20 the Modular Equation
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Class Number One Problem for Imaginary Quadratic Fields
MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM JEREMY BOOHER Gauss found 9 imaginary quadratic fields with class number one, and in the early 19th century conjectured he had found all of them. It turns out he was correct, but it took until the mid 20th century to prove this. Theorem 1. Let K be an imaginary quadratic field whose ring of integers has class number one. Then K is one of p p p p p p p p Q(i); Q( −2); Q( −3); Q( −7); Q( −11); Q( −19); Q( −43); Q( −67); Q( −163): There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner's paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner's proof was essentially correct and in fact equiv- alent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number + one problem to finding special points on the modular curve Xns(n). For certain values of n, it is feasible to find all of these points. He remarks that when \N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner." Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner's argument, as modernized in Cox [7], then explain Serre's strategy. -
Arxiv:1810.08742V1 [Math.CV] 20 Oct 2018 Centroid of the Points Zi and Ei = Zi − C
SOME REMARKS ON THE CORRESPONDENCE BETWEEN ELLIPTIC CURVES AND FOUR POINTS IN THE RIEMANN SPHERE JOSE´ JUAN-ZACAR´IAS Abstract. In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in the Hesse normal form. In this setting, we give an alternative proof of the equivalence betweeen the Edwards and the Jacobi normal forms. Also, we give a geometric construction of the cross ratios for 4-point sets in general position. Introduction A complex elliptic curve is by definition a compact Riemann surface of genus 1. By the uniformization theorem, every elliptic curve is conformally equivalent to an algebraic curve given by a cubic polynomial in the form 2 3 3 2 (1) E : y = 4x − g2x − g3; with ∆ = g2 − 27g3 6= 0; this is called the Weierstrass normal form. For computational or geometric pur- poses it may be necessary to find a Weierstrass normal form for an elliptic curve, which could have been given by another equation. At best, we could predict the right changes of variables in order to transform such equation into a Weierstrass normal form, but in general this is a difficult process. A different method to find the normal form (1) for a given elliptic curve, avoiding any change of variables, requires a degree 2 meromorphic function on the elliptic curve, which by a classical theorem always exists and in many cases it is not difficult to compute. -
Dessins D'enfants in N = 2 Generalised Quiver Theories Arxiv
Dessins d'Enfants in N = 2 Generalised Quiver Theories Yang-Hui He1 and James Read2 1Department of Mathematics, City University, London, Northampton Square, London EC1V 0HB, UK; School of Physics, NanKai University, Tianjin, 300071, P.R. China; Merton College, University of Oxford, OX1 4JD, UK [email protected] 2Merton College, University of Oxford, OX1 4JD, UK [email protected] Abstract We study Grothendieck's dessins d'enfants in the context of the N = 2 super- symmetric gauge theories in (3 + 1) dimensions with product SU (2) gauge groups arXiv:1503.06418v2 [hep-th] 15 Sep 2015 which have recently been considered by Gaiotto et al. We identify the precise con- text in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group. 1 Contents 1 Introduction3 2 Dramatis Personæ6 2.1 Skeleton Diagrams . .6 2.2 Moduli Spaces . .8 2.3 BPS Quivers . .9 2.4 Quadratic Differentials and Graphs on Gaiotto Curves . 11 2.5 Dessins d'Enfants and Belyi Maps . 12 2.6 The Modular Group and Congruence Subgroups . 14 3 A Web of Correspondences 15 3.1 Quadratic Differentials and Seiberg-Witten Curves . 16 3.2 Trajectories on Riemann Surfaces and Ideal Triangulations . 16 3.3 Constructing BPS Quivers . 20 3.4 Skeleton Diagrams and BPS Quivers . 21 3.5 Strebel Differentials and Ribbon Graphs . 23 3.5.1 Ribbon Graphs from Strebel Differentials . -
Lectures on Modular Forms. Fall 1997/98
Lectures on Modular Forms. Fall 1997/98 Igor V. Dolgachev October 26, 2017 ii Contents 1 Binary Quadratic Forms1 2 Complex Tori 13 3 Theta Functions 25 4 Theta Constants 43 5 Transformations of Theta Functions 53 6 Modular Forms 63 7 The Algebra of Modular Forms 83 8 The Modular Curve 97 9 Absolute Invariant and Cross-Ratio 115 10 The Modular Equation 121 11 Hecke Operators 133 12 Dirichlet Series 147 13 The Shimura-Tanyama-Weil Conjecture 159 iii iv CONTENTS Lecture 1 Binary Quadratic Forms 1.1 The theory of modular form originates from the work of Carl Friedrich Gauss of 1831 in which he gave a geometrical interpretation of some basic no- tions of number theory. Let us start with choosing two non-proportional vectors v = (v1; v2) and w = 2 (w1; w2) in R The set of vectors 2 Λ = Zv + Zw := fm1v + m2w 2 R j m1; m2 2 Zg forms a lattice in R2, i.e., a free subgroup of rank 2 of the additive group of the vector space R2. We picture it as follows: • • • • • • •Gv • ••• •• • w • • • • • • • • Figure 1.1: Lattice in R2 1 2 LECTURE 1. BINARY QUADRATIC FORMS Let v v B(v; w) = 1 2 w1 w2 and v · v v · w G(v; w) = = B(v; w) · tB(v; w): v · w w · w be the Gram matrix of (v; w). The area A(v; w) of the parallelogram formed by the vectors v and w is given by the formula v · v v · w A(v; w)2 = det G(v; w) = (det B(v; w))2 = det : v · w w · w Let x = mv + nw 2 Λ. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
Arxiv:2009.05223V1 [Math.NT] 11 Sep 2020 Fdegree of Eaeitrse Nfidn Function a finding in Interested Are We Hoe 1.1
COUNTING ELLIPTIC CURVES WITH A RATIONAL N-ISOGENY FOR SMALL N BRANDON BOGGESS AND SOUMYA SANKAR Abstract. We count the number of rational elliptic curves of bounded naive height that have a rational N-isogeny, for N ∈ {2, 3, 4, 5, 6, 8, 9, 12, 16, 18}. For some N, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack. 1. Introduction ′ Let E be an elliptic curve over Q. An isogeny φ : E E between two elliptic curves is said to be cyclic ¯ → of degree N if Ker(φ)(Q) ∼= Z/NZ. Further, it is said to be rational if Ker(φ) is stable under the action of the absolute Galois group, GQ. A natural question one can ask is, how many elliptic curves over Q have a rational cyclic N-isogeny? Henceforth, we will omit the adjective ‘cyclic’, since these are the only types of isogenies we will consider. It is classically known that for N 10 and N = 12, 13, 16, 18, 25, there are infinitely many such elliptic curves. Thus we order them by naive≤ height. An elliptic curve E over Q has a unique minimal Weierstrass equation y2 = x3 + Ax + B where A, B Z and gcd(A3,B2) is not divisible by any 12th power. Define the naive height of E to be ht(E) = max A∈3, B 2 . -
Remarks on Codes from Modular Curves: MAGMA Applications
Remarks on codes from modular curves: MAGMA applications∗ David Joyner and Salahoddin Shokranian 3-30-2004 Contents 1 Introduction 2 2 Shimura curves 3 2.1 Arithmeticsubgroups....................... 3 2.2 Riemannsurfacesasalgebraiccurves . 4 2.3 An adelic view of arithmetic quotients . 5 2.4 Hecke operators and arithmetic on X0(N) ........... 8 2.5 Eichler-Selbergtraceformula. 10 2.6 The curves X0(N)ofgenus1 .................. 12 3 Codes 14 3.1 Basicsonlinearcodes....................... 14 3.2 SomebasicsonAGcodes. 16 arXiv:math/0403548v1 [math.NT] 31 Mar 2004 3.3 SomeestimatesonAGcodes. 17 4 Examples 19 4.1 Weight enumerators of some elliptic codes . 20 4.2 Thegeneratormatrix(apr´esdesGoppa) . 22 5 Concluding comments 24 ∗This paper is a modification of the paper [JS]. Here we use [MAGMA], version 2.10, instead of MAPLE. Some examples have been changed and minor corrections made. MSC 2000: 14H37, 94B27,20C20,11T71,14G50,05E20,14Q05 1 1 INTRODUCTION 2 1 Introduction Suppose that V is a smooth projective variety over a finite field k. An important problem in arithmetical algebraic geometry is the calculation of the number of k-rational points of V , V (k) . The work of Goppa [G] and others have shown its importance in geometric| | coding theory as well. We refer to this problem as the counting problem. In most cases it is very hard to find an explicit formula for the number of points of a variety over a finite field. When the variety is a “Shimura variety” defined by certain group theoret- ical conditions (see 2 below), methods from non-abelian harmonic analysis on groups can be used§ to find an explicit solution for the counting problem. -
Complex Analysis
Complex Analysis Andrew Kobin Fall 2010 Contents Contents Contents 0 Introduction 1 1 The Complex Plane 2 1.1 A Formal View of Complex Numbers . .2 1.2 Properties of Complex Numbers . .4 1.3 Subsets of the Complex Plane . .5 2 Complex-Valued Functions 7 2.1 Functions and Limits . .7 2.2 Infinite Series . 10 2.3 Exponential and Logarithmic Functions . 11 2.4 Trigonometric Functions . 14 3 Calculus in the Complex Plane 16 3.1 Line Integrals . 16 3.2 Differentiability . 19 3.3 Power Series . 23 3.4 Cauchy's Theorem . 25 3.5 Cauchy's Integral Formula . 27 3.6 Analytic Functions . 30 3.7 Harmonic Functions . 33 3.8 The Maximum Principle . 36 4 Meromorphic Functions and Singularities 37 4.1 Laurent Series . 37 4.2 Isolated Singularities . 40 4.3 The Residue Theorem . 42 4.4 Some Fourier Analysis . 45 4.5 The Argument Principle . 46 5 Complex Mappings 47 5.1 M¨obiusTransformations . 47 5.2 Conformal Mappings . 47 5.3 The Riemann Mapping Theorem . 47 6 Riemann Surfaces 48 6.1 Holomorphic and Meromorphic Maps . 48 6.2 Covering Spaces . 52 7 Elliptic Functions 55 7.1 Elliptic Functions . 55 7.2 Elliptic Curves . 61 7.3 The Classical Jacobian . 67 7.4 Jacobians of Higher Genus Curves . 72 i 0 Introduction 0 Introduction These notes come from a semester course on complex analysis taught by Dr. Richard Carmichael at Wake Forest University during the fall of 2010. The main topics covered include Complex numbers and their properties Complex-valued functions Line integrals Derivatives and power series Cauchy's Integral Formula Singularities and the Residue Theorem The primary reference for the course and throughout these notes is Fisher's Complex Vari- ables, 2nd edition. -
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. -
Zeros of Differential Polynomials in Real Meromorphic Functions
