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Class Group Computations in Number Fields and Applications to Cryptology Alexandre Gélin
Class group computations in number fields and applications to cryptology Alexandre Gélin To cite this version: Alexandre Gélin. Class group computations in number fields and applications to cryptology. Data Structures and Algorithms [cs.DS]. Université Pierre et Marie Curie - Paris VI, 2017. English. NNT : 2017PA066398. tel-01696470v2 HAL Id: tel-01696470 https://tel.archives-ouvertes.fr/tel-01696470v2 Submitted on 29 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité Informatique École Doctorale Informatique, Télécommunications et Électronique (Paris) Présentée par Alexandre GÉLIN Pour obtenir le grade de DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE Calcul de Groupes de Classes d’un Corps de Nombres et Applications à la Cryptologie Thèse dirigée par Antoine JOUX et Arjen LENSTRA soutenue le vendredi 22 septembre 2017 après avis des rapporteurs : M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern devant le jury composé de : M. Karim BELABAS Professeur, Université de Bordeaux M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern M. -
SUMMER SCHOOL 2019 in the Later Half of This Course, We Will Mainly
LECTURE NOTES ON ANT (PART II)- SUMMER SCHOOL 2019 YE TIAN In the later half of this course, we will mainly study quadratic fields and cyclotomic fields. For quadratic fields, we will introduce Gauss' reduction theory of quadratic forms to study ideal class groups and unit group; and use Eisenstein series to give meromorphic continuation of Dedekind zeta functions and residue formula; and complex multiplication gives the explicit construction of class fields of imaginary quadratic fields. For cyclotomic field, we introduce Stickelberger' s theorem and Thaine's theorem, and the proof of Catalan's conjecture as an application. D × For a quadratic field K of fundamental discriminant D, let χD = · :(Z=DZ) −! {±1g be the associated quadratic character of conductor jDj. Sometime we also write ClD; hD; D for the ideal class group ClK , ideal class number hK , and fundamental units K (when D > 0). Contents 1. Lecture I: The equation y2 + 14 = x3 1 2. Lecture II-III: Primes of form p = x2 + ny2 and Class Field Theory4 3. Lecture IV: Archimedes' Cattle Prolem and Pell's equation8 4. Lecture V: Eisenstein series and Class Number Formula of Quadratic Fields 11 4.1. CNF of Imaginary Quadratic Fields 13 4.2. CNF for Real Quadratic Fields 13 5. Catalan's Conjecture 14 5.1. Cassels' result: elementary number theory 14 5.2. Selmer group and the element [x − ζ] 15 5.3. Stickelberger Theorem and Minus argument 17 5.4. Thaine's Theorem and Plus argument 18 References 19 1. Lecture I: The equation y2 + 14 = x3 We review what we have studied in the last two weeks via the following Diophantine equation 2 3 y + 14 = x ; with x; y 2 Z: p p Let K = Q( −14) and O = Z[ −14] its ring of integers. -
Units and Class Groups in Number Theory and Algebraic Geometry
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 3, May 1982 UNITS AND CLASS GROUPS IN NUMBER THEORY AND ALGEBRAIC GEOMETRY BY SERGE LANG1 CONTENTS 1. Introductory remarks: curves and Riemann surfaces 2. Cyclotomic Fields (a) The basic objects (b) Plus eigenspaces (c) A topological interlude (d) Minus eigenspaces (e) The cyclotomic tower 3. Modular curves 4. The Wiles and Mazur-Wiles connection 5. Geometric class field theory 6. Complex multiplication and modular units 7. The Coates-Wiles connection 8. Stark units 9. Higher regulators: number fields 10. Higher regulators: curves (da capo) §1. Introductory remarks Determining unramified coverings over various base spaces is a classical activity, which can take place in many contexts: topological, complex analytic, algebraic, and arithmetic. The abelian coverings are simpler to handle than the non-abelian ones, and in these lectures, we shall concentrate on abelian cases. Furthermore, the base space will have mostly dimension 1. It turns out that the study of the arithmetic case is inextricably inter twined with that of the other cases, in many ways. Thus even though I (for example) start being motivated by the arithmetic case, I come eventually to a consideration of the other cases. Presented as the Colloquium Lectures at the eighty-fifth summer meeting of the American Mathematical Society, in Pittsburgh, August 17-20, 1981; received by the editors July 8, 1981. 1980 Mathematics Subject Classification. Primary 12A35, 12A90; Secondary 14G25, 10D12. Supported by NSF Grant. © 1982 American Mathematical Society 0273-0979/81 /0000-0276/$ 17.00 253 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 254 SERGE LANG "Motivation", by the way, is a very relative term. -
