A Riemann Surfaces and Theta Functions
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LOOKING BACKWARD: from EULER to RIEMANN 11 at the Age of 24
LOOKING BACKWARD: FROM EULER TO RIEMANN ATHANASE PAPADOPOULOS Il est des hommes auxquels on ne doit pas adresser d’´eloges, si l’on ne suppose pas qu’ils ont le goˆut assez peu d´elicat pour goˆuter les louanges qui viennent d’en bas. (Jules Tannery, [241] p. 102) Abstract. We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017. AMS Mathematics Subject Classification: 01-02, 01A55, 01A67, 26A42, 30- 03, 33C05, 00A30. Keywords: Bernhard Riemann, function of a complex variable, space, Rie- mannian geometry, trigonometric series, zeta function, differential geometry, elliptic integral, elliptic function, Abelian integral, Abelian function, hy- pergeometric function, topology, Riemann surface, Leonhard Euler, space, integration. Contents 1. Introduction 2 2. Functions 10 3. Elliptic integrals 21 arXiv:1710.03982v1 [math.HO] 11 Oct 2017 4. Abelian functions 30 5. Hypergeometric series 32 6. The zeta function 34 7. On space 41 8. Topology 47 9. Differential geometry 63 10. Trigonometric series 69 11. Integration 79 12. Conclusion 81 References 85 Date: October 12, 2017. -
Dynamical Systems/Syst`Emes Dynamiques SIMPLE
Dynamical systems/Systemes` dynamiques SIMPLE EXPONENTIAL ESTIMATE FOR THE NUMBER OF REAL ZEROS OF COMPLETE ABELIAN INTEGRALS Dmitri Novikov, Sergeı˘ Yakovenko The Weizmann Institute of Science (Rehovot, Israel) Abstract. We show that for a generic real polynomial H = H(x, y) and an arbitrary real differential 1-form ω = P (x, y) dx + Q(x, y) dy with polynomial coefficients of degree 6 d, the number of ovals of the foliation H = const, which yield the zero value of the complete H Abelian integral I(t) = H=t ω, grows at most as exp OH (d) as d → ∞, where OH (d) depends only on H. This result essentially improves the previous double exponential estimate obtained in [1] for the number of zero ovals on a bounded distance from singular level curves of H. La borne simplement exponentielle pour le nombre de zeros´ reels´ isoles´ des integrales completes` abeliennes´ R´esum´e. Soit H = H(x, y) un polynˆome r´eel de deux variables et ω = P (x, y) dx + Q(x, y) dy une forme diff´erentielle quelconque avec des coefficients polynomiaux r´eels de degr´e d. Nous montrons que le nombre des ovales (c’est-a-dire les composantes compactes connexes) des courbes de niveau H = const, telles que l’int´egrale de la forme s’annule, est au plus exp OH (d) quand d → ∞, o`u OH (d) ne d´epend que du polynˆome H. Ce th´eor`emeam´eliore de mani`ere essentielle la borne doublement exponentielle obtenue dans [1]. Nous ´etablissons aussi les bornes sup´erieures exponentielles similaires pour le nom- bre de z´eros isol´espour les enveloppes polynomiales des ´equations diff´erentielles ordinaires lin´eaires irr´eductibles ou presque irr´eductibles. -
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. -
Abelian Varieties and Theta Functions Associated to Compact Riemannian Manifolds; Constructions Inspired by Superstring Theory
ABELIAN VARIETIES AND THETA FUNCTIONS ASSOCIATED TO COMPACT RIEMANNIAN MANIFOLDS; CONSTRUCTIONS INSPIRED BY SUPERSTRING THEORY. S. MULLER-STACH,¨ C. PETERS AND V. SRINIVAS MATH. INST., JOHANNES GUTENBERG UNIVERSITAT¨ MAINZ, INSTITUT FOURIER, UNIVERSITE´ GRENOBLE I ST.-MARTIN D'HERES,` FRANCE AND TIFR, MUMBAI, INDIA Resum´ e.