LIBRARY UNIVERSITY OF CALIFORNIA.
Deceived /fa/ OSS , i8g</
f *
tAcce&sion No. *7 ft
£0 .
OF
TTNIVERSI'
ABEL'S THEOEEM AND THE ALLIED THEOKY
INCLUDING THE
THEOKY OF THE THETA FUNCTIONS
: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE.
OSIaggoto : 263, ARGYLE STREET.
P. A. BROCKHAUS. Hork: THE MACMILLAN COMPANY.
ABEL'S THEOREM
AND THE
ALLIED THEOEY
INCLUDING THE THEORY OF THE
THETA FUNCTIONS
OF THE
UNIVERSITY
BY
H. F. BAKER, M.A.
FELLOW AND LECTURER OF ST JOHN'S COLLEGE, UNIVERSITY LECTURER IN MATHEMATICS.
CAMBRIDGE: AT THE UNIVERSITY PRESS.
1897
[All Rights reserved]
PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS.
To 4
^w-
PREFACE.
IT may perhaps be fairly stated that no better guide can be found to the analytical developments of Pure Mathematics during the last seventy years than a study of the problems presented by the subject whereof this volume treats. This book is published in the hope that it may be found worthy to form the basis for such study. It is also hoped that the book may be serviceable to those who use it for a first introduction to the subject. And an endeavour has been made to point out what ^are conceived to be the most artistic ways of formally developing the theory regarded as complete.
The matter is arranged primarily with a view to obtaining perfectly general, and not merely illustrative, theorems, in an order in which they can be immediately utilised for the subsequent theory; particular results, however interesting, or important in special applications, which are not an integral portion of the continuous argument of the book, are introduced only so far as they appeared necessary to explain the general results, mainly in the examples, or are postponed, or are excluded altogether. The sequence and scope of ideas to which this has led will be clear from an examination of the table of Contents. fc-
The methods of Riemann, as far as they are explained in books on the general theory of functions, are provisionally regarded as fundamental ; but precise references .are given for all results assumed, and great pains have been taken, in the theory of algebraic functions and their integrals, and in the analytic theory of theta functions, to provide for alternative developments of the theory. If it is desired to dispense with Riemann's existence theorems, the theory of algebraic functions may be founded either on the arithmetical ideas introduced by Kronecker and by Dedekind and Weber ; or on the quasi-geometrical ideas associated with the theory of adjoint polynomials ; while in any case it does not appear to be convenient to avoid reference to either class of ideas. It is believed that, save for some points in the periodicity of Abelian integrals, all that is necessary to the former ele mentary development will be found in Chapters IV. and VII., in connection with which the reader may consult the recent -paper of Hensel, Acta Mathematica, xvm. (1894), and also the papers of Kronecker and of B. • b
V] PREFACE.
Dedekind and Weber, Grelle's Journal, xci., xcn. (1882). And it is hoped that what is necessary for the development of the theory from the elemen tary geometrical point of view will be understood from Chapter VI., in connection with which the reader may consult the Abel'sche Functionen of Clebsch and Gordan (Leipzig, 1866) and the paper of Noether, Mathematische Annalen, vii. (1873). In the theory of Riemann's theta functions, the formulae which are given relatively to the £ and g>- functions, and the general formulae given near the end of Chapter XIV., will provide sufficient indications of how the theta functions can be algebraically denned ; the reader may consult Noether, Mathematische Annalen, xxxvn. (1890), and Klein and Burkhardt, ibid. xxxn. — xxxvi. In Chapters XV., XVII., and XIX., and in Chapters XVIII. and XX., are given the beginnings of that analytical theory of theta. functions from which, -in conjunction with the general theory of functions of several independent variables, so much is to be hoped ; the latter theory is however excluded from this volume.
To the reader who does not desire to follow the development of this volume consecutively through, the following course may perhaps be sug gested; Chapters I., II., III. (in part), IV., VI. (to § 98), VIII., IX., X., XL (in part), XVIII. (in part), XII. , XV. (in. part); it is also possible to begin with the analytical theory of theta functions, reading in order Chapters XV., XVI., XVII., XIX., XX.
The footnotes throughout the volume are intended to contain the mention of all authorities used in its preparation ; occasionally the hazardous plan of adding to the lists of references during the passage of the sheets through the press, has been adopted ; for references omitted, and for refer ences improperly placed, only mistake can be pleaded. Complete lists of papers are given in the valuable report of Brill and Noether, " Die Entwicklung der Theorie der algebraischen Functionen in alterer und neuerer Zeit," Jahresbericht der Deutschen Mathematiker-Vereinigung, Dritter Band, 1892 — 3 (Berlin. Reimer, 1894); this report unfortunately appeared only after the first seventeen chapters of this volume, with the exception of Chapter XL, and parts of VIL, were in manuscript ; its plan is somewhat different from that of this volume, and it will be of advantage to the reader to consult it. Other books which have appeared during the progress of this volume, too late to effect large modifications, have not been consulted. The examples throughout the volume are intended to serve several different purposes ; to provide practice in the ideas involved in the general theory ; to suggest the steps of alternative developments without interrupting the line of reasoning in the text; and to place important consequences which are not utilised, if at all, till much subsequently, in their proper connection.
For my first interest in the subject of this volume, I desire to acknowledge my obligations to the generous help given to me during Gottingen vacations,
PREFACE. Vll
on two occasions, by Professor Felix Klein. In the preparation of the book I have been largely indebted to his printed publications ; the reader is recommended to consult also his lithographed lectures, especially the one dealing with Riemann surfaces. In the final revision of the sheets in their passage through the press, I have received help from several friends. Mr A. E. H. Love, Fellow and Lecturer of St John's College, has read the proofs of the volume ; in the removal of obscurities of expression and in the correction of press, his untiring assistance has been of great value to me. Mr J. Harkness, Professor of Mathematics at Bryn Mawr College, Pennsylvania, has read the proofs from Chapter XV. onwards; many faults, undetected by Mr Love or myself, have yielded to his perusal ; and I have been greatly helped by his sympathy in the subject-matter of the volume. To both these friends I am under obligations not easy to discharge. My gratitude is also due to Professor Forsyth for the generous interest he has taken in the book from its commencement. While, it should be added, the task carried through by the Staff of the University Press deserves more than the usual word of acknowledgment.
This book has a somewhat ambitious aim ; and it has been written under the constant pressure of other work. It cannot but be that .many defects will be found in it. But the author hopes it will be sufficient to shew that the subject offers for exploration a country of which the vastness is equalled by the fascination.
ST JOHN'S COLLEGE, CAMBRIDGE. April 26, 1897.
CONTENTS. CHAPTER I.
