, au travers de regards portés sur une histoire à long terme impliquant des travaux d’auteurs comme Lagrange, Laplace, Cauchy ou Hermite, deux philosophies internes sur la signification de la généralité indissociables d’idéaux disciplinaires opposant algèbre et arithmétique. En questionnant les identités culturelles de telles pratiques cet article vise à enrichir l’histoire de l’algèbre linéaire, souvent abordée dans le cadre de problématiques liées à l’émergence de structures et par l’intermédiaire de l’histoire d’une théorie, d’une notion ou d’un mode de raisonnement. -
Kummer, Regular Primes, and Fermat's Last Theorem
e H a r v a e Harvard College r d C o l l e Mathematics Review g e M a t h e m Vol. 2, No. 2 Fall 2008 a t i c s In this issue: R e v i ALLAN M. FELDMAN and ROBERTO SERRANO e w , Arrow’s Impossibility eorem: Two Simple Single-Profile Versions V o l . 2 YUFEI ZHAO , N Young Tableaux and the Representations of the o . Symmetric Group 2 KEITH CONRAD e Congruent Number Problem A Student Publication of Harvard College Website. Further information about The HCMR can be Sponsorship. Sponsoring The HCMR supports the un- found online at the journal’s website, dergraduate mathematics community and provides valuable high-level education to undergraduates in the field. Sponsors http://www.thehcmr.org/ (1) will be listed in the print edition of The HCMR and on a spe- cial page on the The HCMR’s website, (1). Sponsorship is available at the following levels: Instructions for Authors. All submissions should in- clude the name(s) of the author(s), institutional affiliations (if Sponsor $0 - $99 any), and both postal and e-mail addresses at which the cor- Fellow $100 - $249 responding author may be reached. General questions should Friend $250 - $499 be addressed to Editors-In-Chief Zachary Abel and Ernest E. Contributor $500 - $1,999 Fontes at [email protected]. Donor $2,000 - $4,999 Patron $5,000 - $9,999 Articles. The Harvard College Mathematics Review invites Benefactor $10,000 + the submission of quality expository articles from undergrad- uate students. -
Methodology and Metaphysics in the Development of Dedekind's Theory
Methodology and metaphysics in the development of Dedekind’s theory of ideals Jeremy Avigad 1 Introduction Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical fac- tions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the oppos- ite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. -
The History of the Formulation of Ideal Theory
The History of the Formulation of Ideal Theory Reeve Garrett November 28, 2017 1 Using complex numbers to solve Diophantine equations From the time of Diophantus (3rd century AD) to the present, the topic of Diophantine equations (that is, polynomial equations in 2 or more variables in which only integer solutions are sought after and studied) has been considered enormously important to the progress of mathematics. In fact, in the year 1900, David Hilbert designated the construction of an algorithm to determine the existence of integer solutions to a general Diophantine equation as one of his \Millenium Problems"; in 1970, the combined work (spanning 21 years) of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson showed that no such algorithm exists. One such equation that proved to be of interest to mathematicians for centuries was the \Bachet equa- tion": x2 +k = y3, named after the 17th century mathematician who studied it. The general solution (for all values of k) eluded mathematicians until 1968, when Alan Baker presented the framework for constructing a general solution. However, before this full solution, Euler made some headway with some specific examples in the 18th century, specifically by the utilization of complex numbers. Example 1.1 Consider the equation x2 + 2 = y3. (5; 3) and (−5; 3) are easy to find solutions, but it's 2 not obviousp whetherp or not there are others or whatp they might be. Euler realized by factoring x + 2 as (x + 2i)(x − 2i) and then using the facts that Z[ 2i] is a UFD andp the factorsp given are relatively prime that the solutions above are the only solutions,p namelyp because (x + 2i)(x − 2i) being a cube forces each of these factors to be a cube (i.e. -
Point Sets in Projective Spaces and Theta Functions
