ABELIAN FUNCTIONS Abel's theorem and the allied theory of theta functions

Transcription

1 ABELIAN FUNCTIONS Abel's theorem and the allied theory of theta functions H. F Baker St John's College, Cambridge CAMBRIDGE UNIVERSITY PRESS

2 CHAPTER I. THE SUBJECT OF INVESTIGATION. I Fundamental algebraic irrationality 1 2, 3 The places and infinitesimal on a Riemann surface... 1, 2 4, 5 The theory unaltered by rational transformation The invariance of the deficiency in rational transformation; if a rational function exists of order 1, the surface is of zero deficiency 7, 8 7, 8 The greatest number of irremoveable parameters is 3p , 10 9, 10 The geometrical statement of the theory 11, 12 II Generality of Riemann's methods 12, 13 CHAPTEE II. THE FUNDAMENTAL FUNCTIONS ON A RIEMANN SURFACE. 12 Riemann's existence theorem provisionally regarded as fundamental Notation for normal elementary integral of second kind Notation for normal elementary integral of third kind Choice of normal integrals of the first kind Meaning of the word period. General remarks , Examples of the integrals, and of the places of the surface Periods of the normal elementary integrals of the second kind The integral of the second kind arises by differentiation from the integral of the third kind 22, Expression of a rational function by integrals of the second kind Special rational functions, which are invariant in rational transformation 25, Riemann normal integrals depend on mode of dissection of the surface 26 CHAPTER III. THE INFINITIES OF RATIONAL FUNCTIONS. 23 The interdependence of the poles of a rational function , 25 Condition that specified places be the poles of a rational function General form of Weierstrass's gap theorem 31, Provisional statement of the Riemann-Roch theorem... 33, 34

3 vi CONTENTS. 28, 29 Cases when the poles coalesce; the p critical integers.. 34, Simple anticipatory geometrical illustration 36, The (p-l)p(p +1) places which are the poles of rational functions of order less than p There are at least 2p + 2 such places which are distinct Statement of the Riemann-Roch theorem, with examples CHAPTER IV. SPECIFICATION OF A GENERAL FORM OF RIEMANN'S INTEGRALS. 38 Explanations in regard to Integral Rational Functions.. 47, Definition of dimension ; fundamental set of functions for the expression of rational functions Illustrative example for a surface of four sheets , The sum of the dimensions of the fundamental set of functions is p + n-1 54, Fundamental set for the expression of integral functions.. 55, Principal properties of the fundamental set of integral functions Definition of derived set of special functions <f> 0, tf> u..., < _, Algebraical form of elementary integral of the third kind, whose infinities are ordinary places; and of integrals of the first kind Algebraical form of elementary integral of the third kind in general Algebraical form of integral of the second kind, independently deduced The discriminant of the fundamental set of integral functions Deduction of the expression of a certain fundamental rational function in the general case The algebraical results of thia chapter are sufficient to replace Riemann's existence theorem 78, 79 CHAPTER V. CERTAIN FORMS OF THE FUNDAMENTAL EQUATION OF THE RIEMANN SURFACE. 51 Contents of the chapter When p>l, existence of rational function of the second order involves a (1, 1) correspondence Existence of rational function of the second order involves the hyperelliptic equation , 57 Fundamental integral functions and integrals of the first kind Examples Number of irremoveable parameters in the hyperelliptic equation ; transformation to the canonical form Weierstrass's canonical equation for any deficiency

4 Vll Actual formation of the equation , 68 Illustrations of the theory of integral functions for Weierstrass's canonical equation The method can be considerably generalised Hensel's determination of the fundamental integral functions CHAPTER VI. GEOMETRICAL INVESTIGATIONS. 80 Comparison of the theory of rational functions with the theory of intersections of curves Introductory indications of elementary form of theory The method to be followed in this chapter Treatment of infinity. Homogeneous variables might be used. 118, Grade of an integral polynomial; number of terms ; generalised zeros 120, Adjoint polynomials ; definition of the index of a singular place Pliicker's equations; connection with theory of discriminant. 123, , 90 Expression of rational functions by adjoint polynomials , Expression of integral of the first kind Number of terms in an adjoint polynomial; determination of elementary integral of the third kind Linear systems of adjoint polynomials ; reciprocal theorem.. 133, , 95 Definitions of set, lot, sequent, equivalent sets, coresidual sets , 97 Theorem of coresidual sets; algebraic basis of the theorem A rational function of order less than p+\ is expressible by <fypolynomials , 100 Criticism of the theory; Cayley's theorem Rational transformation by means of ^-polynomials Application of special sets The hyperelliptic surface; transformation to canonical form Whole rational theory can be represented by means of the invariant ratios of (^-polynomials ; number of relations connecting these Elementary considerations in regard to curves in space CHAPTER VII. COORDINATION OF SIMPLE ELEMENTS. TRANSCENDENTAL UNIFORM FUNCTIONS. 120 Scope of the chapter Notation for integrals of the first kind , 123 The function $ (#, a; z, c lt...,c p ) expressed by Riemann integrals 170, Derivation of a certain prime function Applications of this function to rational functions and integrals 173

5 viii CONTENTS The function \ c (%, a; z, c); its utility for the expression of rational functions The derived prime function E{x, z); used to express rational functions , 131 Algebraic expression of the functions ty (x, a; z, c u...,c p ), ^{x,a; 2, e) , Examples of these functions; they determine algebraic expressions for the elementary integrals , 134 Derivation of a canonical integral of the third kind; for which interchange of argument and parameter holds; its algebraic expression ; its relation with Riemann's elementary normal integral Algebraic theorem equivalent to interchange of argument and parameter Elementary canonical integral of the second kind , Applications. Canonical integral of the third kind deduced from the function yfr(x,a; z,c x,...,o p ). Modification for the function ^ (x, a; z, 0) Associated integrals of first and second kind. Further canonical integrals. The algebraic theory of the hyperelliptic integrals in one formula 193, , 140 Deduction of Weierstrass's and Riemann's relations for periods of integrals of the first and second kind Either form is equivalent to the other Alternative proofs of Weierstrass's and Riemann's period relations 199, Expression of uniform transcendental function by the function ty{x, a; 2, c) , 145 Mittag-Leffler's theorem Expression of uniform transcendental function in prime factors General form of interchange of argument and parameter, after Abel 206 CHAPTER VIII. ABEL'S THEOREM. ABEL'S DIFFERENTIAL EQUATIONS Approximative description of Abel's theorem Enunciation of the theorem The general theorem reduced to two simpler theorems.. 211, , 154 Proof and analytical statement of the theorem Remark; statement in terms of polynomials The disappearance of the logarithm on the right side of the equation Applications of the theorem. Abel's own proof , 159 The number of algebraically independent equations given by the theorem. Inverse of Abel's theorem 160, 161 Integration of Abel's differential equations 162 Abel's theorem proved quite similarly for curves in space.

6 IX CHAPTER IX. JACOBI'S INVERSION PROBLEM. 163 Statement of the problem Uniqueness of any solution The necessity of using congruences and not equations , 167 Avoidance of functions with infinitesimal periods , , 169 Proof of the existence of a solution Formation of functions with which to express the solution; connection with theta functions CHAPTER X. RIEMANN'S-THETA FUNCTIONS. GENERAL THEORY. 173 Sketch of the history of the introduction of theta functions Convergence. Notation. Introduction of matrices , , 176 Periodicity of the theta functions. Odd and even functions Number of zeros is p Position of the zeros in the simple case 253, The places m^,..., m p Position of the zeros in general 256, Identical vanishing of the theta functions 258, , 183 Fundamental properties. Geometrical interpretation of the places m 1,...,m p Geometrical developments; special inversion problem; contact curves Solution of Jacobi's inversion problem by quotients of theta functions 274, Theory of the identical vanishing of the theta function. Expression of (^-polynomials by theta functions General form of theta function. Fundamental formulae. Periodicity Introduction of the f functions. Generalisation of an elliptic formula Difference of two f functions expressed by algebraic integrals and rational functions Development. Expression of single f function by algebraic integrals , 198 Introduction of the 0> functions. Expression by rational functions CHAPTER XI. THE HYPERELLIPTIC CASE OF RIEMANN'S THETA FUNCTIONS. 199 Hyperelliptic case illustrates the general theory The places m l,...,m p. The rule for half periods , , 202 Fundamental set of characteristics defined by branch places