THE SUBJECT OF INVESTIGATION. §§ PAGES
1 Fundamental algebraic irrationality 1
2, 3 The places and infinitesimal on a Riemann surface . . . 1, 2
4, 5 The theory unaltered by rational transformation . .' 3 — 6 6 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 - 3 . . 9, 10
9, 10 The geometrical statement of the theory 11, 12
11 Generality of Riemann's methods 12, 13
CHAPTER II.
THE FUNDAMENTAL FUNCTIONS ON A RlEMANN SURFACE.
12 Riemann's existence theorem provisionally regarded as fundamental 14
13 Notation for normal elementary integral of second kind . . 15
14 Notation for normal elementary integral of third kind ... 15
15 Choice of normal integrals of the first kind 16
16 Meaning of the word period. General remarks . . . ^. 16, 17
17 Examples of the integrals, and of the places of the surface . 18 — 20
18 Periods of the normal elementary integrals of the second kind . 21
19 The integral of the second kind arises by differentiation from the
integral of the third kind 22, 23
20 Expression of a rational function by integrals of the second kind . 24
21 Special rational functions, which are invariant in rational trans
formation . 25, 26
22 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 . . 27 24, 25 Condition that specified places be the poles of a rational function . 28 — 30
26 General form of Weierstrass's gap theorem 31, 32
27 Provisional statement of the Riemann-Roch theorem ... 33, 34
K CONTENTS.
§§ PAGES
28, 29 Cases when the poles coalesce ; the p critical integers . . 34, 35
30 Simple anticipatory geometrical illustration ...... 36, 37
31 — 33 The (p-l)p(p + l) places which are the poles of rational functions •
of order less than p + l 38 — 40
34 — 36 There are at least 2jo + 2 such places which are distinct . . 41 — 44
37 Statement of the Riemann-Roch theorem, with examples . . 44 — 46
CHAPTER IV. SPECIFICATION OF A GENERAL FORM OF RIEMANN'S INTEGRALS.
38 Explanations in regard to Integral Rational Functions . . 47, 48
39 Definition of dimension ; fundamental set of functions for the
expression of rational functions 48 — 52
40 Illustrative example for a surface of four sheets . . . . 53, 54
41 The sum of the dimensions of the fundamental set of functions
is p + n-l 54, 55
42 Fundamental set for the expression of integral functions . . 55, 56
43 Principal properties of the fundamental set of integral functions . 57 — 60
44 Definition of derived set of special functions 00, 0j, ..., 4>n_l . 61 — 64
45 Algebraical form of elementary integral of the third kind, whose
infinities are ordinary places ; and of integrals of the first
kind . . . . 65 — 68
46 Algebraical form of elementary, integral of the third kind in general 68 — 70
47 Algebraical form of integral of the second kind, independently
deduced 71—73
48 The discriminant of the fundamental set of integral functions . 74
49 Deduction of the expression of a certain fundamental rational
function in the general case 75 — 77
50 The algebraical results of this 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 80
52 When p>l, existence of rational function of the second order
involves a (1, 1) correspondence 81
53—55 Existence of rational function of the second order involves the
hyperelliptic equation 81 — 84
56, 57 Fundamental integral functions and integrals of the first kind . 85 — 86
58 Examples 87
59 Number of irremoveable parameters in the hyperelliptic equation ;
transformation to the canonical form 88 — 89
60—63 Weierstrass's canonical equation for any deficiency . . . 90—92
CONTENTS.
XI
§§
64—66 67, 68
69—71 72—79
Actual formation of the equation . . .
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
PAGES
93—98
99—101 102—104 105—112
CHAPTER VI.
GEOMETRICAL INVESTIGATIONS.
80 Comparison of the theory of rational functions with the theory
of intersections of curves . . . . ... . 113
81 — 83 Introductory indications of elementary form of theory . . . 113—116
84 The method to be followed in this chapter 117
85 Treatment of infinity. Homogeneous variables might be used . 118,- 119
86 Grade of an integral polynomial ; number of terms ; generalised
zeros . 120, 121
87 Adjoint polynomials ; definition of the index of a singular place . 122
88 Pliicker's equations ; connection with theory of discriminant • . 123, 124 89, 90 Expression of rational functions by adjoint polynomials . . . 125, 126
91 Expression of integral of the first kind . . . . . 127
92 Number of terms in an adjoint polynomial ; determination of
elementary integral of the third kind . . ... . 128 — 132
93 Linear systems of adjoint polynomials ; reciprocal theorem . . 133, 134 94, 95 Definitions of set, lot, sequent, equivalent sets, coresidual sets . 135 96, 97 Theorem of coresidual sets ; algebraic basis of the theorem . . 136
98 A rational function of order less than p + 1 is expressible by <£-
polynomials , 137
99, 100 Criticism of the theory; Cayley's theorem . .' ' .'"', , . 138—141
101 — 104 Rational transformation by means of (^-polynomials . . . 142—146
105 — 108 Application of special sets 147 — 151
109 The hyperelliptic surface ; transformation to canonical form . 152
1.10 — 114 Whole rational theory can be represented by means of the invari ant ratios of (^-polynomials ; number of relations connecting
these 153—159
115 — 119 Elementary considerations in regard to curves in space . . 160 — 167
CHAPTER VII.
COORDINATION OF SIMPLE ELEMENTS. TRANSCENDENTAL UNIFORM
FUNCTIONS.
Scope of the chapter 168
Notation for integrals of the first kind . . . . . 169
The function ^ (x, a; z, cl5 ..., cp) expressed by Riemann integrals 170, 171
Derivation of a certain prime function 172
Applications of this function to rational functions and integrals 173
Xll CONTENTS.
§§ PAGES
126—128 The function ^(x,a-, z, c) ; its utility for the expression of
rational functions 174 — 176
129 The derived prime function E(x,z); used to express rational
functions 177
130, 131 Algebraic expression of the functions ^ (x, a ; z, clt ...,cp),
ty{x, a; z,c) 177, 178
132 Examples of these functions; they determine algebraic expres sions for the elementary integrals 179 — 182
133, 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 182—185
135 Algebraic theorem equivalent to interchange of argument and
parameter 185
136 Elementary canonical integral of the second kind . . . 186, 187
137 Applications. Canonical integral of the third kind deduced from
the function ^(.v,a; z,c^ ...,cp). Modification for the func tion ty(x, a; z, c) 188—192
138 Associated integrals of first and second kind. Further canonical
integrals. The algebraic theory of the hyperelliptic integrals
in one formula. . . 193, 194
139, 140 Deduction of Weierstrass's and Riemann's relations for periods
of integrals of the first and second kind .... 195 — 197
141 Either form is equivalent to the other 198
142 Alternative proofs of Weierstrass's and Riemann's period relations 199, 200
143 ' Expression of uniform transcendental function by the function
ty(x, a; z, c) ' . . . . 201
144, 145 Mittag-Lefner's theorem . . . ... . . 202 204
146 Expression of uniform transcendental function in prime factors 205
147 General form of interchange of argument and parameter, after
Abel 206
CHAPTER VIII.
