DOCSLIB.ORG

Complex Multiplication of Abelian Surfaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

COMPLEX MULTIPLICATION OF ABELIAN SURFACES Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof. mr. P.F. van der Heijden, volgens besluit van het College voor Promoties te verdedigen op dinsdag 1 juni 2010 klokke 15:00 uur door Theodorus Cornelis Streng geboren te IJsselstein in 1982 Samenstelling van de promotiecommissie: Promotor prof. dr. Peter Stevenhagen Overige leden prof. dr. Gunther Cornelissen (Universiteit Utrecht) prof. dr. Bas Edixhoven prof. dr. David R. Kohel (Universit´ede la M´editerran´ee) prof. dr. Hendrik W. Lenstra Jr. dr. Ronald van Luijk Complex multiplication of abelian surfaces Marco Streng Marco Streng Complex multiplication of abelian surfaces ISBN-13 / EAN: 978-90-5335-291-5 AMS subj. class.: 11G15, 14K22 NUR: 921 c Marco Streng, Leiden 2010 [email protected] Typeset using LaTeX Printed by Ridderprint, Ridderkerk Asteroids, of which a screen shot is shown on page 188, is due to Atari, 1979. The cover illustration shows the complex curve C : y2 = x5 + 1 in the coordinates (Re x; Im x; Re y). Its Jacobian J(C) is an abelian surface with complex multiplication by Z[ζ5] induced by the curve automor- phism ζ5 :(x; y) 7! (ζ5x; y). The colored curves are the real locus of C and its images under hζ5i. The illustration was created using Sage [70] and Tachyon. Contents Contents 5 Introduction 9 I Complex multiplication 17 1 Kronecker's Jugendtraum . 17 2 CM-fields . 18 3 CM-types . 20 4 Complex multiplication . 21 5 Complex abelian varieties . 23 5.1 Complex tori and polarizations . 23 5.2 Ideals and polarizations . 24 5.3 Another representation of the ideals . 27 6 Jacobians of curves . 28 7 The reflex of a CM-type . 30 8 The type norm . 32 9 The main theorem of complex multplication . 33 10 The class fields of quartic CM-fields . 35 II Computing Igusa class polynomials 39 1 Introduction . 39 2 Igusa class polynomials . 41 2.1 Igusa invariants . 42 2.2 Alternative definitions . 43 3 Abelian varieties with CM . 44 3.1 The general algorithm . 45 3.2 Quartic CM-fields . 46 3.3 Implementation details . 47 4 Symplectic bases . 49 4.1 A symplectic basis for Φ(a)............ 49 6 Contents 4.2 A symplectic basis for (z; b)............ 51 5 The fundamental domain of the Siegel upper half space 52 5.1 The genus-1 case . 52 5.2 The fundamental domain for genus two . 55 5.3 The reduction algorithm for genus 2 . 57 5.4 Identifying points on the boundary . 62 6 Bounds on the period matrices . 64 6.1 The bound on the period matrix . 64 6.2 A good pair (z; b)................. 65 7 Theta constants . 67 7.1 Igusa invariants in terms of theta constants . 68 7.2 Bounds on the theta constants . 70 7.3 Evaluating Igusa invariants . 72 7.4 Evaluating theta constants . 74 8 The degree of the class polynomials . 76 9 Denominators . 76 9.1 The bounds of Goren and Lauter . 77 9.2 The bounds of Bruinier and Yang . 80 9.3 Counterexample to a conjectured formula . 82 10 Recovering a polynomial from its roots . 82 10.1 Polynomial multiplication . 82 10.2 Recovering a polynomial from its roots . 84 10.3 Recognizing rational coefficients . 86 11 The algorithm . 87 III The irreducible components of the CM locus 91 1 The moduli space of CM-by-K points . 92 2 The irreducible components of CMK;Φ ......... 92 3 Computing the irreducible components . 94 4 The CM method . 98 5 Double roots . 101 IV Abelian varieties with prescribed embedding degree 105 1 Introduction . 105 2 Weil numbers yielding prescribed embedding degrees . 107 3 Performance of the algorithm . 112 4 Constructing abelian varieties with given Weil numbers 117 5 Numerical examples . 119 Contents 7 V Abelian surfaces with p-rank 1 123 1 Introduction . 123 2 Characterization of abelian surfaces with p-rank one . 125 3 Existence of suitable Weil numbers . 127 4 The algorithms . 130 5 Constructing curves with given Weil numbers . 136 6 A sufficient and necessary condition . 137 7 Factorization of class polynomials mod p ........ 141 8 Examples . 143 Appendix 145 1 The Fourier expansion of Igusa invariants . 147 2 An alternative algorithm for enumerating CM varieties 151 2.1 Reduced pairs (z; b)................ 151 2.2 Real quadratic fields . 153 2.3 Analysis of Algorithm 2.5 . 156 2.4 Generalization of Spallek's formula . 157 3 Experimental results . 159 3.1 Good absolute Igusa invariants . 159 3.2 Asymptotics of bit sizes . 163 Bibliography 167 List of notation 177 Index of terms 179 Index of people 182 Nederlandse samenvatting 185 1 Priemgetallen . 185 2 Een probleem uit de getaltheorie . 185 3 De oplossing . 186 4 Een variant op het probleem . 187 5 Fietsbanden . 188 6 Elliptische krommen . 189 7 Pinpassen en slimme prijskaartjes . 190 8 Dubbele donuts . 192 9 Wat staat er in dit proefschrift? . 192 Dankwoord / Acknowledgements 195 Curriculum vitae 197 Introduction The theory of complex multiplication makes it possible to construct certain class fields and abelian varieties. The main theme of this thesis is making these constructions explicit for the case where the abelian varieties have dimension 2. Elliptic curves over finite fields One-dimensional abelian varieties are known as elliptic curves, which in most cases can be represented as a curve in the (x; y)-plane given by y2 = x3 + ax + b (0.1) for some choice of parameters a, b in a field k. Elliptic curves come with a natural (abelian) group law, which can be described completely geometrically. In the representation (0.1), the unit element of the group is an extra point O at infinity, and three points P; Q; R satisfy P + Q + R = O in the group if and only if they are collinear. For k = R, this looks as follows. P P + Q Q R The group law can be given by algebraic equations, and we can define elliptic curves over any field k. If k has characteristic different from 2 and 3, which we assume from now on for simplicity, then this is done by taking a and b in k. If we do this for a finite field k, then the group E(k) 10 Introduction of points defined over k is finite. Indeed, the number of elements #E(k) of E(k) can be computed simply by testing for every x-coordinate in k whether x3 + ax + b is a square in k. If the order q = #k of k gets large, then this method of point counting takes too much time. However, there are faster methods based on the properties of the Frobenius endomorphism F :(x; y) 7! (xq; yq) of E. The points in E(k) are exactly those points over an algebraic closure of k that are left invariant by F . In particular, they are the points in the kernel of the endomorphism (F − id), where subtraction takes place in the ring of endomorphisms End(E) of E. It is known that F is (as an element of the endomorphism ring) a root of a quadratic Weil polynomial f = X2 − tX + q 2 Z[X]; (0.2) and that we have #E(k) = deg(F − id) = f(1) = q + 1 − t: p The trace of Frobenius t is bounded in size by jtj ≤ 2 q, and indicates to which extent #E(k) differs from the number q+1 of points on a straight line. Schoof realized in 1985 that the reductions (t mod l) at small primes l can be computed by looking at the action of F on the l-torsion points of E, and that this allows one to compute the number t, and therefore #E(k), efficiently. This yields a polynomial time algorithm that, for large q, is much faster than the exponential time method of direct point counting. Cryptography Suppose one has a finite group G in which the group operation can be efficiently implemented, but the discrete logarithm problem is thought to be hard. This means that given x; y 2 G, finding an integer m such that y = xm holds is hard. Then the Diffie-Hellman key exchange protocol from 1976 allows one to agree upon a cryptographic key in such a way that eavesdroppers, who intercept the entire communication, are believed to be unable to derive the key from it. The original example of such a group G is the unit group G = k∗ for a prime finite field k = Fp. Index calculus methods like the number field sieve provide a sub- exponential method for solving the discrete logarithm problem in k∗. To protect the protocol against this algorithm until the year 2030, it is generally recommended to use primes p of over 3000 bits. As for G = k∗, the group order of G = E(k) for an elliptic curve E is of size approximately #k = q. However, it seems that the dis- crete logarithm problem for the group E(k) is harder, as 35 years of The CM method 11 research has not led to a sub-exponential method for it. For this rea- son, the recommended key sizes for achieving the same level of security with elliptic curve cryptography are much smaller: q is recommended to have 256 bits. This difference of a factor 12 in key length is important in practical situations such as on `RFID tags' with limited computing power. The optimal elliptic curves for cryptography are the ones of prime group order, and we will now describe how they can be obtained. The CM method One can construct elliptic curves of prime order over a finite field k by `random curves and point counting', that is, by taking random a's and b's in k and computing #E(k) using Schoof's algorithm until one encounters an elliptic curve of prime order.
Recommended publications
  • 21 the Hilbert Class Polynomial