´ On d´etailleune construction d^ue Witten et Moore-Witten (qui date d'environ 2000) d'une vari´et´eab´elienneprincipalement pola- ris´eeassoci´ee`aune vari´et´ede spin. Le th´eor`emed'indice pour l'op´erateur de Dirac (associ´e`ala structure de spin) implique qu'un accouplement naturel sur le K-groupe topologique prend des valeurs enti`eres.Cet ac- couplement sert commme polarization principale sur le t^oreassoci´e. On place la construction dans un c^adreg´en´eralce qui la relie `ala ja- cobienne de Weil mais qui sugg`ereaussi la construction d'une jacobienne associ´ee`an'importe quelle structure de Hodge polaris´eeet de poids pair. Cette derni`ereconstruction est ensuite expliqu´eeen termes de groupes alg´ebriques,utile pour le point de vue des cat´egoriesTannakiennes. Notre construction depend de param`etres,beaucoup comme dans la th´eoriede Teichm¨uller,mais en g´en´erall'application de p´eriodes n'est que de nature analytique r´eelle. Abstract. We first investigate a construction of principally polarized abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000. The index theorem for the Dirac operator associated to the spin structure implies integrality of a natural skew pairing on the topological K-group. The latter serves as a principal polarization. -
Abelian Solutions of the Soliton Equations and Riemann–Schottky Problems
Russian Math. Surveys 63:6 1011–1022 c 2008 RAS(DoM) and LMS Uspekhi Mat. Nauk 63:6 19–30 DOI 10.1070/RM2008v063n06ABEH004576 Abelian solutions of the soliton equations and Riemann–Schottky problems I. M. Krichever Abstract. The present article is an exposition of the author’s talk at the conference dedicated to the 70th birthday of S. P. Novikov. The talk con- tained the proof of Welters’ conjecture which proposes a solution of the clas- sical Riemann–Schottky problem of characterizing the Jacobians of smooth algebraic curves in terms of the existence of a trisecant of the associated Kummer variety, and a solution of another classical problem of algebraic geometry, that of characterizing the Prym varieties of unramified covers. Contents 1. Introduction 1011 2. Welters’ trisecant conjecture 1014 3. The problem of characterization of Prym varieties 1017 4. Abelian solutions of the soliton equations 1018 Bibliography 1020 1. Introduction The famous Novikov conjecture which asserts that the Jacobians of smooth alge- braic curves are precisely those indecomposable principally polarized Abelian vari- eties whose theta-functions provide explicit solutions of the Kadomtsev–Petviashvili (KP) equation, fundamentally changed the relations between the classical algebraic geometry of Riemann surfaces and the theory of soliton equations. It turns out that the finite-gap, or algebro-geometric, theory of integration of non-linear equa- tions developed in the mid-1970s can provide a powerful tool for approaching the fundamental problems of the geometry of Abelian varieties. The basic tool of the general construction proposed by the author [1], [2]which g+k 1 establishes a correspondence between algebro-geometric data Γ,Pα,zα,S − (Γ) and solutions of some soliton equation, is the notion of Baker–Akhiezer{ function.} Here Γis a smooth algebraic curve of genus g with marked points Pα, in whose g+k 1 neighborhoods we fix local coordinates zα, and S − (Γ) is a symmetric prod- uct of the curve. -
Analytic Vortex Solutions on Compact Hyperbolic Surfaces
Analytic vortex solutions on compact hyperbolic surfaces Rafael Maldonado∗ and Nicholas S. Mantony Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, U.K. August 27, 2018 Abstract We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 1 Introduction Consider the Abelian-Higgs field theory on a background surface M with metric ds2 = Ω(x; y)(dx2+dy2). At critical coupling the static energy functional satisfies a Bogomolny bound 2 1 Z B Ω 2 E = + jD Φj2 + 1 − jΦj2 dxdy ≥ πN; (1) 2 2Ω i 2 where the topological invariant N (the `vortex number') is the number of zeros of Φ counted with multiplicity [1]. In the notation of [2] we have taken e2 = τ = 1. Equality in (1) is attained when the fields satisfy the Bogomolny vortex equations, which are obtained by completing the square in (1). In complex coordinates z = x + iy these are 2 Dz¯Φ = 0;B = Ω (1 − jΦj ): (2) This set of equations has smooth vortex solutions. As we explain in section 2, analytical results are most readily obtained when M is hyperbolic, having constant negative cur- vature K = −1, a case which is of interest in its own right due to the relation between arXiv:1502.01990v1 [hep-th] 6 Feb 2015 hyperbolic vortices and SO(3)-invariant instantons, [3]. -