ABEL'S THEOREM. ABEL'S DIFFERENTIAL EQUATIONS.
148—150 Approximative description of Abel's theorem 207—210
151 Enunciation of the theorem 210
152 The general theorem reduced to two simpler theorems . . 211, 212 153, 154 Proof and analytical statement of the theorem .... 212 214
155 Remark; statement in terms of polynomials . . . . 215
156 The disappearance of the logarithm on the right side of the
equation . . 216
157 Applications of the theorem. Abel's own proof .... 217 222
158, 159 The number of algebraically independent equations given by the
theorem. Inverse of Abel's theorem 222 224
160, 161 Integration of Abel's differential equations " 225 231
162 Abel's theorem proved quite similarly for curves in space . . 231 — 234
CONTENTS. Xlll
CHAPTER IX.
JACOBI'S INVERSION PROBLEM.
§§ PAGES
163 Statement of the problem 235
164 Uniqueness of any solution 236
165 The necessity of using congruences and not equations . . 237 166, 167 Avoidance of functions with infinitesimal periods . . . 238, 239
168, 169 Proof of the existence of a solution 239—241
170 — 172 Formation of functions with which to express the solution;
connection with theta functions . 242—245
CHAPTER X.
RIEMANN'S THETA FUNCTIONS. GENERAL THEORY.
173 Sketch of the history of the introduction of theta functions . 246
174 Convergence. Notation. Introduction of matrices . . . 247, 248 175, 176 Periodicity of the theta functions. Odd and even functions . 249 — 251
177 Number of zeros is p '. 252
178 Position of the zeros in the simple case . . . i 'orf3 253, 254
179 The places TOI} ..., mp 255
180 Position of the zeros in general 256, 257
181 Identical vanishing of the theta functions ..... 258, 259 182, 183 Fundamental properties. Geometrical interpretation of the places
m1,...,mp • . . . . 259—267
184 — 186 Geometrical developments; special inversion problem; contact
curves • ., tj 268 — 273
187 Solution of Jacobi's inversion problem by quotients of theta
functions 274, 275
188 Theory of the identical vanishing of the theta function. Ex
pression of (^-polynomials by theta functions . . . 276 — 282
189—191 General form of theta function. Fundamental formulae. Periodicity 283 — 286
192 Introduction of the f functions. Generalisation of an elliptic formula 287
193 Difference of two f functions expressed by algebraic integrals and
rational functions ....... 288
194 — 196 Development. Expression of single f function by algebraic integrals 289 — 292
197, 198 Introduction of the $ functions. Expression by rational functions 292-295
CHAPTER XI.
THE HYPERELLIPTIC CASE OF RlEMANN'S THETA FUNCTIONS.
199 Hyperelliptic case illustrates the general theory .... 296
200 The places »i1>t.., mp. The rule for half periods . . . 297, 298 201, 202 Fundamental set of characteristics defined by branch places . 299—301
XIV
CONTENTS.
§§ PAGES
203 Notation. General theorems to be illustrated .... 302
204, 205 Tables in illustration of the general theory 303—309
206 — 213 Algebraic expression of quotients of hyperelliptic theta functions.
Solution of hyperelliptic inversion problem . . . . 309 — 317 214, 215 Single £ function expressed by algebraical integrals and rational
functions 318 — 323
216 Rational expression of $> function. Relation to quotients of theta
functions. Solution of inversion problem by g> function . . 323 — 327
217 Rational expression of $> function 327 — 330
218 — 220 Algebraic deduction of addition equation for theta functions
when p = 2; generalisation of the equation tr (u+v) a- (u-v)
= cr2w. o-V(^v-jptt) 330—337
221 Examples for the case p = 2. Qopel's biquadratic relation . . 337 — 342
CHAPTER XII.
A PARTICULAR FORM OF FUNDAMENTAL SURFACE.
222 Chapter introduced as a change of independent variable, and as
introducing a particular prime function .... 343
223—225 Definition of a group of substitutions ; fundamental properties . 343—348
226, 227 Convergence of a series ; functions associated with the group . 349 — 352 228 — 232 The fundamental functions. Comparison with foregoing theory
of this volume 353 — 359
233 — 235 Definition and periodicity of the Schottky prime function . . 359 — 364
236, 237 Its connection with the theta functions 364 — 366
238 A further function connected therewith 367 — 372
239 The hyperelliptic case . . . . .. . «. . . 372, 373
CHAPTER XIII. RADICAL FUNCTIONS.
240 Introductory . . 374
241, 242 Expression of any radical function by Riemann's integrals, and
by theta functions 375, 376
243 Radical functions are a generalisation of rational functions . 377
244, 245 Characteristics of radical functions . . . . . . 378 — 381
246 — 249 Bitangents of a plane quartic curve 381 — 390
250, 251 Solution of the inversion problem by radical functions . . 390 — 392
CHAPTER XIV.
FACTORIAL FUNCTIONS.
252 Statement of results obtained. Notations 393, 394
253 Necessary dissection of the Riemann surface .... 395
254 Definition of a factorial function (including radical function).
Primary and associated systems of factorial functions . . 396, 397
CONTENTS.
XV
§§ PAGES
255 Factorial integrals of the primary and associated systems . . 397, 398
256 Factorial integrals which are everywhere finite, save at the fixed
infinities. Introduction of the numbers or, <r + 1 . . . 399
257 When <r + l>0, there are o- + l everywhere finite factorial functions
of the associated system ........ 400
258 Alternative investigation of everywhere finite factorial functions
of the associated system. Theory divisible according to the
values of o- + l and o-' + l 401, 402
259 Expression of these functions by everywhere finite integrals . 403
260 General consideration of the periods of the factorial integrals . 404 261, 262 Riemanri-Roch theorem for factorial functions. When or' + 1=0,
least number of arbitrary poles for fimction of the primary
system is or' + l 405, 406
263 Construction of factorial function of the primary system with
or' + l arbitrary poles . . . . . . . . 406, 407
264, 265 Construction of a factorial integral having only poles. Least number of such poles, for an integral of the primary system,
is o- + 2 407, 410
266 This factorial integral can be simplified, in analogy with Riemann's
elementary integral of the second kind 411
267 Expression of the factorial function with or' + l poles in terms of
the factorial integral with o- + 2 poles. The factorial function
in analogy with the function i\r (x, a; z, clt ..., cp). . . 411 — 413
268 The theory tested by examination of a very particular case . 413 — 419
269 The radical functions as a particular case of factorial functions 419, 420
270 Factorial functions whose factors are any constants, having no
essential singularities ........ 421
271, 272 Investigation of a general formula connecting factorial functions
and theta functions 422 — 426
273 Introduction of the Schottky-Klein prime form, in a certain shape 427 — 430
274 Expression of a theta function in terms of radical functions, as '
a particular case of § 272 . . . . '-. . " . 430
275, 276 The formula of § 272 for the case of rational functions . . 431—433
277 The formula of § 272 applied to define algebraically the hyper-
elliptic theta function, and its theta characteristic . . 433 — 437
278 Expression of any factorial function by simple theta functions ;
examples 437, 433
279 Connection of theory of factorial functions with theory of auto-
morphic forms . 439 442
CHAPTER XV.
