DOCSLIB.ORG

Irene I. Bouw · Ekin Ozman Jennifer Johnson-Leung Rachel Newton Editors Contributions to Number Theory and Arithmetic Geomet

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Irene I. Bouw · Ekin Ozman Jennifer Johnson-Leung Rachel Newton Editors Contributions to Number Theory and Arithmetic Geomet Association for Women in Mathematics Series Irene I. Bouw · Ekin Ozman Jennifer Johnson-Leung Rachel Newton Editors Women in Numbers Europe II Contributions to Number Theory and Arithmetic Geometry Association for Women in Mathematics Series Volume 11 Series Editor Kristin Lauter Microsoft Research Redmond, Washington, USA Association for Women in Mathematics Series Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer’s Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics. Since its inception in 1971, The Association for Women in Mathematics has been a non-profit organization designed to help encourage women and girls to study and pursue active careers in mathematics and the mathematical sciences and to promote equal opportunity and equal treatment of women and girls in the mathematical sciences. Currently, the organization represents more than 3000 members and 200 institutions constituting a broad spectrum of the mathematical community, in the United States and around the world. More information about this series at http://www.springer.com/series/13764 Irene I. Bouw • Ekin Ozman Jennifer Johnson-Leung • Rachel Newton Editors Women in Numbers Europe II Contributions to Number Theory and Arithmetic Geometry 123 Editors Irene I. Bouw Ekin Ozman Institut für Reine Mathematik Department of Mathematics Universität Ulm Bogazici University Ulm, Baden-Württemberg, Germany Istanbul, Istanbul, Turkey Jennifer Johnson-Leung Rachel Newton Department of Mathematics Department of Mathematics and Statistics University of Idaho University of Reading Moscow, ID, USA Reading, UK ISSN 2364-5733 ISSN 2364-5741 (electronic) Association for Women in Mathematics Series ISBN 978-3-319-74997-6 ISBN 978-3-319-74998-3 (eBook) https://doi.org/10.1007/978-3-319-74998-3 Library of Congress Control Number: 2015946869 © The Author(s) and the Association for Women in Mathematics 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Women in Numbers Europe II, Lorentz Center, September 2016 Preface This volume grew out of the second Women in Numbers Europe conference held at the Lorentz Center, Leiden, from 26th to 30th September 2016. This conference was one of an ongoing series of collaboration conferences for women in mathematics, which began with the Women in Numbers conference held at the Banff International Research Station in 2008. This volume includes research reports from projects that were started at the conference, expository papers describing ongoing research begun at the conference, and contributed papers from other women number theorists. All of the papers included were reviewed by anonymous referees who verified them as worthy and valid contributions to these proceedings. We are proud that this volume continues the tradition of supporting and highlighting the contributions of women number theorists at a variety of career stages from graduate students to leading experts in the field. The articles collected here span a wide range of topics in contemporary number theory and arithmetic geometry such as arithmetic dynamics, failure of local-global principles, geometry in positive characteristic, and heights of algebraic integers. The use of tools from algebra, analysis, and geometry, as well as computational methods, exemplifies the wealth of techniques available to modern researchers in number theory. Several papers in this volume stem from collaborations between authors with different mathematical backgrounds and expertise. These papers naturally explore connections between different branches of mathematics and combine different points of view. Another common theme is the provision of explicit examples to complement new theorems and to provide evidence for conjectures. These concrete examples provide an easily accessible introduction to the field and have the potential to inspire future work. Ulm, Germany Irene I. Bouw Moscow, ID, USA Jennifer Johnson-Leung Reading, UK Rachel Newton Istanbul, Turkey Ekin Ozman vii Acknowledgement We would like to thank the following organizations for their generous support of the conference and this resulting volume: Clay Mathematics Institute, Foundation Compositio Mathematica, Heilbronn Institute for Mathematical Research, National Science Foundation as part of the Advance grant for the Association for Women in Mathematics, Microsoft Research, and the Number Theory Foundation. We are particularly thankful for the additional financial support we received when we were forced to relocate our workshop, originally planned to take place in the Nesin Mathematics Village, due to instability in Turkey. The Lorentz Center was very generous in accommodating our conference at short notice and provided us with extremely valuable organizational support. We also would like to extend our sincere gratitude to the Nesin Mathematics Village for their understanding when we needed to relocate at the very last minute. We are deeply grateful to the referees for their careful and timely review of the papers included in this volume. ix Contents Lower Bounds for Heights in Relative Galois Extensions ................... 