CURVES, JACOBIANS, and CRYPTOGRAPHY 3 Hence We Define in Detail Algebraic Curves, Constant field Extensions, Group Schemes, and Principally Polarized Varieties
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Topological Modes in Dual Lattice Models
Topological Modes in Dual Lattice Models Mark Rakowski 1 Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland Abstract Lattice gauge theory with gauge group ZP is reconsidered in four dimensions on a simplicial complex K. One finds that the dual theory, formulated on the dual block complex Kˆ , contains topological modes 2 which are in correspondence with the cohomology group H (K,Zˆ P ), in addition to the usual dynamical link variables. This is a general phenomenon in all models with single plaquette based actions; the action of the dual theory becomes twisted with a field representing the above cohomology class. A similar observation is made about the dual version of the three dimensional Ising model. The importance of distinct topological sectors is confirmed numerically in the two di- 1 mensional Ising model where they are parameterized by H (K,Zˆ 2). arXiv:hep-th/9410137v1 19 Oct 1994 PACS numbers: 11.15.Ha, 05.50.+q October 1994 DIAS Preprint 94-32 1Email: [email protected] 1 Introduction The use of duality transformations in statistical systems has a long history, beginning with applications to the two dimensional Ising model [1]. Here one finds that the high and low temperature properties of the theory are related. This transformation has been extended to many other discrete models and is particularly useful when the symmetries involved are abelian; see [2] for an extensive review. All these studies have been confined to hypercubic lattices, or other regular structures, and these have rather limited global topological features. Since lattice models are defined in a way which depends clearly on the connectivity of links or other regions, one expects some sort of topological effects generally. -
Automorphisms of a Symmetric Product of a Curve (With an Appendix by Najmuddin Fakhruddin)
Documenta Math. 1181 Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin) Indranil Biswas and Tomas´ L. Gomez´ Received: February 27, 2016 Revised: April 26, 2017 Communicated by Ulf Rehmann Abstract. Let X be an irreducible smooth projective curve of genus g > 2 defined over an algebraically closed field of characteristic dif- ferent from two. We prove that the natural homomorphism from the automorphisms of X to the automorphisms of the symmetric product Symd(X) is an isomorphism if d > 2g − 2. In an appendix, Fakhrud- din proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve. 2000 Mathematics Subject Classification: 14H40, 14J50 Keywords and Phrases: Symmetric product; automorphism; Torelli theorem. 1. Introduction Automorphisms of varieties is currently a very active topic in algebraic geom- etry; see [Og], [HT], [Zh] and references therein. Hurwitz’s automorphisms theorem, [Hu], says that the order of the automorphism group Aut(X) of a compact Riemann surface X of genus g ≥ 2 is bounded by 84(g − 1). The group of automorphisms of the Jacobian J(X) preserving the theta polariza- tion is generated by Aut(X), translations and inversion [We], [La]. There is a universal constant c such that the order of the group of all automorphisms of any smooth minimal complex projective surface S of general type is bounded 2 above by c · KS [Xi]. Let X be a smooth projective curve of genus g, with g > 2, over an al- gebraically closed field of characteristic different from two. -
Associating Abelian Varieties to Weight-2 Modular Forms: the Eichler-Shimura Construction
Ecole´ Polytechnique Fed´ erale´ de Lausanne Master’s Thesis in Mathematics Associating abelian varieties to weight-2 modular forms: the Eichler-Shimura construction Author: Corentin Perret-Gentil Supervisors: Prof. Akshay Venkatesh Stanford University Prof. Philippe Michel EPF Lausanne Spring 2014 Abstract This document is the final report for the author’s Master’s project, whose goal was to study the Eichler-Shimura construction associating abelian va- rieties to weight-2 modular forms for Γ0(N). The starting points and main resources were the survey article by Fred Diamond and John Im [DI95], the book by Goro Shimura [Shi71], and the book by Fred Diamond and Jerry Shurman [DS06]. The latter is a very good first reference about this sub- ject, but interesting points are sometimes eluded. In particular, although most statements are given in the general setting, the book mainly deals with the particular case of elliptic curves (i.e. with forms having rational Fourier coefficients), with little details about abelian varieties. On the other hand, Chapter 7 of Shimura’s book is difficult, according to the author himself, and the article by Diamond and Im skims rapidly through the subject, be- ing a survey. The goal of this document is therefore to give an account of the theory with intermediate difficulty, accessible to someone having read a first text on modular forms – such as [Zag08] – and with basic knowledge in the theory of compact Riemann surfaces (see e.g. [Mir95]) and algebraic geometry (see e.g. [Har77]). This report begins with an account of the theory of abelian varieties needed for what follows. -
AN EXPLORATION of COMPLEX JACOBIAN VARIETIES Contents 1
AN EXPLORATION OF COMPLEX JACOBIAN VARIETIES MATTHEW WOOLF Abstract. In this paper, I will describe my thought process as I read in [1] about Abelian varieties in general, and the Jacobian variety associated to any compact Riemann surface in particular. I will also describe the way I currently think about the material, and any additional questions I have. I will not include material I personally knew before the beginning of the summer, which included the basics of algebraic and differential topology, real analysis, one complex variable, and some elementary material about complex algebraic curves. Contents 1. Abelian Sums (I) 1 2. Complex Manifolds 2 3. Hodge Theory and the Hodge Decomposition 3 4. Abelian Sums (II) 6 5. Jacobian Varieties 6 6. Line Bundles 8 7. Abelian Varieties 11 8. Intermediate Jacobians 15 9. Curves and their Jacobians 16 References 17 1. Abelian Sums (I) The first thing I read about were Abelian sums, which are sums of the form 3 X Z pi (1.1) (L) = !; i=1 p0 where ! is the meromorphic one-form dx=y, L is a line in P2 (the complex projective 2 3 2 plane), p0 is a fixed point of the cubic curve C = (y = x + ax + bx + c), and p1, p2, and p3 are the three intersections of the line L with C. The fact that C and L intersect in three points counting multiplicity is just Bezout's theorem. Since C is not simply connected, the integrals in the Abelian sum depend on the path, but there's no natural choice of path from p0 to pi, so this function depends on an arbitrary choice of path, and will not necessarily be holomorphic, or even Date: August 22, 2008. -
Singularities of Integrable Systems and Nodal Curves
Singularities of integrable systems and nodal curves Anton Izosimov∗ Abstract The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the system turn out to be the coefficients of the characteristic polynomial χ of the Lax matrix, and the solutions are expressed in terms of theta functions related to the curve χ = 0. The aim of the present paper is to show that the possibility to write an integrable system in the Lax form, as well as the algebro-geometric technique related to this possibility, may also be applied to study qualitative features of the system, in particular its singularities. Introduction It is well known that the majority of finite dimensional integrable systems can be written in the form d L(λ) = [L(λ),A(λ)] (1) dt where L and A are matrices depending on the time t and additional parameter λ. The parameter λ is called a spectral parameter, and equation (1) is called a Lax equation with spectral parameter1. The possibility to write a system in the Lax form allows us to solve it explicitly by means of algebro-geometric technique. The algebro-geometric scheme of solving Lax equations can be briefly described as follows. Let us assume that the dependence on λ is polynomial. Then, with each matrix polynomial L, there is an associated algebraic curve C(L)= {(λ, µ) ∈ C2 | det(L(λ) − µE) = 0} (2) called the spectral curve. -
Deligne Pairings and Families of Rank One Local Systems on Algebraic Curves
DELIGNE PAIRINGS AND FAMILIES OF RANK ONE LOCAL SYSTEMS ON ALGEBRAIC CURVES GERARD FREIXAS I MONTPLET AND RICHARD A. WENTWORTH Abstract. For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface. Contents 1. Introduction2 2. Relative Connections and Deligne Pairings5 2.1. Preliminary definitions5 2.2. Gauss-Manin invariant7 2.3. Deligne pairings, norm and trace9 2.4. Metrics and connections 10 3. Trace Connections and Intersection Connections 12 3.1. Weil reciprocity and trace connections 12 3.2. Reformulation in terms of Poincaré bundles 17 3.3. Intersection connections 20 4. Proof of the Main Theorems 23 4.1. The canonical extension: local description and properties 23 4.2. The canonical extension: uniqueness 29 4.3. Variant in the absence of rigidification 31 4.4. Relation between trace and intersection connections 32 arXiv:1507.02920v1 [math.DG] 10 Jul 2015 4.5. Curvatures 34 5. Examples and Applications 37 5.1. Reciprocity for trivial fibrations 37 5.2. Holomorphic extension of analytic torsion 40 5.3. -
Elliptic Factors in Jacobians of Hyperelliptic Curves with Certain Automorphism Groups Jennifer Paulhus
THE OPEN BOOK SERIES 1 ANTS X Proceedings of the Tenth Algorithmic Number Theory Symposium Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups Jennifer Paulhus msp THE OPEN BOOK SERIES 1 (2013) Tenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2013.1.487 Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups Jennifer Paulhus We decompose the Jacobian varieties of hyperelliptic curves up to genus 20, defined over an algebraically closed field of characteristic zero, with reduced automorphism group A4, S4, or A5. Among these curves is a genus-4 curve 2 2 with Jacobian variety isogenous to E1 E2 and a genus-5 curve with Jacobian 5 variety isogenous to E , for E and Ei elliptic curves. These types of results have some interesting consequences for questions of ranks of elliptic curves and ranks of their twists. 1. Introduction Curves with Jacobian varieties that have many elliptic curve factors in their de- compositions up to isogeny have been studied in many different contexts. Ekedahl and Serre found examples of curves whose Jacobians split completely into elliptic curves (not necessarily isogenous to one another)[13] (see also[27],[14, §5]). In genus 2, Cardona showed connections between curves whose Jacobians have two isogenous elliptic curve factors and Q-curves of degree 2 and 3 [5]. There are applications of such curves to ranks of twists of elliptic curves[24], results on torsion[19], and cryptography[12]. Let JX denote the Jacobian variety of a curve X and let represent an isogeny between abelian varieties. -
The Mathematics of Lattices
The Mathematics of Lattices Daniele Micciancio January 2020 Daniele Micciancio (UCSD) The Mathematics of Lattices Jan 2020 1 / 43 Outline 1 Point Lattices and Lattice Parameters 2 Computational Problems Coding Theory 3 The Dual Lattice 4 Q-ary Lattices and Cryptography Daniele Micciancio (UCSD) The Mathematics of Lattices Jan 2020 2 / 43 Point Lattices and Lattice Parameters 1 Point Lattices and Lattice Parameters 2 Computational Problems Coding Theory 3 The Dual Lattice 4 Q-ary Lattices and Cryptography Daniele Micciancio (UCSD) The Mathematics of Lattices Jan 2020 3 / 43 Key to many algorithmic applications Cryptanalysis (e.g., breaking low-exponent RSA) Coding Theory (e.g., wireless communications) Optimization (e.g., Integer Programming with fixed number of variables) Cryptography (e.g., Cryptographic functions from worst-case complexity assumptions, Fully Homomorphic Encryption) Point Lattices and Lattice Parameters (Point) Lattices Traditional area of mathematics ◦ ◦ ◦ Lagrange Gauss Minkowski Daniele Micciancio (UCSD) The Mathematics of Lattices Jan 2020 4 / 43 Point Lattices and Lattice Parameters (Point) Lattices Traditional area of mathematics ◦ ◦ ◦ Lagrange Gauss Minkowski Key to many algorithmic applications Cryptanalysis (e.g., breaking low-exponent RSA) Coding Theory (e.g., wireless communications) Optimization (e.g., Integer Programming with fixed number of variables) Cryptography (e.g., Cryptographic functions from worst-case complexity assumptions, Fully Homomorphic Encryption) Daniele Micciancio (UCSD) The Mathematics of Lattices Jan 2020 4 / 43 Point Lattices and Lattice Parameters Lattice Cryptography: a Timeline 1982: LLL basis reduction algorithm Traditional use of lattice algorithms as a cryptanalytic tool 1996: Ajtai's connection Relates average-case and worst-case complexity of lattice problems Application to one-way functions and collision resistant hashing 2002: Average-case/worst-case connection for structured lattices. -
Intersection Theory
APPENDIX A Intersection Theory In this appendix we will outline the generalization of intersection theory and the Riemann-Roch theorem to nonsingular projective varieties of any dimension. To motivate the discussion, let us look at the case of curves and surfaces, and then see what needs to be generalized. For a divisor D on a curve X, leaving out the contribution of Serre duality, we can write the Riemann-Roch theorem (IV, 1.3) as x(.!Z'(D)) = deg D + 1 - g, where xis the Euler characteristic (III, Ex. 5.1). On a surface, we can write the Riemann-Roch theorem (V, 1.6) as 1 x(!l'(D)) = 2 D.(D - K) + 1 + Pa· In each case, on the left-hand side we have something involving cohomol ogy groups of the sheaf !l'(D), while on the right-hand side we have some numerical data involving the divisor D, the canonical divisor K, and some invariants of the variety X. Of course the ultimate aim of a Riemann-Roch type theorem is to compute the dimension of the linear system IDI or of lnDI for large n (II, Ex. 7.6). This is achieved by combining a formula for x(!l'(D)) with some vanishing theorems for Hi(X,!l'(D)) fori > 0, such as the theorems of Serre (III, 5.2) or Kodaira (III, 7.15). We will now generalize these results so as to give an expression for x(!l'(D)) on a nonsingular projective variety X of any dimension. And while we are at it, with no extra effort we get a formula for x(t&"), where @" is any coherent locally free sheaf. -
Electromagnetic Duality for Children
Electromagnetic Duality for Children JM Figueroa-O'Farrill [email protected] Version of 8 October 1998 Contents I The Simplest Example: SO(3) 11 1 Classical Electromagnetic Duality 12 1.1 The Dirac Monopole ....................... 12 1.1.1 And in the beginning there was Maxwell... 12 1.1.2 The Dirac quantisation condition . 14 1.1.3 Dyons and the Zwanziger{Schwinger quantisation con- dition ........................... 16 1.2 The 't Hooft{Polyakov Monopole . 18 1.2.1 The bosonic part of the Georgi{Glashow model . 18 1.2.2 Finite-energy solutions: the 't Hooft{Polyakov Ansatz . 20 1.2.3 The topological origin of the magnetic charge . 24 1.3 BPS-monopoles .......................... 26 1.3.1 Estimating the mass of a monopole: the Bogomol'nyi bound ........................... 27 1.3.2 Saturating the bound: the BPS-monopole . 28 1.4 Duality conjectures ........................ 30 1.4.1 The Montonen{Olive conjecture . 30 1.4.2 The Witten e®ect ..................... 31 1.4.3 SL(2; Z) duality ...................... 33 2 Supersymmetry 39 2.1 The super-Poincar¶ealgebra in four dimensions . 40 2.1.1 Some notational remarks about spinors . 40 2.1.2 The Coleman{Mandula and Haag{ÃLopusza¶nski{Sohnius theorems .......................... 42 2.2 Unitary representations of the supersymmetry algebra . 44 2.2.1 Wigner's method and the little group . 44 2.2.2 Massless representations . 45 2.2.3 Massive representations . 47 No central charges .................... 48 Adding central charges . 49 1 [email protected] draft version of 8/10/1998 2.3 N=2 Supersymmetric Yang-Mills . -
The Enumerative Geometry of Rational and Elliptic Tropical Curves and a Riemann-Roch Theorem in Tropical Geometry
The enumerative geometry of rational and elliptic tropical curves and a Riemann-Roch theorem in tropical geometry Michael Kerber Am Fachbereich Mathematik der Technischen Universit¨atKaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) vorgelegte Dissertation 1. Gutachter: Prof. Dr. Andreas Gathmann 2. Gutachter: Prof. Dr. Ilia Itenberg Abstract: The work is devoted to the study of tropical curves with emphasis on their enumerative geometry. Major results include a conceptual proof of the fact that the number of rational tropical plane curves interpolating an appropriate number of general points is independent of the choice of points, the computation of intersection products of Psi- classes on the moduli space of rational tropical curves, a computation of the number of tropical elliptic plane curves of given genus and fixed tropical j-invariant as well as a tropical analogue of the Riemann-Roch theorem for algebraic curves. Mathematics Subject Classification (MSC 2000): 14N35 Gromov-Witten invariants, quantum cohomology 51M20 Polyhedra and polytopes; regular figures, division of spaces 14N10 Enumerative problems (combinatorial problems) Keywords: Tropical geometry, tropical curves, enumerative geometry, metric graphs. dedicated to my parents — in love and gratitude Contents Preface iii Tropical geometry . iii Complex enumerative geometry and tropical curves . iv Results . v Chapter Synopsis . vi Publication of the results . vii Financial support . vii Acknowledgements . vii 1 Moduli spaces of rational tropical curves and maps 1 1.1 Tropical fans . 2 1.2 The space of rational curves . 9 1.3 Intersection products of tropical Psi-classes . 16 1.4 Moduli spaces of rational tropical maps . -
The Rank of the Jacobian of Modular Curves: Analytic Methods
THE RANK OF THE JACOBIAN OF MODULAR CURVES: ANALYTIC METHODS BY EMMANUEL KOWALSKI ii Preface The interaction of analytic and algebraic methods in number theory is as old as Euler, and assumes many guises. Of course, the basic algebraic structures are ever present in any modern mathematical theory, and analytic number theory is no exception, but to algebraic geometry in particular it is indebted for the tremendous advances in under- standing of exponential sums over finite fields, since Andr´eWeil’s proof of the Riemann hypothesis for curves and subsequent deduction of the optimal bound for Kloosterman sums to prime moduli. On the other hand, algebraic number theory has often used input from L-functions; not only as a source of results, although few deep theorems in this area are proved with- out some appeal to Tchebotarev’s density theorem, but also as a source of inspiration, ideas and problems. One particular subject in arithmetic algebraic geometry which is now expected to benefit from analytic methods is the study of the rank of the Mordell-Weil group of an elliptic curve, or more generally of an abelian variety, over a number field. The beautiful conjecture of Birch and Swinnerton-Dyer asserts that this deep arithmetic invariant can be recovered from the order of vanishing of the L-function of the abelian variety at the center of the critical strip. This conjecture naturally opens two lines of investigation: to try to prove it, but here one is, in general, hampered by the necessary prerequisite of proving analytic continuation of the L-function up to this critical point, before any attempt can be made; or to take it for granted and use the information and insight it gives into the nature of the rank as a means of exploring further its behavior.