RELATIONS CONNECTING PRODUCTS OF THETA FUNCTIONS — INTRODUCTORY.
280 281
443
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 .... 443 445
XVI CONTENTS.
§§ PAGES
282, 283 Proof that the 22p theta functions with half-integer character istics are linearly independent ...... 446 — 447
284, 285 Definition of general theta function of order r ; its linear expres sion by r1' theta functions. Any p-f2 theta functions of same order, periods, and characteristic connected by a homo geneous polynomial relation ....... 447 — 455
286 Addition theorem for hyperelliptic theta functions, or for the
general case when p<4 ........ 456 — 461
286, 288 Number of linearly independent theta functions of order r which
are all of the same parity ....... 461 — 464
289 Examples. The Gopel biquadratic relation 465 — 470
CHAPTER XVI.
A DIRECT METHOD OF OBTAINING THE EQUATIONS CONNECTING THETA
PRODUCTS.
290 Contents of this chapter 471
291 An addition theorem obtained by multiplying two theta functions . 471 — 474
292 An addition theorem obtained by multiplying four theta functions 474 — 477
293 The general formula obtained by multiplying any number of
theta functions . . . . . . 477 485
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, 487
295 A preliminary lemma . 488
296 Determination of a Gopel group of characteristics . . . 489, 490
297 Determination of a Gopel system of characteristics . . . 490, 491 298, 299 Determination and number of Gopel systems of the same parity 492 — 494 300 — 303 Determination of a fundamental set of Gopel systems . . 494 — 501 304, 305 Statement of results obtained, with the simpler applications . 502 — 504 306 — 308 Number of linearly independent theta functions of the second
order of a particular kind. Explicit mention of an import ant identity 505 — 510
309 — 311 The most important formulae of the chapter. A general addi tion theorem. The g> function expressed by quotients of
theta functions 510 — 516
312 — 317 Other applications of the principles of the chapter. The expres sion of a function 3 (nv) as an integral polynomial of order
«2 in 2" functions $(v) 517—527
CONTENTS.
XV11
CHAPTER XVIII. TRANSFORMATION OF PERIODS, ESPECIALLY LINEAR TRANSFORMATION.
§§ PAGES
318 Bearings of the theory of transformation 528, 529
319 — 323 The general theory of the modification of the period loops on a
Riemann surface 529 — 534
324 Analytical theory of transformation of periods and characteristic
of a theta function 534 — 538
325 Convergence of the transformed function . . . . . 538
326 Specialisation of the formulae, for linear transformation . . 539, 540
327 Transformation of theta characteristics ; of even characteristics ;
of syzygetic characteristics . . . . . . .541, 542
328 Period characteristics and theta characteristics . . . • . 543
329 Determination of a linear transformation to transform any even
characteristic into the zero characteristic .... 544, 545
330, 331 Linear transformation of two azygetic systems of theta charac teristics into one another . . . . . . . . 546 — 550
332 Composition of two transformations of different orders ; supple mentary transformations 551, 552
333, 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 ,.:;./.,. Lrj . 553—557
335 The constant factor for any linear transformation . . . 558, 559
336 Any linear transformation may be associated with a change of
the period loops of a Riemann surface 560, 561
337, 338 Linear transformation of the places mlt ..., mp .•* • . . . 562
339 Linear transformation of the characteristics of a radical function 563, 564
340 Determination of the places Wj, ..., mp upon a Riemann surface
whose mode of dissection is assigned . ..».'..•. .. 565 — 567
341 Linear transformation of quotients of hyperelliptic theta functions 568
342 A convenient form of the period loops in a special hyperelliptic
case. Weierstrass's number notation for half-integer charac teristics . . . ..:;.:,;; .;. • . •..:. . . . 569, 570
CHAPTER XIX.
ON SYSTEMS OF PERIODS AND ON GENERAL JACOBIAN FUNCTIONS.
343
344—350
571
571—579
Scope of this chapter .........
Columns of periods. Exclusion of infinitesimal periods. Expres sion of all period columns by a finite number of columns,
with integer coefficients
351 — 356 Definition of general Jacobian function, and comparison with
theta function 579 588
357—362 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 . . 588 — 598
XV111 CONTENTS.
CHAPTER XX. TRANSFORMATION OF THETA FUNCTIONS.
§§ PAGES
363 Sketch of the results obtained. References to the literature . 599, 600
364, 365 Elementary theory of transformation of second order . . . 600 — 606 366, 367 Investigation of a general formula preliminary to transformation
of odd order 607—610
368, 369 The general theorem for transformation of odd order . . . 611 — 616
370 The general treatment of transformation of the second order . 617—619
371 The two steps in the determination of the constant coefficients 619
372 The first step in the determination of the constant coefficients 619 — 622
373 Remarks and examples in regard to the second step . . . 622 — 624
374 Transformation of periods when the coefficients are not integral 624 — 628
375 Reference to the algebraical applications of the theory . . 628
CHAPTER XXI.
COMPLEX MULTIPLICATION OF THETA FUNCTIONS. CORRESPONDENCE OF POINTS ON A RlEMANN SURFACE.
376 Scope of the chapter . . . 629
377, 378 Necessary conditions for a complex multiplication, or special
transformation, of theta functions ...... 629 — 632
379 — 382 Proof, in one case, that these conditions are sufficient . . 632 — 636
383 Example of the elliptic case 636—639
384 Meaning of an (r, s) correspondence on a Riemann surface . 639, 640
385 Equations necessary for the existence of such a correspondence 640 — 642
386 Algebraic determination of a correspondence existing on a per
fectly general Riemann surface . ... . . . . 642 — 645
387 The coincidences. Examples of the inflections and bitangents of
a plane curve 645 — 648
388 Conditions for a (1, s) correspondence on a special Riemann surface 648, 649
389 When p>l a (1, 1) correspondence is necessarily periodic . . 649, 650
390 And involves a special form of fundamental equation . . 651 391—393 When p>l there cannot be an infinite number of (1, 1) corre spondences 652 — 654
394 Example of the case p = l 654—656
CHAPTER XXII. DEGENERATE ABELIAN INTEGRALS.
