Covers of an Elliptic Curve E and Curves in Exp1
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ON the EXISTENCE of CURVES with Ak-SINGULARITIES on K3 SURFACES
Math: Res: Lett: 18 (2011), no: 00, 10001{10030 c International Press 2011 ON THE EXISTENCE OF CURVES WITH Ak-SINGULARITIES ON K3 SURFACES Concettina Galati and Andreas Leopold Knutsen Abstract. Let (S; H) be a general primitively polarized K3 surface. We prove the existence of irreducible curves in jOS (nH)j with Ak-singularities and corresponding to regular points of the equisingular deformation locus. Our result is optimal for n = 1. As a corollary, we get the existence of irreducible curves in jOS (nH)j of geometric genus g ≥ 1 with a cusp and nodes or a simple tacnode and nodes. We obtain our result by studying the versal deformation family of the m-tacnode. Moreover, using results of Brill-Noether theory on curves of K3 surfaces, we provide a regularity condition for families of curves with only Ak-singularities in jOS (nH)j: 1. Introduction Let S be a complex smooth projective K3 surface and let H be a globally generated line bundle of sectional genus p = pa(H) ≥ 2 and such that H is not divisible in Pic S. The pair (S; H) is called a primitively polarized K3 surface of genus p: It is well-known that the moduli space Kp of primitively polarized K3 surfaces of genus p is non-empty, smooth and irreducible of dimension 19: Moreover, if (S; H) 2 Kp is a very general element (meaning that it belongs to the complement of a countable ∼ union of Zariski closed proper subsets), then Pic S = Z[H]: If (S; H) 2 Kp, we denote S by VnH;1δ ⊂ jOS(nH)j = jnHj the so called Severi variety of δ-nodal curves, defined as the Zariski closure of the locus of irreducible and reduced curves with exactly δ nodes as singularities. -
Real Rank Two Geometry Arxiv:1609.09245V3 [Math.AG] 5
Real Rank Two Geometry Anna Seigal and Bernd Sturmfels Abstract The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. 1 Introduction Low-rank approximation of tensors is a fundamental problem in applied mathematics [3, 6]. We here approach this problem from the perspective of real algebraic geometry. Our goal is to give an exact semi-algebraic description of the set of tensors of real rank two and to characterize its boundary. This complements the results on tensors of non-negative rank two presented in [1], and it offers a generalization to the setting of arbitrary varieties, following [2]. A familiar example is that of 2 × 2 × 2-tensors (xijk) with real entries. Such a tensor lies in the closure of the real rank two tensors if and only if the hyperdeterminant is non-negative: 2 2 2 2 2 2 2 2 x000x111 + x001x110 + x010x101 + x011x100 + 4x000x011x101x110 + 4x001x010x100x111 −2x000x001x110x111 − 2x000x010x101x111 − 2x000x011x100x111 (1) −2x001x010x101x110 − 2x001x011x100x110 − 2x010x011x100x101 ≥ 0: If this inequality does not hold then the tensor has rank two over C but rank three over R. To understand this example geometrically, consider the Segre variety X = Seg(P1 × P1 × P1), i.e. the set of rank one tensors, regarded as points in the projective space P7 = 2 2 2 7 arXiv:1609.09245v3 [math.AG] 5 Apr 2017 P(C ⊗ C ⊗ C ). -
Examples from Complex Geometry
Examples from Complex Geometry Sam Auyeung November 22, 2019 1 Complex Analysis Example 1.1. Two Heuristic \Proofs" of the Fundamental Theorem of Algebra: Let p(z) be a polynomial of degree n > 0; we can even assume it is a monomial. We also know that the number of zeros is at most n. We show that there are exactly n. 1. Proof 1: Recall that polynomials are entire functions and that Liouville's Theorem says that a bounded entire function is in fact constant. Suppose that p has no roots. Then 1=p is an entire function and it is bounded. Thus, it is constant which means p is a constant polynomial and has degree 0. This contradicts the fact that p has positive degree. Thus, p must have a root α1. We can then factor out (z − α1) from p(z) = (z − α1)q(z) where q(z) is an (n − 1)-degree polynomial. We may repeat the above process until we have a full factorization of p. 2. Proof 2: On the real line, an algorithm for finding a root of a continuous function is to look for when the function changes signs. How do we generalize this to C? Instead of having two directions, we have a whole S1 worth of directions. If we use colors to depict direction and brightness to depict magnitude, we can plot a graph of a continuous function f : C ! C. Near a zero, we'll see all colors represented. If we travel in a loop around any point, we can keep track of whether it passes through all the colors; around a zero, we'll pass through all the colors, possibly many times. -