Abelian Integrals and the Genesis of Abel's Greatest Discovery
Abelian integrals and the genesis of Abel's greatest discovery Christian F. Skau Department of Mathematical Sciences, Norwegian University of Science and Technology March 20, 2020 Draft P(x) R(x) = Q(x) 3x 7 − 18x 6 + 46x 5 − 84x 4 + 127x 3 − 122x 2 + 84x − 56 = (x − 2)3(x + i)(x − i) 4x 4 − 12x 3 + 7x 2 − x + 1 = 3x 2 + 7 + (x − 2)3(x + i)(x − i) (−1) 3 4 i=2 (−i=2) = 3x 2 + 7 + + + + + (x − 2)3 (x − 2)2 x − 2 x + i x − i Draft Z 1=2 −3 R(x) dx = x 3 + 7x + + + 4 log (x − 2) (x − 2)2 x − 2 i −i + log (x + i) + log (x − i) 2 2 i −i = Re(x) + 4 log (x − 2) + log (x + i) + log (x − i) 2 2 The integral of a rational function R(x) is a rational function Re(x) plus a sum of logarithms of the form log(x + a), a 2 C, multiplied with constants. Draft eix − e−ix 1 p sin x = arcsin x = log ix + 1 − x 2 2i i Trigonometric functions and the inverse arcus-functions can be expressed by (complex) exponential and logarithmic functions. Draft A rational function R(x; y) of x and y is of the form P(x; y) R(x; y) = ; Q(x; y) where P and Q are polynomials in x and y. So, P i j aij x y R(x; y) = P k l bkl x y Similarly we define a rational function of x1; x2;:::; xn. -
Mirror Symmetry of Abelian Variety and Multi Theta Functions
1 Mirror symmetry of Abelian variety and Multi Theta functions by Kenji FUKAYA (深谷賢治) Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan Table of contents § 0 Introduction. § 1 Moduli spaces of Lagrangian submanifolds and construction of a mirror torus. § 2 Construction of a sheaf from an affine Lagrangian submanifold. § 3 Sheaf cohomology and Floer cohomology 1 (Construction of a homomorphism). § 4 Isogeny. § 5 Sheaf cohomology and Floer cohomology 2 (Proof of isomorphism). § 6 Extension and Floer cohomology 1 (0 th cohomology). § 7 Moduli space of holomorphic vector bundles on a mirror torus. § 8 Nontransversal or disconnected Lagrangian submanifolds. ∞ § 9 Multi Theta series 1 (Definition and A formulae.) § 10 Multi Theta series 2 (Calculation of the coefficients.) § 11 Extension and Floer cohomology 2 (Higher cohomology). § 12 Resolution and Lagrangian surgery. 2 § 0 Introduction In this paper, we study mirror symmetry of complex and symplectic tori as an example of homological mirror symmetry conjecture of Kontsevich [24], [25] between symplectic and complex manifolds. We discussed mirror symmetry of tori in [12] emphasizing its “noncom- mutative” generalization. In this paper, we concentrate on the case of a commutative (usual) torus. Our result is a generalization of one by Polishchuk and Zaslow [42], [41], who studied the case of elliptic curve. The main results of this paper establish a dictionary of mirror symmetry between symplectic geometry and complex geometry in the case of tori of arbitrary dimension. We wrote this dictionary in the introduction of [12]. We present the argument in a way so that it suggests a possibility of its generalization. -
Theta Function Review G = 1 Case