395 Example of the property to be considered . . . . 657
396 Weierstrass's theorem. The property involves a transformation
leading to a theta function which breaks into factors . . 657, 658
CONTENTS.
XIX
397 Weierstrass's and Picard's theorem. The property involves a
linear transformation leading to T^'2 = l/r.
398 Existence of one degenerate integral involves another (p = 2) 399, 400 Connection with theory of special transformation, when p — Z . 401 — 403 Determination of necessary form of fundamental equation.
Eeferences
PAGES
658, 659
659 660, 661
661—663
404
APPENDIX I.
ON ALGEBRAIC CURVES IN SPACE.
Formal proof that an algebraic curve in space is an interpreta tion of the relations connecting three rational functions on a Riemann surface (cf. § 162) 664,
665
APPENDIX II.
ON MATRICES.
405 — 410 Introductory explanations . . . ; ., . , ... 666 669
411 — 415 Decomposition of an Abelian matrix into simpler ones . ' . 669 674
416 A particular result ... 674
417, 418 Lemmas , . . 675
419, 420 Proof of results assumed in §§ 396, 397 .... .'. 675, 676
INDEX OF AUTHORS QUOTED . TABLE OF SOME FUNCTIONAL SYMBOLS SUBJECT INDEX
677, 678
679 680—684
ADDENDA. CORRIGENDA.
PAGE LINE
6, 2, for bb^da, read (tb*~lda.
8, 22, for deficiency 1, read deficiency 0.
11, 12, for 2n-2+p, read 2n-2 + 2p.
16, § 16, 4, for called, read applied to.
dx . dx
18, 25, for — , read — .
x y
37, 31, for in, read is.
38, 3, for surfaces, read surface. 43, 20, for w, read w.
56, 22, for (x-af~\ read (x -a)P-A+1.
61, 24, add or g{ (x, y).
66, 22, for r'-l, read Tj'-l.
70, 14, for rr+l, read rr+l.
73, 28, for x'^'^''2 sl5 2, read x~2r'~2 slt j.
81. The argument of § 52 supposes p>l.
104, § 72. See also Hensel, Crelle, cxv. (1895).
114, 3 from the bottom, add here.
137. To the references, add, Macaulay, Proc. Lon. Math. Soc., xxvi. p. 495.
157. See also Kraus, Math. Annal. xvi. (1879).
166. See also Zeuthen, Ann. d. Mat. 2a Ser., t. in. (1869).
189, 21, for xii, read xi.
196, 23, for \h, read \h. 24, for \h, read \h.
197, 24, for A, read B.
198, 5, for ^(w')"1^, read y(u')~lu.
18, for fourth minus sign, read sign of equality.
206, 4, supply dz, after third integral sign: the summation is from k = 2, fc'=0.
5, supply dz, after first integral sign. 8, for $(X)l<t>(X), read 0'(*)/0(X). 247, 11. Positive means >0. The discriminant must not vanish.
6 from bottom. Cf. p. 531, notef. 282, 11, for ft, read O.
284, 18, the equation is httP = iriP + bP'.
316, 3 from the bottom, for u, read UQ.
320, heading, destroy full stop.
327, 23, for Pi(xp), read /J.J(XP).
340. Further references are given in the report of Brill and Noether (see
Preface), p. 473. 342. For various notations for characteristics see the references in the report of
Brill and Noether, p. 519. 379, 16, for T(II, ritp, read v^-", vpx'a.
420, 18, read ...characteristic, other than the zero characteristic, as the sum of two
different odd half-integer characteristics in
441, 15, for one, read in turn every combination.
533, 13. The relation had been given by Frobenius.
557, 15, for .w2, read w-?.
575, 20, for from, read for.
587, 8 and 11 ; the quantity is AeA.
In this volume no account is given of the differential equations satisfied by the theta functions, or of their expansion in integral powers of the arguments. The following refer ences may be useful : Wiltheiss, Crelle, xcix., Math. Annal. xxix., xxxi., xxxin., Gotting. Nachr., 1889, p. 381; Pascal, Gotting. Xachr., 1889, pp. 416, 547, Ann. di Mat., Ser. 2% t. xvii.; Burkhardt (and Klein), Math. Annal. xxxn. The case p — 2 is considered in Krause, Transf. Hyperellip. Functionen.-
The following books of recent appearance, not referred to in the text, may be named here. (1) The completion of Picard, Traite d'Analyse, (2) Jordan, Cours d'Analyse, t. n. (1894), (3) Appell and Goursat, Theorie des Fonctions algebriques et de leurs integrates (1895), (4) Stahl, Theorie der AbeVschen Functionen (1896).
CHAPTER I.
ADDITIONAL CORRECTIONS FOR BAKER'S ABELIAN FUNCTIONS.
PAGE LINE
138, 14, from the bottom, for greater, read less.
219, 12, 13, from the bottom, for r, read R.
315, 6, from the bottom, for f, read f.
316, 5, from the bottom, for u, read u0.
317, 4, from the bottom, for a, vanishes, &j , read, respectively, bl , is infinite, a. 333, 3, for the first + , read - .
333, 3, 7, 8, from the bottom, for A, read V.
334, 6, 7, from the bottom, for pt, pj, read p$, pf'2.
335, 12, from the bottom, for A, read \ .
340, 6, from the bottom, for Gopel, read Kummer. Supply also the reference,
Weber, Crelle LXXXIV. (1878), p. 341.
359, 1, after periods, add and let ^ (u) = @ (u) + @(u + u').
5, for $>, read ^ ; for iir, read Ziir.
9, for P + iQ, read (P + iQ) [(£>' (u) + $' (v)], where u, v are the arguments occurring in the denominator ; and similarly for P-iQ ; and add to the function
the term 4P f (u) - — , f (w')l, where f(w) is Weierstrass's function.
367, 5, from the bottom, for m, read p.
444, 16, for x, read u.
445, 14, for n, read p.
457, 14, from the bottom, supply the reference, § 181.
615, 5, for xviii., read xvn.
665, 6, from the bottom, add, which may be taken to be linear polynomials in
x only.
sheet. Or the sheets may wind into one another : in which case we shall regard this winding point (or branch point) as constituting one place : this place belongs then indifferently to either sheet ; the sheets here merge into one another. In the first case, if a be the value of x for which the sheets just touch, supposed for convenience of statement to be finite, and x a value
* For references see Chap. II. § 12, note.
t Such a point is called by Riemann "ein sich aufhebender Verzweigungspunkt " : Gesam- melte Werke (1876), p. 105.
B. 1
ADDENDA. CORRIGENDA.
PAGE LINE
^
6, 2, for db^da, read db^~da.
