Riemann Surfaces
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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. -
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. -
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. -
The Universal Covering Space of the Twice Punctured Complex Plane
The Universal Covering Space of the Twice Punctured Complex Plane Gustav Jonzon U.U.D.M. Project Report 2007:12 Examensarbete i matematik, 20 poäng Handledare och examinator: Karl-Heinz Fieseler Mars 2007 Department of Mathematics Uppsala University Abstract In this paper we present the basics of holomorphic covering maps be- tween Riemann surfaces and we discuss and study the structure of the holo- morphic covering map given by the modular function λ onto the base space C r f0; 1g from its universal covering space H, where the plane regions C r f0; 1g and H are considered as Riemann surfaces. We develop the basics of Riemann surface theory, tie it together with complex analysis and topo- logical covering space theory as well as the study of transformation groups on H and elliptic functions. We then utilize these tools in our analysis of the universal covering map λ : H ! C r f0; 1g. Preface First and foremost, I would like to express my gratitude to the supervisor of this paper, Karl-Heinz Fieseler, for suggesting the project and for his kind help along the way. I would also like to thank Sandra for helping me out when I got stuck with TEX and Daja for mathematical feedback. Contents 1 Introduction and Background 1 2 Preliminaries 2 2.1 A Quick Review of Topological Covering Spaces . 2 2.1.1 Covering Maps and Their Basic Properties . 2 2.2 An Introduction to Riemann Surfaces . 6 2.2.1 The Basic Definitions . 6 2.2.2 Elementary Properties of Holomorphic Mappings Between Riemann Surfaces . -
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 = ∅. -
The Riemann-Roch Theorem
THE RIEMANN-ROCH THEOREM GAL PORAT Abstract. These are notes for a talk which introduces the Riemann-Roch Theorem. We present the theorem in the language of line bundles and discuss its basic consequences, as well as an application to embeddings of curves in projective space. 1. Algebraic Curves and Riemann Surfaces A deep theorem due to Riemann says that every compact Riemann surface has a nonconstant meromorphic function. This leads to an equivalence {smooth projective complex algebraic curves} !{compact Riemann surfaces}. Topologically, all such Riemann surfaces are orientable compact manifolds, which are all genus g surfaces. Example 1.1. (1) The curve 1 corresponds to the Riemann sphere. PC (2) The (smooth projective closure of the) curve C : y2 = x3 − x corresponds to the torus C=(Z + iZ). Throughout the talk we will frequentely use both the point of view of algebraic curves and of Riemann surfaces in the context of the Riemann-Roch theorem. 2. Divisors Let C be a complex curve. A divisor of C is an element of the group Div(C) := ⊕P 2CZ. There is a natural map deg : Div(C) ! Z obtained by summing the coordinates. Functions f 2 K(C)× give rise to divisors on C. Indeed, there is a map div : K(C)× ! Div(C), given by setting X div(f) = vP (f)P: P 2C One can prove that deg(div(f)) = 0 for f 2 K(C)×; essentially this is a consequence of the argument principle, because the sum over all residues of any differential of a compact Riemann surface is equal to 0. -
Moduli Spaces of Riemann Surfaces
MODULI SPACES OF RIEMANN SURFACES ALEX WRIGHT Contents 1. Different ways to build and deform Riemann surfaces 2 2. Teichm¨uller space and moduli space 4 3. The Deligne-Mumford compactification of moduli space 11 4. Jacobians, Hodge structures, and the period mapping 13 5. Quasiconformal maps 16 6. The Teichm¨ullermetric 18 7. Extremal length 24 8. Nielsen-Thurston classification of mapping classes 26 9. Teichm¨uller space is a bounded domain 29 10. The Weil-Petersson K¨ahlerstructure 40 11. Kobayashi hyperbolicity 46 12. Geometric Shafarevich and Mordell 48 References 57 These are course notes for a one quarter long second year graduate class at Stanford University taught in Winter 2017. All of the material presented is classical. For the softer aspects of Teichm¨ullertheory, our reference was the book of Farb-Margalit [FM12], and for Teichm¨ullertheory as a whole our comprehensive ref- erence was the book of Hubbard [Hub06]. We also used Gardiner and Lakic's book [GL00] and McMullen's course notes [McM] as supple- mental references. For period mappings our reference was the book [CMSP03]. For Geometric Shafarevich our reference was McMullen's survey [McM00]. The author thanks Steve Kerckhoff, Scott Wolpert, and especially Jeremy Kahn for many helpful conversations. These are rough notes, hastily compiled, and may contain errors. Corrections are especially welcome, since these notes will be re-used in the future. 2 WRIGHT 1. Different ways to build and deform Riemann surfaces 1.1. Uniformization. A Riemann surface is a topological space with an atlas of charts to the complex plane C whose translation functions are biholomorphisms. -
Introduction to Complex Analysis Michael Taylor
Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26. -
The Measurable Riemann Mapping Theorem