A Review on Elliptic Curve Cryptography for Embedded Systems
International Journal of Computer Science & Information Technology (IJCSIT), Vol 3, No 3, June 2011 A REVIEW ON ELLIPTIC CURVE CRYPTOGRAPHY FOR EMBEDDED SYSTEMS Rahat Afreen 1 and S.C. Mehrotra 2 1Tom Patrick Institute of Computer & I.T, Dr. Rafiq Zakaria Campus, Rauza Bagh, Aurangabad. (Maharashtra) INDIA [email protected] 2Department of C.S. & I.T., Dr. B.A.M. University, Aurangabad. (Maharashtra) INDIA [email protected] ABSTRACT Importance of Elliptic Curves in Cryptography was independently proposed by Neal Koblitz and Victor Miller in 1985.Since then, Elliptic curve cryptography or ECC has evolved as a vast field for public key cryptography (PKC) systems. In PKC system, we use separate keys to encode and decode the data. Since one of the keys is distributed publicly in PKC systems, the strength of security depends on large key size. The mathematical problems of prime factorization and discrete logarithm are previously used in PKC systems. ECC has proved to provide same level of security with relatively small key sizes. The research in the field of ECC is mostly focused on its implementation on application specific systems. Such systems have restricted resources like storage, processing speed and domain specific CPU architecture. KEYWORDS Elliptic curve cryptography Public Key Cryptography, embedded systems, Elliptic Curve Digital Signature Algorithm ( ECDSA), Elliptic Curve Diffie Hellman Key Exchange (ECDH) 1. INTRODUCTION The changing global scenario shows an elegant merging of computing and communication in such a way that computers with wired communication are being rapidly replaced to smaller handheld embedded computers using wireless communication in almost every field. This has increased data privacy and security requirements. -
Contents 5 Elliptic Curves in Cryptography
Cryptography (part 5): Elliptic Curves in Cryptography (by Evan Dummit, 2016, v. 1.00) Contents 5 Elliptic Curves in Cryptography 1 5.1 Elliptic Curves and the Addition Law . 1 5.1.1 Cubic Curves, Weierstrass Form, Singular and Nonsingular Curves . 1 5.1.2 The Addition Law . 3 5.1.3 Elliptic Curves Modulo p, Orders of Points . 7 5.2 Factorization with Elliptic Curves . 10 5.3 Elliptic Curve Cryptography . 14 5.3.1 Encoding Plaintexts on Elliptic Curves, Quadratic Residues . 14 5.3.2 Public-Key Encryption with Elliptic Curves . 17 5.3.3 Key Exchange and Digital Signatures with Elliptic Curves . 20 5 Elliptic Curves in Cryptography In this chapter, we will introduce elliptic curves and describe how they are used in cryptography. Elliptic curves have a long and interesting history and arise in a wide range of contexts in mathematics. The study of elliptic curves involves elements from most of the major disciplines of mathematics: algebra, geometry, analysis, number theory, topology, and even logic. Elliptic curves appear in the proofs of many deep results in mathematics: for example, they are a central ingredient in the proof of Fermat's Last Theorem, which states that there are no positive integer solutions to the equation xn + yn = zn for any integer n ≥ 3. Our goals are fairly modest in comparison, so we will begin by outlining the basic algebraic and geometric properties of elliptic curves and motivate the addition law. We will then study the behavior of elliptic curves modulo p: ultimately, there is a fairly strong analogy between the structure of the points on an elliptic curve modulo p and the integers modulo n. -
An Efficient Approach to Point-Counting on Elliptic Curves
mathematics Article An Efficient Approach to Point-Counting on Elliptic Curves from a Prominent Family over the Prime Field Fp Yuri Borissov * and Miroslav Markov Department of Mathematical Foundations of Informatics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria; [email protected] * Correspondence: [email protected] Abstract: Here, we elaborate an approach for determining the number of points on elliptic curves 2 3 from the family Ep = fEa : y = x + a (mod p), a 6= 0g, where p is a prime number >3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for the quantities of interest-reduced modulo p. It allows to advance an efficient technique to compute the 2 six