8, 22, for deficiency 1, read deficiency 0.
11, 12, for 2n-2+p, read 2n-2 + 2p.
16, § 16, 4, for called, read applied to.
dx , da;
18, 25, for — , read — .
x y
37, 31, for in, read is.
38, 3, for surfaces, read surface. 43, 20, for w, read w.
56, 22, for (x-af~\ read (x-a)<>-*+1.
fil 24. add or n, (x. u\.
587, 8 and 11 ; the quantity is AeA.
In this volume no account is given of the differential equations satisfied by the theta functions, or of their expansion in integral powers of the arguments. The following refer ences may be useful : Wiltheiss, Crelle, xcix., Hath. Annal. xxix., xxxi., xxxm., Gotting. Nachr., 1889, p. 381; Pascal, Gotting. Nachr., 1889, pp. 416, 547, Ann. di Mat., Ser. 2% t. xvii.; Burkhardt (and Klein), Math. Annal. xxxn. The case p = 2 is considered in Krause, Transf. Hyperellip. Functionen.-
The following books of recent appearance, not referred to in the text, may be named here. (1) The completion of Picard, Traite d'Anatyse, (2) Jordan, Cours d'Analyse, t. 11. (1894), (3) Appell and Goursat, Theorie des Fonctions algebriques et de leurs integrates (1895), (4) Stahl, Theorie der AbeVschen Functionen (1896).
CHAPTER I.
THE SUBJECT OF INVESTIGATION.
1. THIS book is concerned with a particular development of the theory of the algebraic irrationality arising when a quantity y is defined in terms of a quantity x by means of an equation of the form
a0yn + atf1'1 +...+ an^y + an = 0,
wherein a0, al} ...,an are rational integral polynomials in x. The equation is supposed to be irreducible ; that is, the left-hand side cannot be written as the product of other expressions of the same rational form.
2. Of the various means by which this dependence may be represented, that invented by Riemann, the so-called Riemann surface, is throughout regarded as fundamental. Of this it is not necessary to give an account here*. But the sense in which we speak of a place of a Riemann surface must be explained. To a value of the independent variable x there will in general correspond n distinct values of the dependent variable y — represented by as many places, lying in distinct sheets of the surface. For some values of x two of these n values of y may happen to be equal : in that case the corresponding sheets of the surface may behave in one of two ways. Either they may just touch at one point without having any further connexion in the immediate neighbourhood of the point t : in which case we shall regard the point where the sheets touch as constituting two places, one in each sheet. Or the sheets may wind into one another : in which case we shall regard this winding point (or branch point) as constituting one place : this place belongs then indifferently to either sheet ; the sheets here merge into one another. In the first case, if a be the value of x for which the sheets just touch, supposed for convenience of statement to be finite, and x a value
* For references see Chap. II. § 12, note.
t Such a point is called by Riemann "ein sich aufhebender Verzweigungspunkt " : Gesam- melte Werke (1876), p. 105.
B. 1
2 THE PLACES OF A RIEMANN SURFACE. [2
very near to a, and if b be the value of y at each of the two places, also supposed finite, and ylt yz be values of y very near to b, represented by points in the two sheets very near to the point of contact of the two sheets, each of 3/1 — 6, yz — b can be expressed as a power-series in x — a with integral exponents. In the second case with a similar notation each of 2/1 — 6, y2 — 6 can be expressed as a power-series in (x — a)* with integral exponents. In the first case a small closed curve can be drawn on either of the two sheets considered, to enclose the point at which the sheets touch :
and the value of the integral •= — . Id log (x - a) taken round this closed curve
will be 1 ; hence, adopting a definition given by Riemann*, we shall say that x — a is an infinitesimal of the first order at each of the places. In the second case the attempt to enclose the place by a curve leads to a curve lying partly in one sheet and partly in the other; in fact, in order that the curve may be closed it must pass twice round the branch place. In this
case the integral ^ — . Id log [(x — a)*] taken round the closed curve will be 1 :
and we speak of (x — a}*- as an infinitesimal of the first order at the place. In either case, if t denote the infinitesimal, x and y are uniform functions of t in the immediate neighbourhood of the place ; conversely, to each point on the surface in the immediate neighbourhood of the place there corre sponds uniformly a certain value of if. The quantity t effects therefore a conformal representation of this neighbourhood upon a small simple area in the plane of t, surrounding t — 0.
3. This description of a simple case will make the general case clear. In general for any finite value of x, x = a, there may be several, say k, branch points J; the number of sheets that wind at these branch points may be denoted by w1+l,w.2+l, . .., wk+ 1 respectively, where
(w1 + 1) + (w, + l) + ...+(wk+l) = n,
so that the case of no branch point is characterised by a zero value of the corresponding w. For instance in the first case above, notwithstanding that two of the n values of y are the same, each of w1} w.2, ...,Wk is zero and k is equal to n : and in the second case above, the values are k = n — 1, wr = 1, w.2 = 0, w3 = 0, . . . , wk = 0. In the general case each of these k branch points is called a
place, and at these respective places the quantities (x - a)w>+l, ..., (x— a)wt+l
* Gesammelte Werke (1876), p. 96.
+ The limitation to the immediate neighbourhood involves that t is not necessarily a rational function of x, y.
It may be remarked that a rational function of x and y can be found whose behaviour in the neighbourhood of the place is the same as that of t. See for example Hamburger, Zeitschrift f. Math, und Phys. Bd. 16, 1871 ; Stolz, Math. Ann. 8, 1874 ; Harkness and Morley, Theory of Functions, p. 141.
t Cf. Forsyth, Theory of Functions, p. 171. Prym, Crelle, Bd. 70.
4] TRANSFORMATION OF THE EQUATION. 3
are infinitesimals of the first order. For the infinite value of x we shall similarly have n or a less number of places and as many infinitesimals, say
-_
+1, ..., (-r'+1, where (Wl + l)+ ... +(w,. + I) = n. And as in the par-
xj \x/
ticular cases discussed above, the infinitesimal t thus defined for every place of the surface has the two characteristics that for the immediate neighbour hood of the place x and y are uniquely expressible thereby (in series of integral powers), and conversely t is a uniform function of position on the surface in this neighbourhood. Both these are expressed by saying that t effects a reversible conformal representation of this neighbourhood upon a simple area enclosing t = 0. It is obvious of course that quantities other than t have the same property.
A place of the Riemann surface will generally be denoted by a single letter. And in fact a place (x, y} will generally be called the place x. When we have occasion to speak of the (n or less) places where the inde pendent variable x has the same value, a different notation will be used.