Cyclotomic Fields with Applications
Cyclotomic Fields with applications G Eric Moorhouse CYCLOTOMIC FIELDS WITH APPLICATIONS Lecture Notes for Math 5590 Fall 2018 G. Eric Moorhouse University of Wyoming c 2018 ii Preface These notes were written during the summer of 2018, while planning a graduate course for the Fall 2018 semester. The theme was chosen to appeal to students of varying backgrounds, some with interest primarily in number theory, and others more interested in combinatorics, graph theory and finite geometry. As it happens, my research has led me into areas of overlap between these two areas, with cyclotomic fields arising as a common theme. And so beyond the immediate goal of appealing to students with multiple interests, this course was conceived also as a way of crystallizing in my mind some of the finer points of the theory of cyclotomic fields, many of which I had less familiarity with. Students were referred primarily to Washington's book [Wa] for further details on cyclotomic fields, and various other sources as needed. The realities of my teaching environment (perhaps yours too?) mean that I now apportion less lecture time on theory and proofs, with more on examples and applications, than when I first began teaching. In the grand tradition of mathematics, these applications arise largely in ::: (wait for it!) ::: other areas of mathematics. (That may not be strictly true; but as usual, our description of these applications has been rather simplified, sometimes oversimplified, to their mathematical essence, for the sake of brevity.) These applications include algorithms for fast arithmetic with polynomials and integers; constructions and nonexistence results for Hadamard matrices, difference sets, and designs, particularly nets finite affine and projective planes; spectra of Cayley graphs and digraphs over abelian groups; counting solutions to equations over finite fields; the MacWilliams relations for error-correcting codes; Dirichlet's theorem on primes on arithmetic progressions; and mutually unbiased bases (quantum information theory). -
THE ANALYTIC CLASS NUMBER FORMULA and L-FUNCTIONS 1. Overview 3 2. L-Functions: Analytic Properties 5 2.1. Analytic Continuation
THE ANALYTIC CLASS NUMBER FORMULA AND L-FUNCTIONS AKSHAY VENKATESH NOTES BY TONY FENG CONTENTS 1. Overview 3 2. L-functions: analytic properties 5 2.1. Analytic continuation 5 2.2. Epstein ζ functions 7 2.3. The Dedekind zeta function 10 2.4. Generalizations 14 3. L-functions: arithmetic properties 18 3.1. Rationality 18 3.2. p-adic continuation. 18 3.3. Abstract sequence spaces 19 3.4. Application to rationality 21 4. p-adic L-functions 25 4.1. Analyticity for real quadratic fields 25 4.2. Totally real fields 30 5. Hurwitz zeta functions 35 5.1. Interlude on analysis 35 5.2. Hurwitz zeta functions 36 5.3. Explicit evaluations 38 6. Artin L-functions 41 6.1. Motivation 41 6.2. Artin’s conjecture 42 6.3. The conductor-discriminant formula 44 6.4. Analytic properties 46 6.5. Positive characteristic speculation 49 7. Stark’s conjectures 51 7.1. The class number formula 51 7.2. Aside: how to compute L(s,ρ) 52 7.3. Stark’s conjectures 54 7.4. Compatibility with class number formula 58 7.5. Imaginary quadratic fields 61 8. Class numbers of cyclotomic fields 67 1 Math 263C 2015 8.1. Reformulating Stark’s conjecture 67 8.2. Stickelberger’s Theorem 69 8.3. Herbrand’s Theorem 76 9. Converse to Herbrand’s Theorem 79 9.1. Ribet’s proof 79 9.2. Cyclotomic units 80 9.3. Converse to Herbrand 88 9.4. Euler Systems 89 2 Math 263C 2015 1. OVERVIEW Let K be a number field. -
Bernoulli Numbers, Hurwitz Numbers, P-Adic L-Functions and Kummer's Criterion
