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13 Elliptic Function
13 Elliptic Function Recall that a function g : C → Cˆ is an elliptic function if it is meromorphic and there exists a lattice L = {mω1 + nω2 | m, n ∈ Z} such that g(z + ω)= g(z) for all z ∈ C and all ω ∈ L where ω1,ω2 are complex numbers that are R-linearly independent. We have shown that an elliptic function cannot be holomorphic, the number of its poles are finite and the sum of their residues is zero. Lemma 13.1 A non-constant elliptic function f always has the same number of zeros mod- ulo its associated lattice L as it does poles, counting multiplicites of zeros and orders of poles. Proof: Consider the function f ′/f, which is also an elliptic function with associated lattice L. We will evaluate the sum of the residues of this function in two different ways as above. Then this sum is zero, and by the argument principle from complex analysis 31, it is precisely the number of zeros of f counting multiplicities minus the number of poles of f counting orders. Corollary 13.2 A non-constant elliptic function f always takes on every value in Cˆ the same number of times modulo L, counting multiplicities. Proof: Given a complex value b, consider the function f −b. This function is also an elliptic function, and one with the same poles as f. By Theorem above, it therefore has the same number of zeros as f. Thus, f must take on the value b as many times as it does 0. -
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. -
Generalization of a Theorem of Hurwitz
J. Ramanujan Math. Soc. 31, No.3 (2016) 215–226 Generalization of a theorem of Hurwitz Jung-Jo Lee1,∗ ,M.RamMurty2,† and Donghoon Park3,‡ 1Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea e-mail: [email protected] 2Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada e-mail: [email protected] 3Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, South Korea e-mail: [email protected] Communicated by: R. Sujatha Received: February 10, 2015 Abstract. This paper is an exposition of several classical results formulated and unified using more modern terminology. We generalize a classical theorem of Hurwitz and prove the following: let 1 G (z) = k (mz + n)k m,n be the Eisenstein series of weight k attached to the full modular group. Let z be a CM point in the upper half-plane. Then there is a transcendental number z such that ( ) = 2k · ( ). G2k z z an algebraic number Moreover, z can be viewed as a fundamental period of a CM elliptic curve defined over the field of algebraic numbers. More generally, given any modular form f of weight k for the full modular group, and with ( )π k /k algebraic Fourier coefficients, we prove that f z z is algebraic for any CM point z lying in the upper half-plane. We also prove that for any σ Q Q ( ( )π k /k)σ = σ ( )π k /k automorphism of Gal( / ), f z z f z z . 2010 Mathematics Subject Classification: 11J81, 11G15, 11R42. -
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. -
ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi)
Math 213a (Fall 2021) Yum-Tong Siu 1 ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi) Significance of Elliptic Functions. Elliptic functions and their associated theta functions are a new class of special functions which play an impor- tant role in explicit solutions of real world problems. Elliptic functions as meromorphic functions on compact Riemann surfaces of genus 1 and their associated theta functions as holomorphic sections of holomorphic line bun- dles on compact Riemann surfaces pave the way for the development of the theory of Riemann surfaces and higher-dimensional abelian varieties. Two Approaches to Elliptic Function Theory. One approach (which we call the approach of Abel and Jacobi) follows the historic development with motivation from real-world problems and techniques developed for solving the difficulties encountered. One starts with the inverse of an elliptic func- tion defined by an indefinite integral whose integrand is the reciprocal of the square root of a quartic polynomial. An obstacle is to show that the inverse function of the indefinite integral is a global meromorphic function on C with two R-linearly independent primitive periods. The resulting dou- bly periodic meromorphic functions are known as Jacobian elliptic functions, though Abel was actually the first mathematician who succeeded in inverting such an indefinite integral. Nowadays, with the use of the notion of a Rie- mann surface, the inversion can be handled by using the fundamental group of the Riemann surface constructed to make the square root of the quartic polynomial single-valued. The great advantage of this approach is that there is vast literature for the properties of the Jacobain elliptic functions and of their associated Jacobian theta functions. -
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. -
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. -