cardinalities associated with the family Ep, for p ≡ 1 (mod 3), whose complexity is O˜ (log p), thus improving the best-known algorithmic solution with almost an order of magnitude. Keywords: elliptic curve over Fp; Hasse bound; high-order residue modulo prime 1. Introduction The elliptic curves over finite fields play an important role in modern cryptography. We refer to [1] for an introduction concerning their cryptographic significance (see, as well, Citation: Borissov, Y.; Markov, M. An the pioneering works of V. Miller and N. Koblitz from 1980’s [2,3]). Briefly speaking, the Efficient Approach to Point-Counting advantage of the so-called elliptic curve cryptography (ECC) over the non-ECC is that it on Elliptic Curves from a Prominent requires smaller keys to provide the same level of security. Family over the Prime Field Fp. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
Diophantine Problems in Polynomial Theory
Diophantine Problems in Polynomial Theory by Paul David Lee B.Sc. Mathematics, The University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The College of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) August 2011 c Paul David Lee 2011 Abstract Algebraic curves and surfaces are playing an increasing role in mod- ern mathematics. From the well known applications to cryptography, to computer vision and manufacturing, studying these curves is a prevalent problem that is appearing more often. With the advancement of computers, dramatic progress has been made in all branches of algebraic computation. In particular, computer algebra software has made it much easier to find rational or integral points on algebraic curves. Computers have also made it easier to obtain rational parametrizations of certain curves and surfaces. Each algebraic curve has an associated genus, essentially a classification, that determines its topological structure. Advancements on methods and theory on curves of genus 0, 1 and 2 have been made in recent years. Curves of genus 0 are the only algebraic curves that you can obtain a rational parametrization for. Curves of genus 1 (also known as elliptic curves) have the property that their rational points have a group structure and thus one can call upon the massive field of group theory to help with their study. Curves of higher genus (such as genus 2) do not have the background and theory that genus 0 and 1 do but recent advancements in theory have rapidly expanded advancements on the topic. -
25 Modular Forms and L-Series
18.783 Elliptic Curves Spring 2015 Lecture #25 05/12/2015 25 Modular forms and L-series As we will show in the next lecture, Fermat's Last Theorem is a direct consequence of the following theorem [11, 12]. Theorem 25.1 (Taylor-Wiles). Every semistable elliptic curve E=Q is modular. In fact, as a result of subsequent work [3], we now have the stronger result, proving what was previously known as the modularity conjecture (or Taniyama-Shimura-Weil conjecture). Theorem 25.2 (Breuil-Conrad-Diamond-Taylor). Every elliptic curve E=Q is modular. Our goal in this lecture is to explain what it means for an elliptic curve over Q to be modular (we will also define the term semistable). This requires us to delve briefly into the theory of modular forms. Our goal in doing so is simply to understand the definitions and the terminology; we will omit all but the most straight-forward proofs. 25.1 Modular forms Definition 25.3. A holomorphic function f : H ! C is a weak modular form of weight k for a congruence subgroup Γ if f(γτ) = (cτ + d)kf(τ) a b for all γ = c d 2 Γ. The j-function j(τ) is a weak modular form of weight 0 for SL2(Z), and j(Nτ) is a weak modular form of weight 0 for Γ0(N). For an example of a weak modular form of positive weight, recall the Eisenstein series X0 1 X0 1 G (τ) := G ([1; τ]) := = ; k k !k (m + nτ)k !2[1,τ] m;n2Z 1 which, for k ≥ 3, is a weak modular form of weight k for SL2(Z). -
Lectures on the Combinatorial Structure of the Moduli Spaces of Riemann Surfaces
LECTURES ON THE COMBINATORIAL STRUCTURE OF THE MODULI SPACES OF RIEMANN SURFACES MOTOHICO MULASE Contents 1. Riemann Surfaces and Elliptic Functions 1 1.1. Basic Definitions 1 1.2. Elementary Examples 3 1.3. Weierstrass Elliptic Functions 10 1.4. Elliptic Functions and Elliptic Curves 13 1.5. Degeneration of the Weierstrass Elliptic Function 16 1.6. The Elliptic Modular Function 19 1.7. Compactification of the Moduli of Elliptic Curves 26 References 31 1. Riemann Surfaces and Elliptic Functions 1.1. Basic Definitions. Let us begin with defining Riemann surfaces and their moduli spaces. Definition 1.1 (Riemann surfaces). A Riemann surface is a paracompact Haus- S dorff topological space C with an open covering C = λ Uλ such that for each open set Uλ there is an open domain Vλ of the complex plane C and a homeomorphism (1.1) φλ : Vλ −→ Uλ −1 that satisfies that if Uλ ∩ Uµ 6= ∅, then the gluing map φµ ◦ φλ φ−1 (1.2) −1 φλ µ −1 Vλ ⊃ φλ (Uλ ∩ Uµ) −−−−→ Uλ ∩ Uµ −−−−→ φµ (Uλ ∩ Uµ) ⊂ Vµ is a biholomorphic function. Remark. (1) A topological space X is paracompact if for every open covering S S X = λ Uλ, there is a locally finite open cover X = i Vi such that Vi ⊂ Uλ for some λ. Locally finite means that for every x ∈ X, there are only finitely many Vi’s that contain x. X is said to be Hausdorff if for every pair of distinct points x, y of X, there are open neighborhoods Wx 3 x and Wy 3 y such that Wx ∩ Wy = ∅. -
Elliptic Curve Cryptography and Digital Rights Management
Lecture 14: Elliptic Curve Cryptography and Digital Rights Management Lecture Notes on “Computer and Network Security” by Avi Kak ([email protected]) March 9, 2021 5:19pm ©2021 Avinash Kak, Purdue University Goals: • Introduction to elliptic curves • A group structure imposed on the points on an elliptic curve • Geometric and algebraic interpretations of the group operator • Elliptic curves on prime finite fields • Perl and Python implementations for elliptic curves on prime finite fields • Elliptic curves on Galois fields • Elliptic curve cryptography (EC Diffie-Hellman, EC Digital Signature Algorithm) • Security of Elliptic Curve Cryptography • ECC for Digital Rights Management (DRM) CONTENTS Section Title Page 14.1 Why Elliptic Curve Cryptography 3 14.2 The Main Idea of ECC — In a Nutshell 9 14.3 What are Elliptic Curves? 13 14.4 A Group Operator Defined for Points on an Elliptic 18 Curve 14.5 The Characteristic of the Underlying Field and the 25 Singular Elliptic Curves 14.6 An Algebraic Expression for Adding Two Points on 29 an Elliptic Curve 14.7 An Algebraic Expression for Calculating 2P from 33 P 14.8 Elliptic Curves Over Zp for Prime p 36 14.8.1 Perl and Python Implementations of Elliptic 39 Curves Over Finite Fields 14.9 Elliptic Curves Over Galois Fields GF (2n) 52 14.10 Is b =0 a Sufficient Condition for the Elliptic 62 Curve6 y2 + xy = x3 + ax2 + b to Not be Singular 14.11 Elliptic Curves Cryptography — The Basic Idea 65 14.12 Elliptic Curve Diffie-Hellman Secret Key 67 Exchange 14.13 Elliptic Curve Digital Signature Algorithm (ECDSA) 71 14.14 Security of ECC 75 14.15 ECC for Digital Rights Management 77 14.16 Homework Problems 82 2 Computer and Network Security by Avi Kak Lecture 14 Back to TOC 14.1 WHY ELLIPTIC CURVE CRYPTOGRAPHY? • As you saw in Section 12.12 of Lecture 12, the computational overhead of the RSA-based approach to public-key cryptography increases with the size of the keys. -
Foundations of Algebraic Geometry Class 41
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the “clear denominators” theorem 5 Welcome back! Let's now use what we have developed to study something explicit — curves. Our motivating question is a loose one: what are the curves, by which I mean nonsingular irreducible separated curves, finite type over a field k? In other words, we'll be dealing with geometry, although possibly over a non-algebraically closed field. Here is an explicit question: are all curves (say reduced, even non-singular, finite type over given k) isomorphic? Obviously not: some are affine, and some (such as P1) are not. So to simplify things — and we'll further motivate this simplification in Class 42 — are all projective curves isomorphic? Perhaps all nonsingular projective curves are isomorphic to P1? Once again the answer is no, but the proof is a bit subtle: we've defined an invariant, the genus, and shown that P1 has genus 0, and that there exist nonsingular projective curves of non-zero genus. Are all (nonsingular) genus 0 curves isomorphic to P1? We know there exist nonsingular genus 1 curves (plane cubics) — is there only one? If not, “how many” are there? In order to discuss interesting questions like these, we'll develop some theory. We first show a useful result that will help us focus our attention on the projective case. 1. NORMALIZATION I now want to tell you how to normalize a reduced Noetherian scheme, which is roughly how best to turn a scheme into a normal scheme.