Astérisque IGOR DOLGACHEV DAVID ORTLAND Point sets in projective spaces and theta functions Astérisque, tome 165 (1988) <http://www.numdam.org/item?id=AST_1988__165__1_0> © Société mathématique de France, 1988, 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/ 165 ASTÉRISQUE 1988 POINT SETS IN PROJECTIVE SPACES AND THETA FUNCTIONS Igor DOLGACHEV and David ORTLAND SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE A.M.S. Subjects Classification : 14-02, 14C05, 14C21, 14D25, 14E07, 14H10, 14H40, 14H45, 14J26, 14J25, 14J30, 14K25, 14N99, 14J50, 20C30, 17B20, 17B67. TO HILARY AND NATASHA Table of Contents Introduction 3 I. CROSS-RATIO FUNCTIONS 1. The variety Pmn (first definition) 7 2. Standard monomials .9 3. Examples 13 II. GEOMETRIC INVARIANT THEORY 1. Pmn (second definition) 21 2. A criterion of semi-stability.... 22 3. Most special point sets 2.6 4. Examples 31 III. ASSOCIATED POINT SETS 1. The association 33 2. Geometric properties of associated point sets 41 3. Self-associated point sets .43 IV. BLOWING-UPS OF POINT SETS 1. Infinitely near point sets 53 2. Analysis of stability in m .57 V. GENERALIZED DEL PEZZO VARIETIES 1. The Neron-Severi bilattice 63 2. -
Kummer Surfaces: 200 Years of Study
Kummer Surfaces: 200 Years of Study Igor Dolgachev The fascinating story about the Kummer surface starts from material where double refraction occurs: a ray of light the discovery by Augustin-Jean Fresnel in 1822 of the equa- splits into two, traveling at the same speed along different tion describing the propagation of light in an optically paths. The speed of light may depend on the coordinates biaxial crystal [Fre]. Biaxial crystals are an example of 푥 = (푥1, 푥2, 푥3) of a point and the unit direction vector 휉 = (휉1, 휉2, 휉3). The propagation of light is described by Igor Dolgachev is a professor of mathematics, emeritus, at the University of the speed 푣(푥, 휉) at 푥 in the direction 휉. We say that the Michigan. His email address is [email protected]. matter is homogeneous if 푣 does not depend on 푥 and we The article is based on the author’s Oliver Club talk at Cornell University de- livered on October 10, 2019, exactly 121 years since the first meeting of the say that it is isotropic if it does not depend on 휉. For exam- club, at which then the faculty member John Hutchinson spoke. As we will see, ple, while a student, James Maxwell described a lens that Hutchinson contributed significantly to the theory of Kummer surfaces. reminded him of the eyes of a fish. Through his fish eye, Communicated by Notices Associate Editor Angela Gibney. he found that light is inhomogeneous but isotropic, bend- For permission to reprint this article, please contact: ing in arcs whose shape depends on where they start and [email protected]. -
L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field. -
Geometric Aspects on Humbert-Edge's Curves of Type 5
GEOMETRIC ASPECTS ON HUMBERT-EDGE’S CURVES OF TYPE 5, KUMMER SURFACES AND HYPERELLIPTIC CURVES OF GENUS 2 ABEL CASTORENA AND JUAN BOSCO FR´IAS-MEDINA Abstract. In this work we study the Humbert-Edge’s curves of type 5, defined as a complete intersection of four diagonal quadrics in P5. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert- Edge’s curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of n−1 hyperelliptic curves of genus g = 2 and the moduli space of Humbert-Edge’s curves of type n ≥ 5 where n is an odd number. Keywords: Intersection of quadrics; Curves with automorphisms, Jacobian variety. 1. Introduction W.L. Edge began the study of Humbert’s curves in [5], such curves are defined as canonical curves in P4 that are complete intersection of three diagonal quadrics. A natural generalization of Humbert’s curve was later introduced by Edge in [7]: irreducible, non-degenerated and non-singular curves on Pn that are the complete intersection of n − 1 diagonal quadrics. One important feature of Humbert-Edge’s curves of type n noted by Edge in [7] is that each one can be write down in a normal form. Indeed, we can assume that the n − 1 quadrics in Pn are given by n i 2 Qi = ajxj , i =0,...,n − 1 j=0 X where aj ∈ C for all j ∈ {0,...,n} and aj 6= ak if j 6= k. -
Dedekind's 1871 Version of the Theory of Ideals∗