KLEIN's EVANSTON LECTURES. the Evanston Colloquium : Lectures on Mathematics, Deliv Ered from Aug
1894] KLEIN'S EVANSTON LECTURES, 119 KLEIN'S EVANSTON LECTURES. The Evanston Colloquium : Lectures on mathematics, deliv ered from Aug. 28 to Sept. 9,1893, before members of the Congress of Mai hematics held in connection with the World's Fair in Chicago, at Northwestern University, Evanston, 111., by FELIX KLEIN. Reported by Alexander Ziwet. New York, Macniillan, 1894. 8vo. x and 109 pp. THIS little volume occupies a somewhat unicrue position in mathematical literature. Even the Commission permanente would find it difficult to classify it and would have to attach a bewildering series of symbols to characterize its contents. It is stated as the object of these lectures " to pass in review some of the principal phases of the most recent development of mathematical thought in Germany" ; and surely, no one could be more competent to do this than Professor Felix Klein. His intimate personal connection with this develop ment is evidenced alike by the long array of his own works and papers, and by those of the numerous pupils and followers he has inspired. Eut perhaps even more than on this account is he fitted for this task by the well-known comprehensiveness of his knowledge and the breadth of view so -characteristic of all his work. In these lectures there is little strictly mathematical reason ing, but a great deal of information and expert comment on the most advanced work done in pure mathematics during the last twenty-five years. Happily this is given with such freshness and vigor of style as makes the reading a recreation. -
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. -
VECTOR BUNDLES OVER a REAL ELLIPTIC CURVE 3 Admit a Canonical Real Structure If K Is Even and a Canonical Quaternionic Structure If K Is Odd
VECTOR BUNDLES OVER A REAL ELLIPTIC CURVE INDRANIL BISWAS AND FLORENT SCHAFFHAUSER Abstract. Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we deter- mine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and degree d on the curve. When r and d are coprime, we describe the topology of the real locus and give a modular interpretation of its points. We also study, for arbitrary rank and degree, the moduli space of indecom- posable vector bundles of rank r and degree d, and determine its isomorphism class as a real algebraic variety. Contents 1. Introduction 1 1.1. Notation 1 1.2. The case of genus zero 2 1.3. Description of the results 3 2. Moduli spaces of semi-stable vector bundles over an elliptic curve 5 2.1. Real elliptic curves and their Picard varieties 5 2.2. Semi-stable vector bundles 6 2.3. The real structure of the moduli space 8 2.4. Topologyofthesetofrealpointsinthecoprimecase 10 2.5. Real vector bundles of fixed determinant 12 3. Indecomposable vector bundles 13 3.1. Indecomposable vector bundles over a complex elliptic curve 13 3.2. Relation to semi-stable and stable bundles 14 3.3. Indecomposable vector bundles over a real elliptic curve 15 References 17 arXiv:1410.6845v2 [math.AG] 8 Jan 2016 1. Introduction 1.1. Notation. In this paper, a real elliptic curve will be a triple (X, x0, σ) where (X, x0) is a complex elliptic curve (i.e., a compact connected Riemann surface of genus 1 with a marked point x0) and σ : X −→ X is an anti-holomorphic involution (also called a real structure). -
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). -
NOTES on ELLIPTIC CURVES Contents 1
NOTES ON ELLIPTIC CURVES DINO FESTI Contents 1. Introduction: Solving equations 1 1.1. Equation of degree one in one variable 2 1.2. Equations of higher degree in one variable 2 1.3. Equations of degree one in more variables 3 1.4. Equations of degree two in two variables: plane conics 4 1.5. Exercises 5 2. Cubic curves and Weierstrass form 6 2.1. Weierstrass form 6 2.2. First definition of elliptic curves 8 2.3. The j-invariant 10 2.4. Exercises 11 3. Rational points of an elliptic curve 11 3.1. The group law 11 3.2. Points of finite order 13 3.3. The Mordell{Weil theorem 14 3.4. Isogenies 14 3.5. Exercises 15 4. Elliptic curves over C 16 4.1. Ellipses and elliptic curves 16 4.2. Lattices and elliptic functions 17 4.3. The Weierstrass } function 18 4.4. Exercises 22 5. Isogenies and j-invariant: revisited 22 5.1. Isogenies 22 5.2. The group SL2(Z) 24 5.3. The j-function 25 5.4. Exercises 27 References 27 1. Introduction: Solving equations Solving equations or, more precisely, finding the zeros of a given equation has been one of the first reasons to study mathematics, since the ancient times. The branch of mathematics devoted to solving equations is called Algebra. We are going Date: August 13, 2017. 1 2 DINO FESTI to see how elliptic curves represent a very natural and important step in the study of solutions of equations. Since 19th century it has been proved that Geometry is a very powerful tool in order to study Algebra.