CHAPTER 1 The Measurable Riemann Mapping Theorem 1.1. Conformal structures on Riemann surfaces Throughout, “smooth” will always mean C 1. All surfaces are assumed to be smooth, oriented and without boundary. All diffeomorphisms are assumed to be smooth and orientation-preserving. It will be convenient for our purposes to do local computations involving metrics in the complex-variable notation. Let X be a Riemann surface and z x iy be a holomorphic local coordinate on X. The pair .x; y/ can be thoughtD C of as local coordinates for the underlying smooth surface. In these coordinates, a smooth Riemannian metric has the local form E dx2 2F dx dy G dy2; C C where E; F; G are smooth functions of .x; y/ satisfying E > 0, G > 0 and EG F 2 > 0. The associated inner product on each tangent space is given by @ @ @ @ a b ; c d Eac F .ad bc/ Gbd @x C @y @x C @y D C C C Äc a b ; D d where the symmetric positive definite matrix ÄEF (1.1) D FG represents in the basis @ ; @ . In particular, the length of a tangent vector is f @x @y g given by @ @ p a b Ea2 2F ab Gb2: @x C @y D C C Define two local sections of the complexified cotangent bundle T X C by ˝ dz dx i dy WD C dz dx i dy: N WD 2 1 The Measurable Riemann Mapping Theorem These form a basis for each complexified cotangent space. The local sections @ 1 Â @ @ Ã i @z WD 2 @x @y @ 1 Â @ @ Ã i @z WD 2 @x C @y N of the complexified tangent bundle TX C will form the dual basis at each point. -
Notes on Riemann Surfaces
NOTES ON RIEMANN SURFACES DRAGAN MILICIˇ C´ 1. Riemann surfaces 1.1. Riemann surfaces as complex manifolds. Let M be a topological space. A chart on M is a triple c = (U, ϕ) consisting of an open subset U M and a homeomorphism ϕ of U onto an open set in the complex plane C. The⊂ open set U is called the domain of the chart c. The charts c = (U, ϕ) and c′ = (U ′, ϕ′) on M are compatible if either U U ′ = or U U ′ = and ϕ′ ϕ−1 : ϕ(U U ′) ϕ′(U U ′) is a bijective holomorphic∩ ∅ function∩ (hence ∅ the inverse◦ map is∩ also holomorphic).−→ ∩ A family of charts on M is an atlas of M if the domains of charts form a covering of MA and any two charts in are compatible. Atlases and of M are compatibleA if their union is an atlas on M. This is obviously anA equivalenceB relation on the set of all atlases on M. Each equivalence class of atlases contains the largest element which is equal to the union of all atlases in this class. Such atlas is called saturated. An one-dimensional complex manifold M is a hausdorff topological space with a saturated atlas. If M is connected we also call it a Riemann surface. Let M be an Riemann surface. A chart c = (U, ϕ) is a chart around z M if z U. We say that it is centered at z if ϕ(z) = 0. ∈ ∈Let M and N be two Riemann surfaces. A continuous map F : M N is a holomorphic map if for any two pairs of charts c = (U, ϕ) on M and d =−→ (V, ψ) on N such that F (U) V , the mapping ⊂ ψ F ϕ−1 : ϕ(U) ϕ(V ) ◦ ◦ −→ is a holomorphic function. -
6. the Riemann Sphere It Is Sometimes Convenient to Add a Point at Infinity ∞ to the Usual Complex Plane to Get the Extended Complex Plane
6. The Riemann sphere It is sometimes convenient to add a point at infinity 1 to the usual complex plane to get the extended complex plane. Definition 6.1. The extended complex plane, denoted P1, is simply the union of C and the point at infinity. It is somewhat curious that when we add points at infinity to the reals we add two points ±∞ but only only one point for the complex numbers. It is rare in geometry that things get easier as you increase the dimension. One very good way to understand the extended complex plane is to realise that P1 is naturally in bijection with the unit sphere: Definition 6.2. The Riemann sphere is the unit sphere in R3: 2 3 2 2 2 S = f (x; y; z) 2 R j x + y + z = 1 g: To make a correspondence with the sphere and the plane is simply to make a map (a real map not a function). We define a function 2 3 F : S − f(0; 0; 1)g −! C ⊂ R as follows. Pick a point q = (x; y; z) 2 S2, a point of the unit sphere, other than the north pole p = N = (0; 0; 1). Connect the point p to the point q by a line. This line will meet the plane z = 0, 2 f (x; y; 0) j (x; y) 2 R g in a unique point r. We then identify r with a point F (q) = x + iy 2 C in the usual way. F is called stereographic projection. -
Riemann Mapping Theorem (4/10 - 4/15)
Math 752 Spring 2015 Riemann Mapping Theorem (4/10 - 4/15) Definition 1. A class F of continuous functions defined on an open set G is called a normal family if every sequence of elements in F contains a subsequence that converges uniformly on compact subsets of G. The limit function is not required to be in F. By the above theorems we know that if F ⊆ H(G), then the limit function is in H(G). We require one direction of Montel's theorem, which relates normal families to families that are uniformly bounded on compact sets. Theorem 1. Let G be an open, connected set. Suppose F ⊂ H(G) and F is uniformly bounded on each compact subset of G. Then F is a normal family. ◦ Proof. Let Kn ⊆ Kn+1 be a sequence of compact sets whose union is G. Construction for these: Kn is the intersection of B(0; n) with the comple- ment of the (open) set [a=2GB(a; 1=n). We want to apply the Arzela-Ascoli theorem, hence we need to prove that F is equicontinuous. I.e., for " > 0 and z 2 G we need to show that there is δ > 0 so that if jz − wj < δ, then jf(z) − f(w)j < " for all f 2 F. (We emphasize that δ is allowed to depend on z.) Since every z 2 G is an element of one of the Kn, it suffices to prove this statement for fixed n and z 2 Kn. By assumption F is uniformly bounded on each Kn.