The genus 1 case - review Theta Function Review g = 1 case We recall the main de¯nitions of theta functions in the 1-dim'l case: De¯nition 0 Let· ¿ 2¸C such that Im¿ > 0: For "; " real numbers and z 2 C then: " £ (z;¿) = "0 n ¡ ¢ ¡ ¢ ¡ ¢ ³ ´o P 1 " " ² t ²0 l²Z2 exp2¼i 2 l + 2 ¿ l + 2 + l + 2 z + 2 The series· is uniformly¸ and absolutely convergent on compact subsets " C £ H: are called Theta characteristics "0 The genus 1 case - review Theta Function Properties Review for g = 1 case the following properties of theta functions can be obtained by manipulation of the series : · ¸ · ¸ " + 2m " 1. £ (z;¿) = exp¼i f"eg £ (z;¿) and e; m 2 Z "0 + 2e "0 · ¸ · ¸ " " 2. £ (z;¿) = £ (¡z;¿) ¡"0 "0 · ¸ " 3. £ (z + n + m¿; ¿) = "0 n o · ¸ t t 0 " exp 2¼i n "¡m " ¡ mz ¡ m2¿ £ (z;¿) 2 "0 The genus 1 case - review Remarks on the properties of Theta functions g=1 1. Property number 3 describes the transformation properties of theta functions under an element of the lattice L¿ generated by f1;¿g. 2. The same property implies that that the zeros of theta functions are well de¯ned on the torus given· ¸ by C=L¿ : In fact there is only a " unique such 0 for each £ (z;¿): "0 · ¸ · ¸ "i γj 3. Let 0 ; i = 1:::k and 0 ; j = 1:::l and "i γj 2 3 Q "i k θ4 5(z;¿) ³P ´ ³P ´ i=1 "0 k 0 l 0 2 i 3 "i + " ¿ ¡ γj + γ ¿ 2 L¿ Then i=1 i j=1 j Q l 4 γj 5 j=1 θ 0 (z;¿) γj is a meromorphic function on the elliptic curve de¯ned by C=L¿ : The genus 1 case - review Analytic vs. -
Animal Testing
Animal Testing Adrian Dumitrescu∗ Evan Hilscher † Abstract an animal A. Let A′ be the animal such that there is a cube at every integer coordinate within the box, i.e., it A configuration of unit cubes in three dimensions with is a solid rectangular box containing the given animal. integer coordinates is called an animal if the boundary The algorithm is as follows: of their union is homeomorphic to a sphere. Shermer ′ discovered several animals from which no single cube 1. Transform A1 to A1 by addition only. ′ ′ may be removed such that the resulting configurations 2. Transform A1 to A2 . are also animals [6]. Here we obtain a dual result: we ′ give an example of an animal to which no cube may 3. Transform A2 to A2 by removal only. be added within its minimal bounding box such that ′ ′ It is easy to see that A1 can be transformed to A2 . the resulting configuration is also an animal. We also ′ We simply add or remove one layer of A1 , one cube O n present a ( )-time algorithm for determining whether at a time. The only question is, can any animal A be n a configuration of unit cubes is an animal. transformed to A′ by addition only? If the answer is yes, Keywords: Animal, polyomino, homeomorphic to a then the third step above is also feasible. As it turns sphere. out, the answer is no, thus our alternative algorithm is also infeasible. 1 Introduction Our results. In Section 2 we present a construction of an animal to which no cube may be added within its An animal is defined as a configuration of axis-aligned minimal bounding box such that the resulting collection unit cubes with integer coordinates in 3-space such of unit cubes is an animal. -
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs Samir Datta1, Raghav Kulkarni2, Raghunath Tewari3, and N. Variyam Vinodchandran4 1 Chennai Mathematical Institute Chennai, India [email protected] 2 University of Chicago Chicago, USA [email protected] 3 University of Nebraska-Lincoln Lincoln, USA [email protected] 4 University of Nebraska-Lincoln Lincoln, USA [email protected] Abstract We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the prob- lems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class SPL. Since SPL is contained in the logspace counting classes ⊕L (in fact in ModkL for all k ≥ 2), C=L, and PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in FLSPL. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function. 1998 ACM Subject Classification Computational Complexity Keywords and phrases perfect matching, bounded genus graphs, isolation problem Digital Object Identifier 10.4230/LIPIcs.STACS.2011.579 1 Introduction The perfect matching problem and its variations are one of the most well-studied prob- lems in theoretical computer science. -
The Historical Development of Algebraic Geometry∗
The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome.