DOCSLIB.ORG

§2. Elliptic Curves: J-Invariant (Jan 31, Feb 4,7,9,11,14) After

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
§2. Elliptic Curves: J-Invariant (Jan 31, Feb 4,7,9,11,14) After 24 JENIA TEVELEV §2. Elliptic curves: j-invariant (Jan 31, Feb 4,7,9,11,14) After the projective line P1, the easiest algebraic curve to understand is an elliptic curve (Riemann surface of genus 1). Let M = isom. classes of elliptic curves . 1 { } We are going to assign to each elliptic curve a number, called its j-invariant and prove that 1 M1 = Aj . 1 1 So as a space M1 A is not very interesting. However, understanding A ! as a moduli space of elliptic curves leads to some breath-taking mathemat- ics. More generally, we introduce M = isom. classes of smooth projective curves of genus g g { } and M = isom. classes of curves C of genus g with points p , . , p C . g,n { 1 n ∈ } We will return to these moduli spaces later in the course. But first let us recall some basic facts about algebraic curves = compact Riemann surfaces. We refer to [G] and [Mi] for a rigorous and detailed exposition. §2.1. Algebraic functions, algebraic curves, and Riemann surfaces. The theory of algebraic curves has roots in analysis of Abelian integrals. An easiest example is the elliptic integral: in 1655 Wallis began to study the arc length of an ellipse (X/a)2 + (Y/b)2 = 1. The equation for the ellipse can be solved for Y : Y = (b/a) (a2 X2), − and this can easily be differentiated !to find bX Y ! = − . a√a2 X2 − 2 This is squared and put into the integral 1 + (Y !) dX for the arc length. Now the substitution x = X/a results in " ! 1 e2x2 s = a − dx, 1 x2 # $ − between the limits 0 and X/a, where e = 1 (b/a)2 is the eccentricity. − This is the result for the arc length from X = 0 to X/a in the first quadrant, ! beginning at the point (0, b) on the Y -axis. Notice that we can rewrite this integral as a ae2x2 − dx = P (x, y) dx, (1 e2x2)(1 x2) # − − # where P (x, y) is a rational! function and y is a solution of the equation y2 = (1 e2x2)(1 x2). − − This equation defines an elliptic curve! y is an example of an algebraic func- tion. Namely, an algebraic function y = y(x) is a solution of the equation n n 1 y + a1(x)y − + . + an(x) = 0, (2.1.1) MODULI SPACES AND INVARIANT THEORY 25 where ai(x) C(x) are rational functions (ratios of polynomials). Without ∈ loss of generality, we can assume that this equation is irreducible over C(x). For example, we can get nested radicals y(x) = 3 x3 7x√x, although − after Abel and Galois we know that not any algebraic function is a nested ! radical (for n 5)! An Abelian integral is the integral of the form ≥ P (x, y) dx # where P (x, y) is some rational function. All rational functions P (x, y) form a field K, which is finitely generated and of transcendence degree 1 over C (because x and y are algebraically dependent). And vice versa, given a field K such that tr.deg.CK = 1, we can let x be an element transcendent over C. Then K/C(x) is a finitely generated, algebraic (hence finite), and sepa- rable (because we are in characteristic 0) field extension. By a theorem on the primitive element, we have K = C(x, y), where y is a root of an irreducible polynomial (2.1.1). Notice that of course there are infinitely many choices for x and y, thus the equation (2.1.1) is not determined by the field exten- sion. It is not important from the perspective of computing integrals either (we can always do u-substitutions). So on a purely algebraic level we can study isomorphism classes of f.g. field extensions K/C with tr.deg.CK = 1. Clearing denominators in (2.1.1) gives an irreducible affine plane curve 2 C = f(x, y) = 0 A { } ⊂ and its projective completion, an irreducible plane curve in P2. Recall that the word curve here means “of dimension 1”, and dimension of an irreducible affine or projective variety is by definition the transcendence degree of the field of rational functions C(C). So we can restate our moduli problem as understanding birational equivalence classes of irreducible plane curves. Here we use the following definition 2.1.2. DEFINITION. Irreducible (affine or projective) algebraic varieties X and Y are called birationally equivalent if their fields of rational functions C(X) and C(Y ) are isomorphic. More generally, we can consider an arbitrary irreducible affine or projec- tive curve C An or C Pn: ⊂ ⊂ birational equivalence classes of irreducible algebraic curves. This gives the same class of fields, so we are not gaining any new objects. Geometrically, for any such curve a general linear projection n 2 P !!" P is birational onto its image. Let us remind some basic facts related to regular maps (morphisms) and rational maps (see lectures on the Grassmannian 1.8.2 for definitions): 26 JENIA TEVELEV 2.1.3. THEOREM ([X, 2.3.3]). If C is a smooth curve and f : C Pn is a → rational map then f is regular. More generally, if X is a smooth algebraic vari- ety and f : X Pn is a rational map then the indeterminancy locus of f has → codimension 2. 2.1.4. THEOREM ([X, 1.5.2]). If X is a projective variety and f : X Pn is a → regular morphism then f(X) is closed (i.e. also a projective variety). 2.1.5. THEOREM. For any algebraic curve C, there exists a smooth projective curve C! birational to C. Taken together, these facts imply that our moduli problem can be rephrased as the study of isom. classes of smooth projective algebraic curves. 2.1.6. REMARK. Theorem 2.1.5 is proved in [G] by take a plane model C P2 (by projecting Pn P2). • ⊂ !!" compute the normalization C C. • ! → Construction of the normalization in [G] is transcendental: one first con- structs C! as a compact Riemann surface and then invokes a general fact (see below) that it is in fact a projective algebraic curve. Notice however that there exist purely algebraic approaches to desingularization by either (a) algebraic normalization (integral closure in the field of fractions) [X, 2.5.3] or (b) blow-ups [X, 4.4.1] . The analytic approach is to consider Riemann surfaces instead of alge- braic curves. It turns out that this gives the same moduli problem: biholomorphic isom. classes of compact Riemann surfaces. It is easy to show that a smooth algebraic curve is a compact Riemann sur- face. It is harder but not too hard to show that a holomorphic map between two smooth algebraic curves is in fact a regular morphism, for example ant meromorphic function is in fact a rational function. But a really difficult part of the theory is to show that any compact Riemann surface is an al- gebraic curve. It is hard to construct a single meromorphic function, but once this is done the rest is easy. This is done by analysis: to construct a harmonic function on a Riemann surface one (following Klein and Rie- mann): “This is easily done by covering the Riemann surface with tin foil... Suppose the poles of a galvanic battery of a given voltage are placed at the points A1 and A2. A current arises whose potential u is single-valued, con- tinuous, and satisfies the equation ∆u = 0 across the entire surface, except for the points A1 and A2, which are discontinuity points of the function." A modern treatment can be found in [GH], where a much more general Kodaira embedding theorem is discussed. §2.2. Genus. The genus g of a smooth projective algebraic curve can be computed as follows: topologically: the number of handles. • analytically: the dimension of the space of holomorphic differentials. • algebraically: the dimension of the space of rational differentials with- • out poles ω = a dx, where a, x are rational functions on C. MODULI SPACES AND INVARIANT THEORY 27 One also has the following genus formula: 2g 2 = (number of zeros) (number of poles) (2.2.1) − − of any meromorphic (=rational) differential ω. For example, a form ω = dx on P1 at the chart x = 1/y at infinity is dx = d(1/y) = (1/y2)dy. − So it has no zeros and a pole of order 2 at infinity, which agrees with (2.2.1). A smooth plane curve C P2 of degree d has genus ⊂ (d 1)(d 2) g = − − (2.2.2) 2 (d 1)(d 2) (more generally, if C has only nodal singularities then g = − − δ, 2 − where δ is the number of nodes). There is a nice choice of a holomorphic form on C: suppose C A2 is given by the equation f(x, y) = 0. Differen- ∩ x,y tiating this equation shows that dx dy = fy − fx along C, where the first (resp. second) expression is valid at points where x (resp. y) is a holomorphic coordinate. This gives a non-vanishing holomor- phic form ω on C A2. A simple calculation shows that ω has zeros at points ∩ at infinity each of multiplicity d 3. Combined with (2.2.1), this gives − 2g 2 = d(d 3), − − which is equivalent to (2.2.2). §2.3. Divisors on curves. A divisor D is just a linear combination aiPi of points Pi C with integer multiplicities. Its degree is defined as ∈ % deg D = ai. If f is a rational (=meromorphic) function& on C, we can define its divisor (f) = ordP (f)P, P C &∈ where ordP (f) is the order of zeros (or poles) of f at P .
Load more

Recommended publications
  • Nonintersecting Brownian Motions on the Unit Circle: Noncritical Cases
    The Annals of Probability 2016, Vol. 44, No. 2, 1134–1211 DOI: 10.1214/14-AOP998 c Institute of Mathematical Statistics, 2016 NONINTERSECTING BROWNIAN MOTIONS ON THE UNIT CIRCLE By Karl Liechty and Dong Wang1 DePaul University and National University of Singapore We consider an ensemble of n nonintersecting Brownian particles on the unit circle with diffusion parameter n−1/2, which are condi- tioned to begin at the same point and to return to that point after time T , but otherwise not to intersect. There is a critical value of T which separates the subcritical case, in which it is vanishingly un- likely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n , and →∞ in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode pro- cesses in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev´e II equation of size 2 2. The proofs are based on the determinantal × structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral. 1. Introduction. The probability models of nonintersecting Brownian motions have been studied extensively in last decade; see Tracy and Widom (2004, 2006), Adler and van Moerbeke (2005), Adler, Orantin and van Moer- beke (2010), Delvaux, Kuijlaars and Zhang (2011), Johansson (2013), Ferrari and Vet˝o(2012), Katori and Tanemura (2007) and Schehr et al.
    [Show full text]
  • From String Theory and Moonshine to Vertex Algebras
    Preample From string theory and Moonshine to vertex algebras Bong H. Lian Department of Mathematics Brandeis University [email protected] Harvard University, May 22, 2020 Dedicated to the memory of John Horton Conway December 26, 1937 – April 11, 2020. Preample Acknowledgements: Speaker’s collaborators on the theory of vertex algebras: Andy Linshaw (Denver University) Bailin Song (University of Science and Technology of China) Gregg Zuckerman (Yale University) For their helpful input to this lecture, special thanks to An Huang (Brandeis University) Tsung-Ju Lee (Harvard CMSA) Andy Linshaw (Denver University) Preample Disclaimers: This lecture includes a brief survey of the period prior to and soon after the creation of the theory of vertex algebras, and makes no claim of completeness – the survey is intended to highlight developments that reflect the speaker’s own views (and biases) about the subject. As a short survey of early history, it will inevitably miss many of the more recent important or even towering results. Egs. geometric Langlands, braided tensor categories, conformal nets, applications to mirror symmetry, deformations of VAs, .... Emphases are placed on the mutually beneficial cross-influences between physics and vertex algebras in their concurrent early developments, and the lecture is aimed for a general audience. Preample Outline 1 Early History 1970s – 90s: two parallel universes 2 A fruitful perspective: vertex algebras as higher commutative algebras 3 Classification: cousins of the Moonshine VOA 4 Speculations The String Theory Universe 1968: Veneziano proposed a model (using the Euler beta function) to explain the ‘st-channel crossing’ symmetry in 4-meson scattering, and the Regge trajectory (an angular momentum vs binding energy plot for the Coulumb potential).
    [Show full text]
  • Lecture 9 Riemann Surfaces, Elliptic Functions Laplace Equation
    Lecture 9 Riemann surfaces, elliptic functions Laplace equation Consider a Riemannian metric gµν in two dimensions. In two dimensions it is always possible to choose coordinates (x,y) to diagonalize it as ds2 = Ω(x,y)(dx2 + dy2). We can then combine them into a complex combination z = x+iy to write this as ds2 =Ωdzdz¯. It is actually a K¨ahler metric since the condition ∂[i,gj]k¯ = 0 is trivial if i, j = 1. Thus, an orientable Riemannian manifold in two dimensions is always K¨ahler. In the diagonalized form of the metric, the Laplace operator is of the form, −1 ∆=4Ω ∂z∂¯z¯. Thus, any solution to the Laplace equation ∆φ = 0 can be expressed as a sum of a holomorphic and an anti-holomorphid function. ∆φ = 0 φ = f(z)+ f¯(¯z). → In the following, we assume Ω = 1 so that the metric is ds2 = dzdz¯. It is not difficult to generalize our results for non-constant Ω. Now, we would like to prove the following formula, 1 ∂¯ = πδ(z), z − where δ(z) = δ(x)δ(y). Since 1/z is holomorphic except at z = 0, the left-hand side should vanish except at z = 0. On the other hand, by the Stokes theorem, the integral of the left-hand side on a disk of radius r gives, 1 i dz dxdy ∂¯ = = π. Zx2+y2≤r2 z 2 I|z|=r z − This proves the formula. Thus, the Green function G(z, w) obeying ∆zG(z, w) = 4πδ(z w), − should behave as G(z, w)= log z w 2 = log(z w) log(¯z w¯), − | − | − − − − near z = w.