    21 the Hilbert Class Polynomial

    18.783 Elliptic Curves Spring 2015 Lecture #21 04/28/2015 21 The Hilbert class polynomial In the previous lecture we proved that the field of modular functions for Γ0(N) is generated by the functions j(τ) and jN (τ) := j(Nτ), and we showed that C(j; jN ) is a finite extension of C(j). We then defined the classical modular polynomial ΦN (Y ) as the minimal polynomial of jN over C(j), and we proved that its coefficients are integer polynomials in j. Replacing j with a new variable X, we can view ΦN Z[X; Y ] as an integer polynomial in two variables. In this lecture we will use ΦN to prove that the Hilbert class polynomial Y HD(X) := (X − j(E)) j(E)2EllO(C) also has integer coefficients; here D = disc(O) and EllO(C) := fj(E) : End(E) ' Og is the finite set of j-invariants of elliptic curves E=C with complex multiplication (CM) by O. This implies that each j(E) 2 EllO(C) is an algebraic integer, meaning that any elliptic curve E=C with complex multiplication can actually be defined over a finite extension of Q (a number field). This fact is the key to relating the theory of elliptic curves over the complex numbers to elliptic curves over finite fields. 21.1 Isogenies Recall from Lecture 18 that if L1 is a sublattice of L2, and E1 ' C=L1 and E2 ' C=L2 are the corresponding elliptic curves, then there is an isogeny φ: E1 ! E2 whose kernel is isomorphic to the finite abelian group L2=L1.
  • Curve Counting on Abelian Surfaces and Threefolds

    Curve Counting on Abelian Surfaces and Threefolds

    Algebraic Geometry 5 (4) (2018) 398{463 doi:10.14231/AG-2018-012 Curve counting on abelian surfaces and threefolds Jim Bryan, Georg Oberdieck, Rahul Pandharipande and Qizheng Yin Abstract We study the enumerative geometry of algebraic curves on abelian surfaces and three- folds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov{Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasi-modular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by G¨ottsche{Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the gen- erating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, G¨ottsche, and Lange{Sernesi. Further evidence is provided by Donaldson{Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject. 1. Introduction 1.1 Vanishings Let A be a complex abelian variety of dimension d. The Gromov{Witten invariants of A in genus g and class β 2 H2(A; Z) are defined by integration against the virtual class of the moduli space of stable maps M g;n(A; β), Z A ∗ a1 ∗ an τa1 (γ1) ··· τan (γn) = ev1(γ1) 1 ··· evn(γn) n ; g,β vir [M g;n(A,β)] see [PT14] for an introduction.
  • Abelian Varieties with Complex Multiplication and Modular Functions, by Goro Shimura, Princeton Univ

    Abelian Varieties with Complex Multiplication and Modular Functions, by Goro Shimura, Princeton Univ

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 3, Pages 405{408 S 0273-0979(99)00784-3 Article electronically published on April 27, 1999 Abelian varieties with complex multiplication and modular functions, by Goro Shimura, Princeton Univ. Press, Princeton, NJ, 1998, xiv + 217 pp., $55.00, ISBN 0-691-01656-9 The subject that might be called “explicit class field theory” begins with Kro- necker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x)=exp(2πix), i.e., with x Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something2 similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. Nowadays we would add that the reciprocity law describing the Galois group of an abelian extension L/K in terms of ideals of K should also be given explicitly. After K = Q, the next case is that of an imaginary quadratic number field K, with the real torus R/Z replaced by an elliptic curve E with complex multiplication. (Kronecker knew what the result should be, although complete proofs were given only later, by Weber and Takagi.) For simplicity, let be the ring of integers in O K, and let A be an -ideal. Regarding A as a lattice in C, we get an elliptic curve O E = C/A with End(E)= ;Ehas complex multiplication, or CM,by .If j=j(A)isthej-invariant ofOE,thenK(j) is the Hilbert class field of K, i.e.,O the maximal abelian unramified extension of K.
  • Chapter 5 Complex Numbers

    Chapter 5 Complex Numbers

    Chapter 5 Complex numbers Why be one-dimensional when you can be two-dimensional? ? 3 2 1 0 1 2 3 − − − ? We begin by returning to the familiar number line, where I have placed the question marks there appear to be no numbers. I shall rectify this by defining the complex numbers which give us a number plane rather than just a number line. Complex numbers play a fundamental rˆolein mathematics. In this chapter, I shall use them to show how e and π are connected and how certain primes can be factorized. They are also fundamental to physics where they are used in quantum mechanics. 5.1 Complex number arithmetic In the set of real numbers we can add, subtract, multiply and divide, but we cannot always extract square roots. For example, the real number 1 has 125 126 CHAPTER 5. COMPLEX NUMBERS the two real square roots 1 and 1, whereas the real number 1 has no real square roots, the reason being that− the square of any real non-zero− number is always positive. In this section, we shall repair this lack of square roots and, as we shall learn, we shall in fact have achieved much more than this. Com- plex numbers were first studied in the 1500’s but were only fully accepted and used in the 1800’s. Warning! If r is a positive real number then √r is usually interpreted to mean the positive square root. If I want to emphasize that both square roots need to be considered I shall write √r.
  • Hyperelliptic Curves with Many Automorphisms

    Hyperelliptic Curves with Many Automorphisms

    Hyperelliptic Curves with Many Automorphisms Nicolas M¨uller Richard Pink Department of Mathematics Department of Mathematics ETH Z¨urich ETH Z¨urich 8092 Z¨urich 8092 Z¨urich Switzerland Switzerland [email protected] [email protected] November 17, 2017 To Frans Oort Abstract We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication. arXiv:1711.06599v1 [math.AG] 17 Nov 2017 MSC classification: 14H45 (14H37, 14K22) 1 1 Introduction Let X be a smooth connected projective algebraic curve of genus g > 2 over the field of complex numbers. Following Rauch [17] and Wolfart [21] we say that X has many automorphisms if it cannot be deformed non-trivially together with its automorphism group. Given his life-long interest in special points on moduli spaces, Frans Oort [15, Question 5.18.(1)] asked whether the point in the moduli space of curves associated to a curve X with many automorphisms is special, i.e., whether the jacobian of X has complex multiplication. Here we say that an abelian variety A has complex multiplication over a field K if ◦ EndK(A) contains a commutative, semisimple Q-subalgebra of dimension 2 dim A. (This property is called “sufficiently many complex multiplications” in Chai, Conrad and Oort [6, Def. 1.3.1.2].) Wolfart [22] observed that the jacobian of a curve with many automorphisms does not generally have complex multiplication and answered Oort’s question for all g 6 4. In the present paper we answer Oort’s question for all hyperelliptic curves with many automorphisms.
  • GEOMETRY and NUMBERS Ching-Li Chai

    GEOMETRY and NUMBERS Ching-Li Chai

    GEOMETRY AND NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a GEOMETRY AND NUMBERS diophantine equation Counting congruence solutions L-functions and distribution of prime numbers Ching-Li Chai Zeta and L-values Sample of geometric structures and symmetries Institute of Mathematics Elliptic curve basics Academia Sinica Modular forms, modular and curves and Hecke symmetry Complex multiplication Department of Mathematics Frobenius symmetry University of Pennsylvania Monodromy Fine structure in characteristic p National Chiao Tung University, July 6, 2012 GEOMETRY AND Outline NUMBERS Ching-Li Chai Sample arithmetic 1 Sample arithmetic statements statements Diophantine equations Diophantine equations Counting solutions of a diophantine equation Counting solutions of a diophantine equation Counting congruence solutions L-functions and distribution of Counting congruence solutions prime numbers L-functions and distribution of prime numbers Zeta and L-values Sample of geometric Zeta and L-values structures and symmetries Elliptic curve basics Modular forms, modular 2 Sample of geometric structures and symmetries curves and Hecke symmetry Complex multiplication Elliptic curve basics Frobenius symmetry Monodromy Modular forms, modular curves and Hecke symmetry Fine structure in characteristic p Complex multiplication Frobenius symmetry Monodromy Fine structure in characteristic p GEOMETRY AND The general theme NUMBERS Ching-Li Chai Sample arithmetic statements Diophantine equations Counting solutions of a Geometry and symmetry influences diophantine equation Counting congruence solutions L-functions and distribution of arithmetic through zeta functions and prime numbers Zeta and L-values modular forms Sample of geometric structures and symmetries Elliptic curve basics Modular forms, modular Remark. (i) zeta functions = L-functions; curves and Hecke symmetry Complex multiplication modular forms = automorphic representations.
  • Lesson 3: Roots of Unity

    Lesson 3: Roots of Unity

    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M3 PRECALCULUS AND ADVANCED TOPICS Lesson 3: Roots of Unity Student Outcomes . Students determine the complex roots of polynomial equations of the form 푥푛 = 1 and, more generally, equations of the form 푥푛 = 푘 positive integers 푛 and positive real numbers 푘. Students plot the 푛th roots of unity in the complex plane. Lesson Notes This lesson ties together work from Algebra II Module 1, where students studied the nature of the roots of polynomial equations and worked with polynomial identities and their recent work with the polar form of a complex number to find the 푛th roots of a complex number in Precalculus Module 1 Lessons 18 and 19. The goal here is to connect work within the algebra strand of solving polynomial equations to the geometry and arithmetic of complex numbers in the complex plane. Students determine solutions to polynomial equations using various methods and interpreting the results in the real and complex plane. Students need to extend factoring to the complex numbers (N-CN.C.8) and more fully understand the conclusion of the fundamental theorem of algebra (N-CN.C.9) by seeing that a polynomial of degree 푛 has 푛 complex roots when they consider the roots of unity graphed in the complex plane. This lesson helps students cement the claim of the fundamental theorem of algebra that the equation 푥푛 = 1 should have 푛 solutions as students find all 푛 solutions of the equation, not just the obvious solution 푥 = 1. Students plot the solutions in the plane and discover their symmetry.
  • The Twelfth Problem of Hilbert Reminds Us, Although the Reminder Should

    The Twelfth Problem of Hilbert Reminds Us, Although the Reminder Should

    Some Contemporary Problems with Origins in the Jugendtraum Robert P. Langlands The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate devel• opments. The first of these, the theory of class fields or of abelian extensions of number fields, attained what was pretty much its final form early in this century. The second, the algebraic theory of elliptic curves and, more generally, of abelian varieties, has been for fifty years a topic of research whose vigor and quality shows as yet no sign of abatement. The third, the theory of automorphic functions, has been slower to mature and is still inextricably entangled with the study of abelian varieties, especially of their moduli. Of course at the time of Hilbert these subjects had only begun to set themselves off from the general mathematical landscape as separate theories and at the time of Kronecker existed only as part of the theories of elliptic modular functions and of cyclotomicfields. It is in a letter from Kronecker to Dedekind of 1880,1 in which he explains his work on the relation between abelian extensions of imaginary quadratic fields and elliptic curves with complex multiplication, that the word Jugendtraum appears. Because these subjects were so interwoven it seems to have been impossible to disentangle the different kinds of mathematics which were involved in the Jugendtraum, especially to separate the algebraic aspects from the analytic or number theoretic. Hilbert in particular may have been led to mistake an accident, or perhaps necessity, of historical development for an “innigste gegenseitige Ber¨uhrung.” We may be able to judge this better if we attempt to view the mathematical content of the Jugendtraum with the eyes of a sophisticated contemporary mathematician.
  • Three Aspects of the Theory of Complex Multiplication

    Three Aspects of the Theory of Complex Multiplication

    Three Aspects of the Theory of Complex Multiplication Takase Masahito Introduction In the letter addressed to Dedekind dated March 15, 1880, Kronecker wrote: - the Abelian equations with square roots of rational numbers are ex­ hausted by the transformation equations of elliptic functions with sin­ gular modules, just as the Abelian equations with integral coefficients are exhausted by the cyclotomic equations. ([Kronecker 6] p. 455) This is a quite impressive conjecture as to the construction of Abelian equations over an imaginary quadratic number fields, which Kronecker called "my favorite youthful dream" (ibid.). The aim of the theory of complex multiplication is to solve Kronecker's youthful dream; however, it is not the history of formation of the theory of complex multiplication but Kronecker's youthful dream itself that I would like to consider in the present paper. I will consider Kronecker's youthful dream from three aspects: its history, its theoretical character, and its essential meaning in mathematics. First of all, I will refer to the history and theoretical character of the youthful dream (Part I). Abel modeled the division theory of elliptic functions after the division theory of a circle by Gauss and discovered the concept of an Abelian equation through the investigation of the algebraically solvable conditions of the division equations of elliptic functions. This discovery is the starting point of the path to Kronecker's youthful dream. If we follow the path from Abel to Kronecker, then it will soon become clear that Kronecker's youthful dream has the theoretical character to be called the inverse problem of Abel.
  • Algebraic Geometry Codes Over Abelian Surfaces Containing No Absolutely Irreducible Curves of Low Genus Yves Aubry, Elena Berardini, Fabien Herbaut, Marc Perret

    Algebraic Geometry Codes Over Abelian Surfaces Containing No Absolutely Irreducible Curves of Low Genus Yves Aubry, Elena Berardini, Fabien Herbaut, Marc Perret

    Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus Yves Aubry, Elena Berardini, Fabien Herbaut, Marc Perret To cite this version: Yves Aubry, Elena Berardini, Fabien Herbaut, Marc Perret. Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus. Finite Fields and Their Applications, Elsevier, 2021. hal-02100210v2 HAL Id: hal-02100210 https://hal.archives-ouvertes.fr/hal-02100210v2 Submitted on 31 Mar 2021 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. ALGEBRAIC GEOMETRY CODES OVER ABELIAN SURFACES CONTAINING NO ABSOLUTELY IRREDUCIBLE CURVES OF LOW GENUS YVES AUBRY, ELENA BERARDINI, FABIEN HERBAUT AND MARC PERRET Abstract. We provide a theoretical study of Algebraic Geometry codes con- structed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restric- tions of elliptic curves and abelian surfaces which do not admit a principal polarization. 1. Introduction The success of Goppa construction ([5]) of codes over algebraic curves in break- ing the Gilbert-Varshamov bound (see Tsfasman-Vl˘adu¸t-Zink bound in [18]) has been generating much interest over the last forty years.
  • Applications of Complex Multiplication of Elliptic Curves

    Applications of Complex Multiplication of Elliptic Curves

    Applications of Complex Multiplication of Elliptic Curves MASTER THESIS Supervisor: Candidate: Prof. Jean-Marc Massimo CHENAL COUVEIGNES UNIVERSITÀ DEGLI STUDI DI PADOVA UNIVERSITÉ BORDEAUX 1 Facoltà di Scienze MM. FF. NN. U.F.R. Mathématiques et Informatique Academic year 2011-2012 Introduction Elliptic curves represent perhaps one of the most interesting points of contact between mathematical theory and real-world applications. Altough its fundamentals lie in the Algebraic Number Theory and Algebraic Ge- ometry, elliptic curves theory finds many applications in Cryptography and communication security. This connection was first suggested indepen- dently by N. Koblitz and V.S. Miller in the late ’80s, and many efforts have been made to study it thoroughly ever since. The goal of this Master Thesis is to study the complex multiplication of elliptic curves and to consider some applications. We do this by first studying basic Hilbert class field theory; in parallel, we describe elliptic curves over a generic field k, and then we specialize to the cases k = C and k = Fp. We next study the reduction of elliptic curves, we describe the so called Complex Multiplication method and finally we explain Schoof’s algorithm for computing point on elliptic curves over finite fields. As an application, we see how to compute square roots modulo a prime p. Ex- plicit algorithms and full-detailed examples are also given. Thesis Plan Complex Multiplication method (or CM-method, for short) exploit the so called Hilbert class polynomial, and in Chapter 1 we introduce all the tools we need to define it. Therefore we will describe modular forms and the j-function, and we provide an efficient method to compute the latter.
  • On Complex and Noncommutative Tori

    On Complex and Noncommutative Tori

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 4, Pages 973–981 S 0002-9939(05)08244-4 Article electronically published on September 28, 2005 ON COMPLEX AND NONCOMMUTATIVE TORI IGOR NIKOLAEV (Communicated by Michael Stillman) Abstract. The “noncommutative geometry” of complex algebraic curves is studied. As a first step, we clarify a morphism between elliptic curves, or ∗ 2πiθ complex tori, and C -algebras Tθ = {u, v | vu = e uv},ornoncommutative tori. The main result says that under the morphism, isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces. Introduction Noncommutative geometry is a branch of algebraic geometry studying “varieties” over noncommutative rings. The noncommutative rings are usually taken to be rings of operators acting on a Hilbert space [7]. The rudiments of noncommutative geometry can be traced back to F. Klein [3], [4] or even earlier. The fundamental modern treatise [1] gives an account of status and perspective of the subject. ∗ The noncommutative torus Tθ is a C -algebra generated by linear operators u and v on the Hilbert space L2(S1) subject to the commutation relation vu = e2πiθuv, θ ∈ R − Q [11]. The classification of noncommutative tori was given in [2], [8], [11]. Recall that two such tori Tθ,Tθ are Morita equivalent if and only if θ, θ lie in the same orbit of the action of group GL(2, Z) on irrational numbers by linear fractional transformations. It is remarkable that the “moduli problem” for Tθ looks as such for the complex tori Eτ = C/(Z + τZ), where τ is complex modulus.