7 x CONTENTS. 203 Notation. General theorems to be illustrated , 205 Tables in illustration of the general theory Algebraic expression of quotients of hyperelliptic theta functions. Solution of hyperelliptic inversion problem , 215 Single function expressed by algebraical integrals and rational functions Rational expression of (j? function. Relation to quotients of theta functions. Solution of inversion problem by fp function Rational expression of g> function Algebraic deduction of addition equation for theta functions when jo=2; generalisation of the equation <r(u+v) a (u-v) = O%.O- 2 2>.(^-J?K) Examples for the case p = % Gopel's biquadratic relation CHAPTER XII. A PARTICULAR FORM OF FUNDAMENTAL SURFACE. 222 Chapter introduced as a change of independent variable, and as introducing a particular prime function Definition of a group of substitutions; fundamental properties. 226, 227 Convergence of a series; functions associated with the group The fundamental functions. Comparison with foregoing theory of this volume Definition and periodicity of the Schottky prime function. 236, 237 Its connection with the theta functions 238 A further function connected therewith 239 The hyperelliptic case CHAPTER XIII. RADICAL FUNCTIONS. 240 Introductory. 241, 242 Expression of any radical function by Riemann's integrals, and by theta functions 243 Radical functions are a generalisation of rational functions 244, 245 Characteristics of radical functions Bitangents of a plane quartic curve 250, 251 Solution of the inversion problem by radical functions 374 CHAPTER XIV. FACTORIAL FUNCTIONS. 252 Statement of results obtained. Notations 253 Necessary dissection of the Riemann surface Definition of a factorial function (including radical function). Primary and associated systems of factorial functions.

8 XI 255 Factorial integrals of the primary and associated systems.. 397, Factorial integrals which are everywhere finite, save at the fixed infinities. Introduction of the numbers or, o- + l When o-+l>0, there are <r+l everywhere finite factorial functions of the associated system Alternative investigation of everywhere finite factorial functions of the associated system. Theory divisible according to the values of a+1 and o-' + l 401, Expression of these functions by everywhere finite integrals General consideration of the periods of the factorial integrals , 262 Riemann-Roch theorem for factorial functions. When cr' + l=o, least number of arbitrary poles for function of the primary system is ai' + l 405, Construction of factorial function of the primary system with sr' + l arbitrary poles 406, , 265 Construction of a factorial integral having only poles. Least number of such poles, for an integral of the primary system, is O , This factorial integral can be simplified, in analogy with Riemann's elementary integral of the second kind Expression of the factorial function with ta-' +1 poles in terms of the factorial integral with o- + 2 poles. The factorial function in analogy with the function ^ {x, a; z, c lt..., c p ) The theory tested by examination of a very particular case The radical functions as a particular case of factorial functions 419, Factorial functions whose factors are any constants, having no essential singularities , 272 Investigation of a general formula connecting factorial functions and theta functions Introduction of the Schottky-Klein prime form, in a certain shape Expression of a theta function in terms of radical functions, as a particular case of , 276 The formula of 272 for the case of rational functions The formula of 272 applied to define algebraically the hyperelliptic theta function, and its theta characteristic Expression of any factorial function by simple theta functions; examples 437, Connection of theory of factorial functions with theory of automorphic forms CHAPTER XV. RELATIONS CONNECTING PRODUCTS OF THETA FUNCTIONS INTRODUCTORY. 280 Plan of this and the two following chapters A single-valued integral analytical function of p variables, which is periodic in each variable alone, can be represented by a series of exponentials

9 282, , , Proof that the 2* theta functions with half-integer characteristics are linearly independent Definition of general theta function of order r; its linear expression by t*> theta functions. Any p+2 theta functions of same order, periods, and characteristic connected by a homogeneous polynomial relation Addition theorem for hyperelliptic theta functions, or for the general case when p<i Number of linearly independent theta functions of order r which are all of the same parity Examples. The Gopel biquadratic relation PAGES CHAPTER XVI. A DIRECT METHOD OF OBTAINING THE EQUATIONS CONNECTING THETA PRODUCTS. 290 Contents of this chapter An addition theorem obtained by multiplying two theta functions An addition theorem obtained by multiplying four theta functions The general formula obtained by multiplying any number of theta functions CHAPTER XVII. THETA RELATIONS ASSOCIATED WITH CERTAIN GROUPS OF CHARACTERISTICS. 294 Abbreviations. Definition of syzygetic and azygetic. References to literature (see also p. 296) 486, A preliminary lemma Determination of a Gopel group of characteristics , Determination of a G6pel system of characteristics , , 299 Determination and number of GSpel systems of the same parity Determination of a fundamental set of Gopel systems , 305 Statement of results obtained, with the simpler applications Number of linearly independent theta functions of the second order of a particular kind. Explicit mention of an important identity The most important formulae of the chapter. A general addition theorem. The jp function expressed by quotients of theta functions Other applications of the principles of the chapter. The expression of a function $ (nv) as an integral polynomial of order n 2 in 2P functions ${v)