RAC Rev. R. Acad.SAM Cien. Serie A. Mat. VOL. 101 (1), 2007, pp. 1–32 Algebra´ / Algebra Art´ıculo panoramico´ / Survey Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer’s criterion. Alvaro´ Lozano-Robledo Abstract. Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields. N ´umeros de Bernoulli, n ´umeros de Hurwitz, funciones p-adicas´ y criterios de Kummer. Resumen. Sea K = Q(ζp) y sea hp su numero´ de clases. Kummer demostro´ que p divide hp si y solo si p divide el numerador de ciertos numeros´ de Bernoulli. En este art´ıculo panoramico´ tratamos generalizaciones de este tipo de criterio para cuerpos totalmente reales y cuerpos cuadraticos´ imaginarios. Contents 1 Kummer’s criterion 2 1.1 Bernoulli numbers . 3 1.2 The proof of Kummer’s criterion . 5 1.2.1 Cyclotomic fields . 6 2 Rephrasing Kummer 7 2.1 Cyclotomic units . 8 2.2 Decompositions of the full unit group . 9 2.3 Herbrand’s theorem and a converse by Ribet . 10 3 Totally real extensions of Q 11 3.1 Serre’s p-adic L-function for totally real fields . 12 4 The quadratic imaginary case 13 4.1 Hurwitz numbers . 13 4.2 Robert’s criterion . 15 4.3 Elliptic units . 16 4.3.1 Kronecker limit formulas . 17 4.4 The proof of Robert’s criterion . -
Class Numbers of Real Cyclotomic Fields of Prime Conductor
MATHEMATICS OF COMPUTATION Volume 72, Number 242, Pages 913{937 S 0025-5718(02)01432-1 Article electronically published on February 15, 2002 CLASS NUMBERS OF REAL CYCLOTOMIC FIELDS OF PRIME CONDUCTOR RENE´ SCHOOF Abstract. + 1 The class numbers hl of the real cyclotomic fields Q(ζl + ζl− ) + are notoriously hard to compute. Indeed, the number hl is not known for a single prime l 71. In this paper we present a table of the orders of certain ≥ 1 subgroups of the class groups of the real cyclotomic fields Q(ζl + ζl− )forthe primes l<10; 000. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue |on the basis of the Cohen-Lenstra heuristics| that the probability that our table is actually a + table of class numbers hl , is at least 98%. Introduction Let l>2 be a prime number and let ζl denote a primitive l-th root of unity. The ideal class group Cll of the ring of integers of the cyclotomic field Q(ζl)isa finite abelian group of order hl,theclass number of Q(ζl). The group Cll naturally splits into two parts; there is a natural exact sequence + 0 Cl Cll Cl− 0; −→ l −→ −→ l −→ + where Cll denotes the class group of the ring of integers of the real cyclotomic 1 1 + field Q(ζl + ζl− ). Its order, the class number of Q(ζl + ζl− ), is denoted by hl . -
Euler Systems and Kolyvagin Systems Karl Rubin 1
Contents Euler systems and Kolyvagin systems Karl Rubin 1 Euler systems and Kolyvagin systems 3 Lecture 1. Galois cohomology 5 1.1. G-modules. 5 1.2. Characterization of the cohomology groups. 5 1.3. Continuous cohomology. 7 1.4. Change of group. 7 1.5. Selmer groups. 9 1.6. Kummer theory. 10 1.7. The Brauer group. 12 1.8. Local fields and duality. 12 1.9. Unramified and transverse cohomology groups. 13 1.10. Comparing Selmer groups. 15 Lecture 2. Kolyvagin systems 17 2.1. Varying the Selmer structure. 17 2.2. Kolyvagin systems. 18 2.3. Sheaves and monodromy. 19 2.4. Hypotheses on A and F. 20 2.5. The core Selmer group. 21 2.6. Stepping along the Selmer graph. 22 2.7. Choosing useful primes. 23 2.8. The stub subsheaf. 24 2.9. Recovering the group structure of the dual Selmer group. 26 Lecture 3. Kolyvagin systems for p-adic representations. 29 3.1. Cohomology groups for p-adic representations. 29 3.2. Kolyvagin systems for A. 31 3.3. Example: units and ideal class groups. 32 3.4. Example: Tate modules of elliptic curves. 37 Lecture 4. Euler systems 39 4.1. Definition of an Euler system. 39 4.2. Example: the cyclotomic unit Euler system 39 4.3. Euler systems and Kolyvagin systems. 41 Lecture 5. Applications 45 5.1. Cyclotomic units and ideal class groups. 45 i ii LECTURE 0. CONTENTS 5.2. Elliptic curves. 45 5.3. General speculation. 48 Bibliography 51 Euler systems and Kolyvagin systems Karl Rubin IAS/Park City Mathematics Series Volume XX, XXXX Euler systems and Kolyvagin systems Karl Rubin Department of Mathematics, UC Irvine, Irvine CA 92617 USA E-mail address: [email protected] c 2009 American Mathematical Society 3 LECTURE 1 Galois cohomology We begin with a very quick and selective introduction to the facts from group cohomology and Galois cohomology that will be needed for the following lectures.