Dedekind's 1871 version of the theory of ideals¤ Translated by Jeremy Avigad March 19, 2004 Translator's introduction By the middle of the nineteenth century, it had become clear to mathe- maticians that the study of ¯nite ¯eld extensions of the rational numbers is indispensable to number theory, even if one's ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the \integers" in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the or- dinary integers. In 1844, Ernst Kummer observed that unique factorization fails for the cyclotomic integers with exponent 23, i.e. the ring Z[³] of inte- gers of the ¯eld Q(³), where ³ is a primitive twenty-third root of unity. In 1847, he published his theory of \ideal divisors" for cyclotomic integers with prime exponent. This was to remedy the situation by introducing, for each such ring of integers, an enlarged domain of divisors, and showing that each integer factors uniquely as a product of these. He did not actually construct these integers, but, rather, showed how one could characterize their behav- ior qua divisibility in terms of ordinary operations on the associated ring of integers. Richard Dedekind and Leopold Kronecker later took up the task of ex- tending the theory to the integers in arbitrary ¯nite extensions of the ratio- nals. Despite their common influences and goals, however, the theories they ¤Work on this translation has been supported by a New Directions fellowship from the Andrew W. -
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. -
A History of Stickelberger's Theorem
A History of Stickelberger’s Theorem A Senior Honors Thesis Presented in Partial Fulfillment of the Requirements for graduation with research distinction in Mathematics in the undergraduate colleges of The Ohio State University by Robert Denomme The Ohio State University June 8, 2009 Project Advisor: Professor Warren Sinnott, Department of Mathematics 1 Contents Introduction 2 Acknowledgements 4 1. Gauss’s Cyclotomy and Quadratic Reciprocity 4 1.1. Solution of the General Equation 4 1.2. Proof of Quadratic Reciprocity 8 2. Jacobi’s Congruence and Cubic Reciprocity 11 2.1. Jacobi Sums 11 2.2. Proof of Cubic Reciprocity 16 3. Kummer’s Unique Factorization and Eisenstein Reciprocity 19 3.1. Ideal Numbers 19 3.2. Proof of Eisenstein Reciprocity 24 4. Stickelberger’s Theorem on Ideal Class Annihilators 28 4.1. Stickelberger’s Theorem 28 5. Iwasawa’s Theory and The Brumer-Stark Conjecture 39 5.1. The Stickelberger Ideal 39 5.2. Catalan’s Conjecture 40 5.3. Brumer-Stark Conjecture 41 6. Conclusions 42 References 42 2 Introduction The late Professor Arnold Ross was well known for his challenge to young students, “Think deeply of simple things.” This attitude applies to no story better than the one on which we are about to embark. This is the century long story of the generalizations of a single idea which first occurred to the 19 year old prodigy, Gauss, and which he was able to write down in no less than 4 pages. The questions that the young genius raised by offering the idea in those 4 pages, however, would torment the greatest minds in all the of the 19th century. -
A Letter of Hermann Amandus Schwarz on Isoperimetric Problems
Research Collection Journal Article A Letter of Hermann Amandus Schwarz on Isoperimetric Problems Author(s): Stammbach, Urs Publication Date: 2012-03 Permanent Link: https://doi.org/10.3929/ethz-b-000049843 Originally published in: The Mathematical Intelligencer 34(1), http://doi.org/10.1007/s00283-011-9267-7 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Years Ago David E. Rowe, Editor n 1995 (or thereabouts), Eli Maor, then at Loyola A Letter of Hermann University, discovered a little booklet at a used book fair IIin Chicago containing a reprint of a paper published by Hermann Amandus Schwarz in 1884 [Sch1]. Interestingly, it Amandus Schwarz also contained an original letter by Schwarz written in the old German script. The letter, dated January 28, 1884, began with the words: Hochgeehrter Herr Director! Maor could decipher on Isoperimetric enough of the text to realize that it dealt with isoperimetric problems, which further awakened his interest. With the help Problems of Reny Montandon and Herbert Hunziker, he contacted Gu¨nther Frei, who was able to read and transcribe the old URS STAMMBACH German script. Thanks to his detailed knowledge of the history of mathematics of the 19th century, Frei was also able to provide a number of pertinent remarks. Unfortunately, before completing this task, Gu¨nther Frei fell gravely ill. The text of the letter together with his remarks was then given to Years Ago features essays by historians and the present author, who agreed to continue the work.