10 xm CHAPTER XVIII. TRANSFORMATION OF PERIODS, ESPECIALLY LINEAR TRANSFORMATION. 318 Bearings of the theory of transformation 528, The general theory of the modification of the period loops on a Riemann surface Analytical theory of transformation of periods and characteristic of a theta function Convergence of the transformed function Specialisation of the formulae, for linear transformation.. 539, Transformation of theta characteristics ; of even characteristics; of syzygetic characteristics 541, Period characteristics and theta characteristics Determination of a linear transformation to transform any even characteristic into the zero characteristic , , 331 Linear transformation of two azygetic systems of theta characteristics into one another Composition of two transformations of different orders; supplementary transformations 551, , 334 Formation of p + 2 elementary linear transformations by the composition of which every linear transformation can be formed; determination of the constant factors for each of these The constant factor for any linear transformation , Any linear transformation may be associated with a change of the period loops, of a Riemann surface 560, , 338 Linear transformation of the places m^,..., m p Linear transformation of the characteristics of a radical function 563, Determination of the places wij,..., m p upon a Riemann surface whose mode of dissection is assigned Linear transformation of quotients of hyperelliptic theta functions A convenient form of the period loops in a special hyperelliptic case. Weierstrass's number notation for half-integer characteristics 569, 570 CHAPTER XIX. ON SYSTEMS OF PERIODS AND ON GENERAL JACOBIAN FUNCTIONS. 343 Scope of this chapter Columns of periods. Exclusion of infinitesimal periods. Expression of all period columns by a finite number of columns, with integer coefficients Definition of general Jacobian function, and comparison with theta function Expression of Jacobian function by means of theta functions. Any p + 2 Jacobian functions of same periods and parameter connected by a homogeneous polynomial relation

11 CHAPTER XX. TRANSFORMATION OF THETA FUNCTIONS. 363 Sketch of the results obtained. References to the literature. 599, , 365 Elementary theory of transformation of second order , 367 Investigation of a general formula preliminary to transformation of odd order , 369 The general theorem for transformation of odd order The general treatment of transformation of the second order The two steps in the determination of the constant coefficients The first step in the determination of the constant coefficients Remarks and examples in regard to the second step Transformation of periods when the coefficients are not integral Reference to the algebraical applications of the theory CHAPTER XXI. COMPLEX MULTIPLICATION OF THETA FUNCTIONS. OF POINTS ON A RLEMANN SURFACE. CORRESPONDENCE 376 Scope of the chapter , 378 Necessary conditions for a complex multiplication, or special transformation, of theta functions Proof, in one case, that these conditions are sufficient Example of the elliptic case Meaning of an (r, a) correspondence on a Riemann surface. 639, Equations necessary for the existence of such a correspondence Algebraic determination of a correspondence existing on a perfectly general Riemann surface The coincidences. Examples of the inflections and bitangents of a plane curve Conditions for a (1, s) correspondence on a special Riemann surface 648, When p>l a (1, 1) correspondence is necessarily periodic.. 649, And involves a special form of fundamental equation When p>l there cannot be an infinite number of (1, 1) correspondences Example of the case p = l CHAPTER XXII. DEGENERATE ABELIAN INTEGRALS. 395 Example of the property to be considered Weierstrass's theorem. The property involves a transformation leading to a theta function which breaks into factors.. 657, 658

12 XV 3i)7 Weierstrass's and Picard's theorem. The property involves a linear transformation leading to T^2 = l/r 658, )8 Existence of one degenerate integral involves another Q» = 2) (J, 400 Connection with theory of special transformation, when p = , Determination of necessary form of fundamental equation. References APPENDIX 1. ON ALGEBRAIC CURVES IN SPACE. 404 Formal proof that an algebraic curve in space is an interpretation of the relations connecting three rational functions on a Riemann surface (cf. 162) 664, 665 APPENDIX II. ON MATRICES Introductory explanations Decomposition of an Abeliau matrix into simpler ones A particular result , 418 Lemmas , 420 Proof of results assumed in 396, , 676 INDEX OF AUTHORS QUOTED 677, 678 TABLE OF SOME FUNCTIONAL SYMBOLS SUBJECT INDEX