4. We have said that the subject of enquiry in this book is a certain algebraic irrationality. We may expect therefore that the theory is practi cally unaltered by a rational transformation of the variables x, y which is of a reversible character. Without entering here into the theory of such trans formations, which comes more properly later, in connexion with the theory of correspondence, it is necessary to give sufficient explanations to make it clear that the functions to be considered belong to a whole class of Riemann surfaces and are not the exclusive outcome of that one which we adopt initially.
Let £ be any one of those uniform functions of position on the funda mental (undissected) Riemann surface whose infinities are all of finite order. Such functions can be expressed rationally by x and y*. For that reason we shall speak of them shortly as the rational functions of the surface. The order of infinity of such a function at any place of the surface where the function becomes infinite is the same as that of a certain integral power of
the inverse - of the infinitesimal at that place. The sum of these orders of
6
infinity for all the infinities of the function is called the order of the function. The number of places at which the function f assumes any other value a is the same as this order : it being understood that a place at which £ — a is zero in a finite ratio to the rth order of t is counted as r places at which £ is equal to off. Let v be the order of £. Let T? be another rational function of
* Forsyth, Theory of Functions, p. 370.
t For the integral — /dlog(£-a), taken round an infinity of log(£-a), is equal to the
order of zero of £ - a at the place, or to the negative of the order of infinity of £, as the case may be. And the sum of the integrals for all such places is equal to the value round the boundary of the surface— which is zero. Cf. Forsyth, Theonj of Functions, p. 372.
1—2
4 CONDITION OF REVERSIBILITY. [4
order p. Take a plane whose real points represent all the possible values of |f in the ordinary way. To any value of |f, say |f = a, will correspond v positions Xlt ..., Xvon the original Riemann surface, those namely where £ is equal to a : it is quite possible that they lie at less than v places of the surface. The values of 77 at X1} ..., Xv may or may not be different. Let H denote any definite rational symmetrical function of these v values of 77. Then to each position of a in the |f plane will correspond a perfectly unique value of H, namely, H is a one-valued function of £. Moreover, since 77 and |f are rational functions on the original surface, the character of H for values of |f in the immediate neighbourhood of a value a, for which H is infinite, is clearly the same as that of a finite power of ff — a. Hence H is a rational function of |f. Hence, if Hr denote the sum of the products of the values of i] at Xlt ..., Xv, r together, 77 satisfies an equation
r)"-r)"^H1 + r)^H2-...+(-YHv = 0> whose coefficients are rational functions of |f.
It is conceivable that the left side of this equation can be written as the product of several factors each rational in |f and 77. If possible let this be done. Construct over the |f plane the Riemann surfaces corresponding to these irreducible factors, 77 being the dependent variable and the various surfaces lying above one another in some order. It is a known fact, already used in defining the order of a rational function on a Riemann surface, that the values of 77 represented by any one of these superimposed surfaces in clude all possible values — each value in fact occurring the same number of times on each surface. To any place of the original surface, where |f, 77 have definite values, and to the neighbourhood of this place, will correspond there fore a definite place (|f, 77) (and its neighbourhood) on each of these super imposed surfaces. Let 77!, ...,tjr be the values of 77 belonging, on one of these surfaces, to a value of £ : and T?/, ..., r}s' the values belonging to the same value of |f on another of these surfaces. Since for each of these surfaces there are only a finite number of values of £ at which the values of 77 are not all different, we may suppose that all these r values on the one surface are different from one another, and likewise the s values on the other surface. Since each of the pairs of values (|f, 77^, . . . , (|f, r)r) must arise on both these surfaces, it follows that the values 77!, ...,tjr are included among 77/, ..., 77/. Similarly the values T7/, ..., i?/ are included among 77^ ...,77,.. Hence these two sets are the same and r = s. Since this is true for an infinite number of values of |f, it follows that these two surfaces are merely repetitions of one another. The same is true for every such two surfaces. Hence r is a divisor of v and the equation
when reducible, is the v/rih power of a rational equation of order r in 77. It will be sufficient to confine our attention to one of the factors and the (£, 77)
5] CORRESPONDENCE OF TWO SURFACES. 5
surface represented thereby. Let now Xlt . . . , Xv be the places on the original surface where £ has a certain value. Then the values of 77 at Xlt . . , Xv will consist of v/r repetitions of r values, these r values being different from one another except for a finite number of values of £ Thus to any place (f, 77) on one of the v/r derived surfaces will correspond v/r places on the original surface, those namely where the pair (£, 77) take the supposed values. Denote these by PlfPa, — Let Y be any rational symmetrical function of the v/r pairs of values (a}1} y^), (#2, 2/2)» •••> which the fundamental variables a, y of the original surface assume at P1; P2) — Then to any pair of values (£, 77) will correspond only one value of Y — namely, Y is a one-valued function on the (£, 77) surface. It has clearly also only finite orders of infinity. Hence Y is a rational function of £, 77. In particular #u #2, ... are the roots of an equation whose coefficients are rational in £, 77 — as also are yi} yz, ____
There exists therefore a correspondence between the (£, 77) and (x, y)
surfaces — of the kind which we call a (1, - j correspondence: to every place of the (x, y) surface corresponds one place of the (£, 77) surface; to every place of this surface correspond - places of the (x, y) surface.
The case which most commonly arises is that in which the rational irreducible equation satisfied by 77 is of the vih degree in 77: then only one place of the original surface is associated with any place of the new surface. In that case, as will appear, the new surface is as general as the original surface. Many advantages may be expected to accrue from the utilization of that fact. We may compare the case of the reduction of the general equation of a conic to an equation referred to the principal axes of the conic.
5. The following method* is theoretically effective for the expression of x, y in terms of & r,.
Let the rational expression of £, rj in terms of x, y be given by
<£ (x, y) - & (x, y ) = 0, ^ (x, y} - rfX (x, y} = 0,
and let the rational result of eliminating #, y between these equations and the initial equation connecting x, y be denoted by F(£, rj) = 0, each of $, ..., ^, ^denoting integral polynomials. Let two terms of the expression (f>(z, y) — ty(&, y) = 0 be axry*—t-bxr'y*'. This expression and therefore all others involved will be unaltered if «, 6 be replaced by such quantities a + h, b + k, that hxry*=z£kxr'y*'. In a formal sense this changes F(£, rj) into
where X ^ 1, and F is such that all differential coefficients of it in regard to a and b of order less than X are identically zero.
Hence the term within the square brackets in this expression must be zero. If it is possible, choose now r = rf + \ and s = s', so that k=
* Salmon's Higher Algebra (1885), p. 97, § 103.
ALGEBRAICAL FORMULATION. [5
Then we obtain the equation
This is an equation of the form above referred to, by which x is determinate from £ and T]. And y is similarly determinate.