    [Show full text]
  • 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 = ∅.
    [Show full text]
  • Arxiv:1911.10534V3 [Math.AT] 17 Apr 2020 Statement
    THE ANDO-HOPKINS-REZK ORIENTATION IS SURJECTIVE SANATH DEVALAPURKAR Abstract. We show that the map π∗MString ! π∗tmf induced by the Ando-Hopkins-Rezk orientation is surjective. This proves an unpublished claim of Hopkins and Mahowald. We do so by constructing an E1-ring B and a map B ! MString such that the composite B ! MString ! tmf is surjective on homotopy. Applications to differential topology, and in particular to Hirzebruch's prize question, are discussed. 1. Introduction The goal of this paper is to show the following result. Theorem 1.1. The map π∗MString ! π∗tmf induced by the Ando-Hopkins-Rezk orientation is surjective. This integral result was originally stated as [Hop02, Theorem 6.25], but, to the best of our knowledge, no proof has appeared in the literature. In [HM02], Hopkins and Mahowald give a proof sketch of Theorem 1.1 for elements of π∗tmf of Adams-Novikov filtration 0. The analogue of Theorem 1.1 for bo (namely, the statement that the map π∗MSpin ! π∗bo induced by the Atiyah-Bott-Shapiro orientation is surjective) is classical [Mil63]. In Section2, we present (as a warmup) a proof of this surjectivity result for bo via a technique which generalizes to prove Theorem 1.1. We construct an E1-ring A with an E1-map A ! MSpin. The E1-ring A is a particular E1-Thom spectrum whose mod 2 homology is given by the polynomial subalgebra 4 F2[ζ1 ] of the mod 2 dual Steenrod algebra. The Atiyah-Bott-Shapiro orientation MSpin ! bo is an E1-map, and so the composite A ! MSpin ! bo is an E1-map.
    [Show full text]
  • 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.
    [Show full text]
  • Applications of Elliptic Functions in Classical and Algebraic Geometry
    APPLICATIONS OF ELLIPTIC FUNCTIONS IN CLASSICAL AND ALGEBRAIC GEOMETRY Jamie Snape Collingwood College, University of Durham Dissertation submitted for the degree of Master of Mathematics at the University of Durham It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathe- maticians have a good ear; some not... That’s life. JEAN-PIERRE SERRE Jean-Pierre Serre (1926–). Quote taken from Serre (1991). i Contents I Background 1 1 Elliptic Functions 2 1.1 Motivation ............................... 2 1.2 Definition of an elliptic function ................... 2 1.3 Properties of an elliptic function ................... 3 2 Jacobi Elliptic Functions 6 2.1 Motivation ............................... 6 2.2 Definitions of the Jacobi elliptic functions .............. 7 2.3 Properties of the Jacobi elliptic functions ............... 8 2.4 The addition formulæ for the Jacobi elliptic functions . 10 2.5 The constants K and K 0 ........................ 11 2.6 Periodicity of the Jacobi elliptic functions . 11 2.7 Poles and zeroes of the Jacobi elliptic functions . 13 2.8 The theta functions .......................... 13 3 Weierstrass Elliptic Functions 16 3.1 Motivation ............................... 16 3.2 Definition of the Weierstrass elliptic function . 17 3.3 Periodicity and other properties of the Weierstrass elliptic function . 18 3.4 A differential equation satisfied by the Weierstrass elliptic function . 20 3.5 The addition formula for the Weierstrass elliptic function . 21 3.6 The constants e1, e2 and e3 .....................