Zeros of differential polynomials in real meromorphic functions Walter Bergweiler∗, Alex Eremenko† and Jim Langley June 30, 2004 Abstract We investigate when differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if g is a real transcendental meromorphic function, c R 0 n ∈ \{ } and n 3 is an integer, then g0g c has infinitely many non-real zeros.≥ If g has only finitely many poles,− then this holds for n 2. Related results for rational functions g are also considered. ≥ 1 Introduction and results Our starting point is the following result due to Sheil-Small [15] which solved a longstanding conjecture. 2 Theorem A Let f be a real polynomial of degree d. Then f 0 + f has at least d 1 distinct non-real zeros which are not zeros of f . − In the special case that f has only real roots this theorem is due to Pr¨ufer [14, Ch. V, 182]; see [15] for further discussion of the result. The following Theorem B is an analogue of Theorem A for transcendental meromorphic functions. Here “meromorphic” will mean “meromorphic in the complex plane” unless explicitly stated otherwise. A meromorphic function is called real if it maps the real axis R to R . ∪{∞} ∗Supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.), grant no. G-643-117.6/1999 †Supported by NSF grants DMS-0100512 and DMS-0244421. 1 Theorem B Let f be a real transcendental meromorphic function with 2 finitely many poles. Then f 0 + f has infinitely many non-real zeros which are not zeros of f . -
Notes on Riemann's Zeta Function
NOTES ON RIEMANN’S ZETA FUNCTION DRAGAN MILICIˇ C´ 1. Gamma function 1.1. Definition of the Gamma function. The integral ∞ Γ(z)= tz−1e−tdt Z0 is well-defined and defines a holomorphic function in the right half-plane {z ∈ C | Re z > 0}. This function is Euler’s Gamma function. First, by integration by parts ∞ ∞ ∞ Γ(z +1)= tze−tdt = −tze−t + z tz−1e−t dt = zΓ(z) Z0 0 Z0 for any z in the right half-plane. In particular, for any positive integer n, we have Γ(n) = (n − 1)Γ(n − 1)=(n − 1)!Γ(1). On the other hand, ∞ ∞ Γ(1) = e−tdt = −e−t = 1; Z0 0 and we have the following result. 1.1.1. Lemma. Γ(n) = (n − 1)! for any n ∈ Z. Therefore, we can view the Gamma function as a extension of the factorial. 1.2. Meromorphic continuation. Now we want to show that Γ extends to a meromorphic function in C. We start with a technical lemma. Z ∞ 1.2.1. Lemma. Let cn, n ∈ +, be complex numbers such such that n=0 |cn| converges. Let P S = {−n | n ∈ Z+ and cn 6=0}. Then ∞ c f(z)= n z + n n=0 X converges absolutely for z ∈ C − S and uniformly on bounded subsets of C − S. The function f is a meromorphic function on C with simple poles at the points in S and Res(f, −n)= cn for any −n ∈ S. 1 2 D. MILICIˇ C´ Proof. Clearly, if |z| < R, we have |z + n| ≥ |n − R| for all n ≥ R. -
Congruences Between Modular Forms
CONGRUENCES BETWEEN MODULAR FORMS FRANK CALEGARI Contents 1. Basics 1 1.1. Introduction 1 1.2. What is a modular form? 4 1.3. The q-expansion priniciple 14 1.4. Hecke operators 14 1.5. The Frobenius morphism 18 1.6. The Hasse invariant 18 1.7. The Cartier operator on curves 19 1.8. Lifting the Hasse invariant 20 2. p-adic modular forms 20 2.1. p-adic modular forms: The Serre approach 20 2.2. The ordinary projection 24 2.3. Why p-adic modular forms are not good enough 25 3. The canonical subgroup 26 3.1. Canonical subgroups for general p 28 3.2. The curves Xrig[r] 29 3.3. The reason everything works 31 3.4. Overconvergent p-adic modular forms 33 3.5. Compact operators and spectral expansions 33 3.6. Classical Forms 35 3.7. The characteristic power series 36 3.8. The Spectral conjecture 36 3.9. The invariant pairing 38 3.10. A special case of the spectral conjecture 39 3.11. Some heuristics 40 4. Examples 41 4.1. An example: N = 1 and p = 2; the Watson approach 41 4.2. An example: N = 1 and p = 2; the Coleman approach 42 4.3. An example: the coefficients of c(n) modulo powers of p 43 4.4. An example: convergence slower than O(pn) 44 4.5. Forms of half integral weight 45 4.6. An example: congruences for p(n) modulo powers of p 45 4.7. An example: congruences for the partition function modulo powers of 5 47 4.8.