1 Shabnam Akhtari, Kevser Akta¸s, Kirsti D. Biggs, Alia Hamieh, Kathleen Petersen, and Lola Thompson Reductions of Algebraic Integers II ............................................ 19 Antonella Perucca Reductions of One-Dimensional Tori II ....................................... 35 Antonella Perucca On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments ................................................... 39 Nurdagül Anbar, Almasa Odžak, Vandita Patel, Luciane Quoos, Anna Somoza, and Alev Topuzoglu˘ Dynamical Belyi Maps ........................................................... 57 Jacqueline Anderson, Irene I. Bouw, Ozlem Ejder, Neslihan Girgin, Valentijn Karemaker, and Michelle Manes Discriminant Twins .............................................................. 83 Alyson Deines The a-Number of Hyperelliptic Curves ........................................ 107 Sarah Frei Non-ordinary Curves with a Prym Variety of Low p-Rank ................. 117 Turku Ozlum Celik, Yara Elias, Burçin Güne¸s, Rachel Newton, Ekin Ozman, Rachel Pries, and Lara Thomas xi xii Contents Elliptic Fibrations on Covers of the Elliptic Modular Surface of Level 5 ............................................................................. 159 Francesca Balestrieri, Julie Desjardins, Alice Garbagnati, Céline Maistret, Cecília Salgado, and Isabel Vogt On Birch and Swinnerton-Dyer’s Cubic Surfaces ............................ 199 Mckenzie West Lower Bounds for Heights in Relative Galois Extensions Shabnam Akhtari, Kevser Akta¸s, Kirsti D. Biggs, Alia Hamieh, Kathleen Petersen, and Lola Thompson Abstract The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem, we obtain an effective bound for the height of an algebraic number α when the base field K is a number field and K(α)/K is Galois. Our second result establishes an explicit height bound for any nonzero element α which is not a root of unity in a Galois extension F/K, depending on the degree of K/Q and the number of conjugates of α which are multiplicatively independent over K. As a consequence, we obtain a height bound for such α that is independent of the multiplicative independence condition. S. Akhtari () Department of Mathematics, University of Oregon, Eugene, OR, USA e-mail: [email protected] K. Akta¸s Department of Mathematics Education, Gazi University, Ankara, Turkey e-mail: [email protected] K. D. Biggs School of Mathematics, University of Bristol, Clifton, Bristol, UK e-mail: [email protected] A. Hamieh Department of Mathematics and Statistics, The University of Northern British Columbia, Prince George, BC, Canada e-mail:
Load more

Recommended publications
  • Canonical Heights and Division Polynomials 11
    CANONICAL HEIGHTS AND DIVISION POLYNOMIALS ROBIN DE JONG AND J. STEFFEN MULLER¨ Abstract. We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factori- sation at all. The method is based on a recurrence relation for the `division polynomials' associated to hyperelliptic jacobians, and a diophantine approx- imation result due to Faltings. 1. Introduction In [EW] G. Everest and T. Ward show how to approximate to high precision the canonical height of an algebraic point on an elliptic curve E over a number field K with a limit formula using the (recurrence) sequence of division polynomials φn associated to E, and a diophantine approximation result. The φn have natural analogues for jacobians of hyperelliptic curves. In [Uc2] Y. Uchida shows how to obtain recurrence relations for the φn for hyperelliptic jaco- bians of dimension g ≥ 2. Further there exists a suitable analogue of the diophantine approximation result employed by Everest and Ward, proved by G. Faltings. In this paper we derive a limit formula for the canonical height of an algebraic point on a hyperelliptic jacobian from these inputs. We have implemented the resulting method for computing canonical heights in Magma for g = 2. The method does not require geometrical models, neither p-adic nor complex analytic ones. If the curve is defined over Q and the coordinates of the point are integral, then it also requires no factorisation.
    [Show full text]
  • Lectures on Modular Forms 1St Edition Free Download
    FREE LECTURES ON MODULAR FORMS 1ST EDITION PDF Joseph J Lehner | 9780486821405 | | | | | Lectures on Modular Forms - Robert C. Gunning - Google книги In mathematicsa modular form is a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular groupand also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topologysphere packingand string theory. Instead, modular functions are meromorphic that is, they are almost holomorphic except for a set of isolated points. Modular form theory is a special case of the more general theory of automorphic formsand therefore can now be seen as just the most concrete part of a rich theory of discrete groups. Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. The dimensions of these spaces of modular forms can be computed using the Riemann—Roch theorem [2]. A Lectures on Modular Forms 1st edition form of weight k for the modular group. A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions:. The simplest examples from this point of view are the Eisenstein series. Then E k is a modular form of weight k. An even unimodular lattice L in R n is a lattice generated by n Lectures on Modular Forms 1st edition forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer.
    [Show full text]
  • Arithmetical Problems in Number Fields, Abelian Varieties and Modular Forms*
    CONTRIBUTIONS to SCIENCE, 1 (2): 125-145 (1999) Institut d’Estudis Catalans, Barcelona Arithmetical problems in number fields, abelian varieties and modular forms* P. Bayer** et al. Abstract Resum Number theory, a fascinating area in mathematics and one La teoria de nombres, una àrea de la matemàtica fascinant i of the oldest, has experienced spectacular progress in re- de les més antigues, ha experimentat un progrés especta- cent years. The development of a deep theoretical back- cular durant els darrers anys. El desenvolupament d’una ground and the implementation of algorithms have led to base teòrica profunda i la implementació d’algoritmes han new and interesting interrelations with mathematics in gen- conduït a noves interrelacions matemàtiques interessants eral which have paved the way for the emergence of major que han fet palesos teoremes importants en aquesta àrea. theorems in the area. Aquest informe resumeix les contribucions a la teoria de This report summarizes the contribution to number theory nombres dutes a terme per les persones del Seminari de made by the members of the Seminari de Teoria de Teoria de Nombres (UB-UAB-UPC) de Barcelona. Els seus Nombres (UB-UAB-UPC) in Barcelona. These results are resultats són citats en connexió amb l’estat actual d’alguns presented in connection with the state of certain arithmetical problemes aritmètics, de manera que aquesta monografia problems, and so this monograph seeks to provide readers cerca proporcionar al públic lector una ullada sobre algunes with a glimpse of some specific lines of current mathemati- línies específiques de la recerca matemàtica actual. cal research.
    [Show full text]
  • ON COMPUTING BELYI MAPS J. Sijsling & J. Voight
    ON COMPUTING BELYI MAPS J. Sijsling & J. Voight Abstract.— We survey methods to compute three-point branched covers of the projective line, also known as Bely˘ımaps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples. Résumé.— Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtements de la droite projective ramifiés en au plus trois points, connus sous le nom de morphismes de Bely˘ı.Ces méthodes comprennent une approche directe, se ramenant à la solution d’un système d’équations polynomiales ainsi que des méthodes analytiques complexes, de formes modulaires et p-adiques. Ce faisant, nous posons quelques questions et donnons de nombreux exemples. Contents Introduction. 6 1. Background and applications. 7 2. Gröbner techniques. 14 3. Complex analytic methods. 22 4. Modular forms. 28 5. p-adic methods. 38 6. Galois Bely˘ımaps. 42 7. Field of moduli and field of definition. 44 8. Simplification and verification. 50 9. Further topics and generalizations. 52 References. 55 2010 Mathematics Subject Classification.— 11G32, 11Y40. Key words and phrases.— Belyi maps, dessins d’enfants, covers, uniformization, computational algebra. 6 On computing Belyi maps Introduction Every compact Riemann surface X is an algebraic curve over C, and every meromorphic function on X is an algebraic function. This remarkable fact, generalized in the GAGA principle, links the analytic with the algebraic in a fundamental way. A natural problem is then to link this further with arithmetic; to characterize those Riemann surfaces that can be defined by equations over Q and to study the action of the absolute Galois group Gal(Q=Q) on these algebraic curves.
    [Show full text]
  • Elliptic Curve Handbook
    Elliptic Curve Handbook Ian Connell February, 1999 http://www.math.mcgill.ca/connell/ Foreword The first version of this handbook was a set of notes of about 100 pages handed out to the class of an introductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then I have added to the notes, holding to the principle: If I look up a certain topic a year from now I want all the details right at hand, not in an “exercise”, so if I’ve forgotten something I won’t waste time. Thus there is much that an ordinary text would either condense, or relegate to an exercise. But at the same time I have maintained a solid mathematical style with the thought of sharing the handbook. Montreal, August, 1996. 1 Contents 1 Introduction to Elliptic Curves 101 1.1 The a,b,c’s and ∆,j ...................... 101 1.2 Quartic to Weierstrass ..................... 105 1.3 Projective coordinates. ..................... 111 1.4 Cubic to Weierstrass: Nagell’s algorithm ........... 115 1.4.1 Example 1: Selmer curves ............... 117 1.4.2 Example 2: Desboves curves .............. 121 1.4.3 Example 3: Intersection of quadric surfaces ..... 123 1.5 Singular points. ......................... 125 − 1.5.1 Example: No E/Z has ∆ = 1 or 1 .......... 130 1.6 Affine coordinate ring, function field, generic points ..... 132 1.7 The group law: nonsingular case ............... 133 1.7.1 Halving points ..................... 140 1.7.2 The division polynomials ............... 145 1.7.3 Remarks on the group of division points ....... 151 1.8 The group law: singular case ................
    [Show full text]
  • Elliptic Curves in the Modern Age
    Elliptic curves in the modern age (3rd millennium) Jens Berlips Elliptic curves in the modern age (3rd millennium) Jens Berlips August 21, 2006 Copyright c 2006 Jens Berlips. draft date: August, 2006. Typeset by the author with the LATEX 2" Documentation System. All rights reserved. No part of this work may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission. To my grandmother may she rest in peace CONTENTS 7 Contents 1 Introduction 10 2 Credits 12 3 Preliminiaries 13 3.1 Ane space ............................ 13 3.2 Projective space .......................... 14 3.3 From projective to Ane space and vice verse ......... 15 3.4 Notation .............................. 15 3.5 Cubic plane curve ......................... 16 3.5.1 Intersection of curves ................... 17 4 Elliptic Curves 18 4.1 Addition on E(k) ......................... 20 4.2 General theory .......................... 24 4.2.1 Order ........................... 24 4.2.2 Torsion points ....................... 25 4.2.3 Division polynomials ................... 26 5 Practical computational considerations 30 5.1 Binary ladder ........................... 30 6 Dierent parametrization 31 6.1 Ane coordinates ......................... 31 6.2 Projective coordinates ...................... 32 6.3 Montgomery coordinates ..................... 33 7 Finding the order 35 7.1 Schoof's method ......................... 36 7.1.1 The Frobenius endomorphism .............. 36 7.1.2 Division polynomials and Schoof's method ....... 37 7.1.3 Schoof's method explained ................ 37 8 CONTENTS 8 Factorization 40 8.1 Factorization methods ...................... 41 8.2 Pollard p ¡ 1 ........................... 41 8.3 Smooth numbers ......................... 43 8.4 Ideas of factorization ....................... 43 8.5 The general method ......................
    [Show full text]
  • Appendix a Packages for Number Theory
    Appendix A Packages for Number Theory There exist several computer packages which can profitably be used for number-theoretic computations. In this appendix, I will briefly describe the advantages and disadvantages of some of these systems. The oldest general-purpose symbolic algebra packages were written pri­ marily for physicists, and are not well suited for number theory. Furthermore (even for what they were designed for) they are very slow. On the other hand, they have been extensively tested, are have probably less bugs than more re­ cent systems. The two main systems in this category are Macsyma and Reduce. Reduce was extended by G. Collins to handle more efficiently computations linked to polynomials, and this gave rise to Sac-II, with a user interface called AIdes. These were the packages used in the 1970's. A second generation of packages was developed in the 1980's. They are much faster, better adapted to number theory, but having had less use, have more bugs. The two main commercial systems are Mathematica, by Wolfram Research, Inc., and Maple, by the University of Waterloo, Canada. Mathematica has a very nice user interface, and its plotting capabilities, for example on the Macintosh, are su­ perb. Maple is faster and often simpler to use, and has my preference. A third system is widely available, although not commercially released, the Scratch­ pad II system developed by IBM. This is a monster (in the same sense that ADA is a monster as a programming language). A subset of this system has recently been commercially released under the name Axiom.
    [Show full text]
  • Algebraic Curves an Elementary Introduction
    Algebraic Curves An Elementary Introduction Abhijit Das Department of Computer Science and Engineering Indian Institute of Technology Kharagpur August 22, 2011 Part I Affine and Projective Curves Rational Points on Curves Polynomial and Rational Functions on Curves Divisors and Jacobians on Curves Affine Curves K is a field. K is the algebraic closure of K. It is often necessary to assume that K is algebraically closed. Affine plane: K2 = (h, k) h, k K . { | ∈ } For (h, k) K2, the field elements h, k are called affine coordinates. ∈ Affine curve: Defined by a polynomial equation: C : f (X, Y)= 0. It is customary to consider only irreducible polynomials f (X, Y). If f (X, Y) admits non-trivial factors, the curve C is the set-theoretic union of two (or more) curves of smaller degrees. Rational points on C: All points (h, k) K2 such that f (h, k)= 0. ∈ Rational points on C are called finite points. Affine Curves: Examples Straight lines: aX + bY + c = 0. Circles: (X a)2 + (Y b)2 r2 = 0. − − − Conic sections: aX2 + bXY + cY2 + dX + eY + f = 0. Elliptic curves: Defined by the Weierstrass equation: 2 3 2 Y + (a1X + a3)Y = X + a2X + a4X + a6. If char K = 2, 3, this can be simplified as Y2 = X3 + aX + b. Hyperelliptic curves of genus g: Y2 + u(X)Y = v(X) with deg u 6 g, deg v = 2g + 1, and v monic. If char K = 2, this can be simplified as Y2 = w(X) with deg w = 2g + 1 and w monic. Parabolas are hyperelliptic curves of genus 0.
    [Show full text]
  • 8.4 Exceptional Polynomials
    Permutation polynomials 229 diagonal kinds of polynomials in several variables where the preimage points come from subsets of the field rather than the entire field. See Also 8.1 Discusses permutation polynomials in one variable. x8.2 Discusses permutation polynomials in several variables. x8.4 Considers exceptional polynomials. x [624] Considers polynomials whose value sets lie in a subfield. [729] Studies value sets as they relate to Dembowski-Ostrom and planar polynomials. References Cited: [57, 284, 288, 349, 350, 546, 623, 624, 625, 662, 729, 752, 753, 755, 767, 768, 769, 1083, 1219, 1307, 1308, 1363, 1364, 1454, 1934, 2038, 2100, 2743, 2823, 2908, 2916, 2978, 2979] 8.4 Exceptional polynomials Michael E. Zieve, University of Michigan 8.4.1 Fundamental properties 8.4.1 Definition An exceptional polynomial over Fq is a polynomial f Fq[x] which is a permu- 2 tation polynomial on Fqm for infinitely many m. 8.4.2 Remark If f Fq[x] is exceptional over Fqk for some k, then f is exceptional over Fq. 2 8.4.3 Definition A polynomial F (x; y) Fq[x; y] is absolutely irreducible if it is irreducible in ¯ ¯ 2 Fq[x; y], where Fq is an algebraic closure of Fq. 8.4.4 Theorem [662] A polynomial f Fq[x] is exceptional over Fq if and only if every absolutely 2 irreducible factor of f(x) f(y) in Fq[x; y] is a constant times x y. − − 8.4.5 Corollary If f Fq[x] is exceptional, then there are integers 1 < e1 < e2 < < ek such 2 ··· that: f is exceptional over Fqn if and only if n is not divisible by any ei.
    [Show full text]
  • An Upper Bound and Finiteness Criteria for the Galois Group of Weighted
    Comptes Rendus Mathématique Ruichao Jiang, Javad Tavakoli and Yiqiang Zhao An upper bound and finiteness criteria for the Galois group of weighted walks with rational coeYcients in the quarter plane Volume 359, issue 5 (2021), p. 563-576 <https://doi.org/10.5802/crmath.196> © Académie des sciences, Paris and the authors, 2021. Some rights reserved. This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Les Comptes Rendus. Mathématique sont membres du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Comptes Rendus Mathématique 2021, 359, nO 5, p. 563-576 https://doi.org/10.5802/crmath.196 Combinatorics, Probability theory / Combinatoire, Probabilités An upper bound and finiteness criteria for the Galois group of weighted walks with rational coeYcients in the quarter plane Un majorant et des critères de finitude pour le groupe de Galois de marches pondérées avec des coeYcients rationnels dans le quart de plan a , a b Ruichao Jiang , Javad Tavakoli¤ and Yiqiang Zhao a The University of British Columbia Okanagan, Kelowna, BC V1V 1V7, Canada b Carleton University, Ottawa, ON K1S 5B6, Canada E-mail: [email protected] Abstract. Using Mazur’s theorem on torsions of elliptic curves, an upper bound 24 for the order of the finite Galois group H associated with weighted walks in the quarter plane Z2 is obtained. The explicit criterion for H to have order 4 or 6 is rederived by simple geometric arguments. UsingÅ division polynomials, a recursive criterion for H to have order 4m or 4m 2 is also obtained.
    [Show full text]
  • Elliptic Curves
    Chapter 9 Elliptic Curves This is a chapter from version 2.0 of the book “Mathematics of Public Key Cryptogra- phy” by Steven Galbraith, available from http://www.math.auckland.ac.nz/˜sgal018/crypto- book/crypto-book.html The copyright for this chapter is held by Steven Galbraith. This book was published by Cambridge University Press in early 2012. This is the extended and corrected version. Some of the Theorem/Lemma/Exercise numbers may be different in the published version. Please send an email to [email protected] if youfind any mistakes. This chapter summarises the theory of elliptic curves. Since there are already many outstanding textbooks on elliptic curves (such as Silverman [564] and Washington [626]) we do not give all the details. Our focus is on facts relevant for the cryptographic appli- cations, especially those for which there is not already a suitable reference. 9.1 Group law Recall that an elliptic curve over afieldk is given by a non-singular affine Weierstrass equation 2 3 2 E:y +a 1xy+a 3y=x +a 2x +a 4x+a 6 (9.1) wherea 1, a2, a3, a4, a6 k. There is a unique point E on the projective closure that does not lie on the affine∈ curve. O We recall the formulae for the elliptic curve group law with identity element : For O E allP E(k) we haveP+ E = E +P=P so it remains to consider the case where P ,P∈ E(k) are suchO thatP ,PO= . In other words,P andP are affine points and 1 2 ∈ 1 2 6 O E 1 2 so writeP 1 = (x1, y1) andP 2 = (x2, y2).
    [Show full text]
  • PART I. ELLIPTIC CURVES Contents 1 Introduction 1.1 Definition 1.2-3 the Group Law 1.4-6 a Cultural Detour (Optional) 2 Torsion
    PART I. ELLIPTIC CURVES Contents 1 Introduction 1.1 Definition 1.2-3 The group law 1.4-6 A cultural detour (optional) 2 Torsion on elliptic curves 2.1 m-torsion subgroups 2.2 Division polynomials 3 Functions and divisors 3.1-2 Divisor class groups 3.3 The Weil pairing 4 Elliptic curves over finite fields 4.1 Hasse’s theorem 4.2 The Frobenius action on E[m] 4.3 Schoof’s algorithm Some useful references [1] I. Blake, G. Seroussi and N. Smart, Elliptic curves in cryptography, London Mathematical Society Lecture Note Series 265, Cambridge University Press (1999) [2] H. Cohen and G. Frey (eds.), Handbook of elliptic and hyperelliptic curve cryptography, Discrete Mathematics and its Applications 34, Chapman & Hall/CRC (2005) [3] N. Koblitz, Algebraic aspects of cryptography, Algorithms and Computation in Mathematics 3, Springer-Verlag (1997) [4] P. Nguyen, Public-key cryptanalysis, a chapter in I. Luengo, Recent trends in cryptography, Contemporary Mathematics series 477, AMS-RSME (2009) [5] J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag (1986) The most down-to-earth text is [3]. The standard reference on elliptic curves is [5], for which some experience with algebraic geometry is recommended (the first two chapters give a good introduction). The most up-to-date account is the extensive [2], of which a second edition will appear in the near future. Reference [4] is mainly about RSA instead of elliptic curve cryptography. 1 2 PART I. ELLIPTIC CURVES 1. Introduction to elliptic curves 1.1. Definition. Throughout, let k be a field of characteristic not equal to 2 or 3, and let k be an algebraic closure.
    [Show full text]