    [Show full text]
  • Our Mathematical Universe: I. How the Monster Group Dictates All of Physics
    October, 2011 PROGRESS IN PHYSICS Volume 4 Our Mathematical Universe: I. How the Monster Group Dictates All of Physics Franklin Potter Sciencegems.com, 8642 Marvale Drive, Huntington Beach, CA 92646. E-mail: [email protected] A 4th family b’ quark would confirm that our physical Universe is mathematical and is discrete at the Planck scale. I explain how the Fischer-Greiss Monster Group dic- tates the Standard Model of leptons and quarks in discrete 4-D internal symmetry space and, combined with discrete 4-D spacetime, uniquely produces the finite group Weyl E8 x Weyl E8 = “Weyl” SO(9,1). The Monster’s j-invariant function determines mass ratios of the particles in finite binary rotational subgroups of the Standard Model gauge group, dictates Mobius¨ transformations that lead to the conservation laws, and connects interactions to triality, the Leech lattice, and Golay-24 information coding. 1 Introduction 5. Both 4-D spacetime and 4-D internal symmetry space are discrete at the Planck scale, and both spaces can The ultimate idea that our physical Universe is mathematical be telescoped upwards mathematically by icosians to at the fundamental scale has been conjectured for many cen- 8-D spaces that uniquely combine into 10-D discrete turies. In the past, our marginal understanding of the origin spacetime with discrete Weyl E x Weyl E symmetry of the physical rules of the Universe has been peppered with 8 8 (not the E x E Lie group of superstrings/M-theory). huge gaps, but today our increased understanding of funda- 8 8 mental particles promises to eliminate most of those gaps to 6.
    [Show full text]
  • An Eloquent Formula for the Perimeter of an Ellipse Semjon Adlaj
    An Eloquent Formula for the Perimeter of an Ellipse Semjon Adlaj he values of complete elliptic integrals Define the arithmetic-geometric mean (which we of the first and the second kind are shall abbreviate as AGM) of two positive numbers expressible via power series represen- x and y as the (common) limit of the (descending) tations of the hypergeometric function sequence xn n∞ 1 and the (ascending) sequence { } = 1 (with corresponding arguments). The yn n∞ 1 with x0 x, y0 y. { } = = = T The convergence of the two indicated sequences complete elliptic integral of the first kind is also known to be eloquently expressible via an is said to be quadratic [7, p. 588]. Indeed, one might arithmetic-geometric mean, whereas (before now) readily infer that (and more) by putting the complete elliptic integral of the second kind xn yn rn : − , n N, has been deprived such an expression (of supreme = xn yn ∈ + power and simplicity). With this paper, the quest and observing that for a concise formula giving rise to an exact it- 2 2 √xn √yn 1 rn 1 rn erative swiftly convergent method permitting the rn 1 − + − − + = √xn √yn = 1 rn 1 rn calculation of the perimeter of an ellipse is over! + + + − 2 2 2 1 1 rn r Instead of an Introduction − − n , r 4 A recent survey [16] of formulae (approximate and = n ≈ exact) for calculating the perimeter of an ellipse where the sign for approximate equality might ≈ is erroneously resuméd: be interpreted here as an asymptotic (as rn tends There is no simple exact formula: to zero) equality.
    [Show full text]
  • Elliptic Functions and Elliptic Curves: 1840-1870
    Elliptic functions and elliptic curves: 1840-1870 2017 Structure 1. Ten minutes context of mathematics in the 19th century 2. Brief reminder of results of the last lecture 3. Weierstrass on elliptic functions and elliptic curves. 4. Riemann on elliptic functions Guide: Stillwell, Mathematics and its History, third ed, Ch. 15-16. Map of Prussia 1867: the center of the world of mathematics The reform of the universities in Prussia after 1815 Inspired by Wilhelm von Humboldt (1767-1835) Education based on neo-humanistic ideals: academic freedom, pure research, Bildung, etc. Anti-French. Systematic policy for appointing researchers as university professors. Research often shared in lectures to advanced students (\seminarium") Students who finished at universities could get jobs at Gymnasia and continue with scientific research This system made Germany the center of the mathematical world until 1933. Personal memory (1983-1984) Otto Neugebauer (1899-1990), manager of the G¨ottingen Mathematical Institute in 1933, when the Nazis took over. Structure of this series of lectures 2 p 2 ellipse: Apollonius of Perga (y = px − d x , something misses = Greek: elleipei) elliptic integrals, arc-length of an ellipse; also arc length of lemniscate elliptic functions: inverses of such integrals elliptic curves: curves of genus 1 (what this means will become clear) Results from the previous lecture by Steven Wepster: Elliptic functions we start with lemniscate Arc length is elliptic integral Z x dt u(x) = p 4 0 1 − t Elliptic function x(u) = sl(u) periodic with period 2! length of lemniscate. (x2 + y 2)2 = x2 − y 2 \sin lemn" polar coordinates r 2 = cos2θ x Some more results from u(x) = R p dt , x(u) = sl(u) 0 1−t4 sl(u(x)) = x so sl0(u(x)) · u0(x) = 1 so p 0 1 4 p 4 sl (u) = u0(x) = 1 − x = 1 − sl(u) In modern terms, this provides a parametrisation 0 x = sl(u); y = sl (pu) for the curve y = 1 − x4, y 2 = (1 − x4).
    [Show full text]
  • K3 Surfaces, N= 4 Dyons, and the Mathieu Group
    K3 Surfaces, =4 Dyons, N and the Mathieu Group M24 Miranda C. N. Cheng Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural inter- pretation in terms of the dimensions of representations of the largest Mathieu group M24. In this paper we first give further evidence for this possibility by studying the elliptic genus of K3 surfaces twisted by some simple symplectic automorphisms. These partition functions with insertions of elements of M24 (the McKay-Thompson series) give further information about the relevant representation. We then point out that this new “moonshine” for the largest Mathieu group is con- nected to an earlier observation on a moonshine of M24 through the 1/4-BPS spectrum of K3 T 2-compactified type II string theory. This insight on the symmetry× of the theory sheds new light on the gener- alised Kac-Moody algebra structure appearing in the spectrum, and leads to predictions for new elliptic genera of K3, perturbative spec- arXiv:1005.5415v2 [hep-th] 3 Jun 2010 trum of the toroidally compactified heterotic string, and the index for the 1/4-BPS dyons in the d = 4, = 4 string theory, twisted by elements of the group of stringy K3N isometries. 1 1 Introduction and Summary Recently there have been two new observations relating K3 surfaces and the largest Mathieu group M24. They seem to suggest that the sporadic group M24 naturally acts on the spectrum of K3-compactified string theory.
    [Show full text]
  • Around Symmetries of Vertex Operator Algebras
    Around Symmetries of Vertex Operator Algebras Atsushi Matsuo The University of Tokyo Corrected and revised version of the slides presented for Graphs and Groups, Representations and Relations August 9, 2018 Novosibirsk State University A. Matsuo Around Symmetries of Vertex Operator Algebras 1 x1 Vertex algebras and vertex operator algebras | R.E. Borcherds: Vertex algebras, Kac-Moody algebras and the mon- ster, Proc. Nat. Acad. Sci. U.S.A. 83, (1986), 3068{3071. | I. Frenkel, J. Lepowsky, and A. Meurman: Vertex operator algebras and the Monster, Academic Press, New York, 1988. A. Matsuo Around Symmetries of Vertex Operator Algebras 2 Vertex algebras. A vertex algebra is a vector space V , say over C, equipped with X / −n−1 • A bilinear map V × V V ((z)), (a; b) 7! a(n)bz . n2Z • A nonzero vector 1 2 V , called the vacuum vector. satisfying the following axioms: 2 (1) Borcherds( ) identities. The following holds for all p; q; r Z, X1 p (a b) − c i (r+i) (p+q i) i=0 X1 ( )( ) i r r = (−1) a − (b c) − (−1) b − (a c) : i (p+r i) (q+i) (q+r i) (p+i) i=0 (2) Creation property. { 0; (n ≥ 0), a(n)1 = a; (n = −1). A. Matsuo Around Symmetries of Vertex Operator Algebras 3 Vertex operators. A vertex algebra is specified by the triple (V; Y; 1), where X / −1 −n−1 Y : V (End V )[[z; z ]]; a 7! Y (a; z) = a(n)z : n The generating function Y (a; z) is often called the vertex operator, al- though this term is also used in broader context.
    [Show full text]