It will be noticed that the rational expression of xt y by £, rj, when it is possible from the equations
will not be possible, in general, from the first two equations : it is only the places x, y satisfying the equation f(x, y) = Q which are rationally obtainable from the places £, 17 satisfying the equation F(£, r)) = 0. There do exist transformations, rationally reversible, subject to no such restriction. They are those known as Cremona-transformations*. They can be compounded by reapplication of the transformation x : y : I = rj : {• : £»/.
We may give an example of both of these transformations — For the surface
the function £=y2/(^2 + .£ + l) is of order 2, being infinite at the places where x2+z+l = 0, in each case like (x-a)~°, and the function r}=x/y is of order 4, being infinite at the places x*+x+I=0, in each case like (^-a)"t, a being the value of x at the place. From the given equation we immediately find, as the relation connecting £ and 17,
and infer, since the equation formed as in the general statement above should be of order 2 in rj, that this general equation will be
Thence in accordance with that general statement we infer that to each place (£, >;) on the new surface should correspond two places of the original surface : and in fact these are obviously given by the equations
r}^=^/ If however we take
£=y2/(#2
where « is an imaginary cube root of unity, so that 17 is a function of order 3, these equations are reversible independently of the original equation, giving in fact
x = („£ _ wy )/(£ - ^}, y = (m- 2 and we obtain the surface
having a (1, 1) correspondence with the original one.
It ought however to be remarked that it is generally possible to obtain reversible transformations which are not Cremona-transformations.
6. When a surface (x, y) is (1,1) related to a (£, 77) surface, the defi ciencies of the surfaces, as denned by Riemann by means of the connectivity, must clearly be the same.
* See Salmon, Higher Plane Curves (1879), § 362, p. 322.
6] RELATION OF DEF1CIENCES. 7
It is instructive to verify this from another point of view*. — Consider at
how many places on the original surface the function -~ is zero. It is infinite
CLOG
at the places where % is infinite: suppose for simplicity that these are separated places on the original surface or in other words are infinities of the first order, and are not at the branch points of the original surface. At
d£ 1
a pole of £, ,- is infinite twice. It is infinite like — at a branch place (a)
CLOG v
where x — a = tw+l: namely it is infinite ^w = 2n + '2p - 2 times t at the branch places of the original surface. It is zero 2n times at the infinite places of the original surface. There remain therefore 2v + 2n + 2p — 2 — 2n = 2v + 2p — 2
places where ~ is zero. If a branch place of the original surface be a pole
1 -7fc 1
of £, and £ be there infinite like -, -~ is infinite like - — — , namely 2+w
t ax t2 . tw
times : the total number of infinities of -^ will therefore be the same as
dx
7«-
before. Now at a finite place of the original surface where -r = 0, there are
ax
two consecutive places for which £ has the same value. Since - = 1 they can
only arise from consecutive places of the new surface for which £ has the same value. The only consecutive places of a surface for which this is the case are the branch places. Hence f there are 2v+2p — 2 branch places of the new surface. This shews that the new surface is of deficiency p.
When v/r is not equal to 1, the case is different. The consecutive places of the old surface, for which £ has the same value, may either be those arising from consecutive places of the new surface — or may be what we may call accidental coincidences among the v/r places which correspond to one place of the new surface. Conversely, to a branch place of the new surface, characterised by the same value for £ for consecutive placesj, will correspond vjr places on the old surface where £ has the same value for consecutive places. In fact to two very near places of the new surface will correspond v/r pairs each of very near places on the old surface. If then C denote the number of places on the old surface at which two of the v/r places corre sponding to a place on the new surface happen to coincide, and w' the number of branch points of the new surface, we have the equation
'- r
* Compare the interesting geometrical account, Salmon, Higher Plane Curves (1879), p. 326, § 364, and the references there given.
t Forsyth, Tlieory of Functions, p. 348.
:£ Namely, near such a branch place f = a, £ - a is zero of higher order than the first.
8 PARAMETERS NOT REMOVED [6
and if p be the deficiency of the new surface (of r sheets), this leads to the equation
f
(2r + 2pf
from which
Corollary*. If p =p', then C = (2p - 2) (l - -\ . Thus - > 1, so that (7 = 0, and the correspondence is reversible.
We have, herein, excluded the case when some of the poles of £ are of higher than the first order. In that case the new surface has branch places at infinity. The number of finite branch places is correspondingly less. The reader can verify that the general result is unaffected.
Ex. In the example previously given (§ 5) shew that the function £ takes any given value at two points of the original surface (other than the branch places where it is infinite), 17 having the same value for these two points, and that there are six places at which these two places coincide. (These are the place (# = 0, y = 0) and the five places where x= — 2.)
There is one remark of considerable importance which follows from the theory here given. We have shewn that the number of places of the (x, y)
surface which correspond to one place of the (£, 97) surface is - , where v is the
order of £ and r is not greater than v, being the number of sheets of the (f , 77) surface ; hence, if there were a function £ of order 1 the correspondence would be reversible and therefore the original surface would be of deficiency 1.
7. This notion of the transformation of a Riemann surface suggests an inference of a fundamental character.
The original equation contains only a finite number of terms : the original surface depends therefore upon a finite number of constants, namely, the coefficients in the equation. But conversely it is not necessary, in order that the equation be reversibly transformable into another given one, that the equation of the new surface contain as many constants as that of the original surface. For we may hope to be able to choose a transformation whose coefficients so depend on the coefficients of the original equation as to reduce this number. If we speak of all surfaces of which any two are connected by a rational reversible transformation as belonging to the same class f, it becomes a question whether there is any limit to the reduction obtainable, by rational reversible transformation, in the number of constants in the equation of a surface of the class.
* See Weber, Crelle, 76, 345.
t So that surfaces of the same class will be of the same deficiency.
7] BY TRANSFORMATION. 9
It will appear in the course of the book* that there is a limit, and that the various classes of surfaces of given deficiency are of essentially different character according to the least number of constants upon which they depend. Further it will appear, that the most general class of deficiency p is characterised by 3p — 3 constants when p > 1 — the number for p = 1 being one, and for p = 0 none.
For the explanatory purposes of the present Chapter we shall content ourselves with the proof of the following statement — When a surface is reversibly transformed as explained in this Chapter, we cannot, even though we choose the new independent variable £ to contain a very large number of disposeable constants, prescribe the position of all the branch points of the new surface ; there will be 3p — 3 of them whose position is settled by the position of the others. Since the correspondence is reversible we may regard the new surface as fundamental, equally with the original surface. We infer therefore that the original surface depends on 3p — 3 parameters — or on less, for the 3/> — 3 undetermined branch points of the new surface may have mutually dependent positions.
In order to prove this statement we recall the fact that a function of order Q contains^ Q—p + l linearly entering constants when its poles are prescribed: it may contain more for values of Q<2p — 1, but we shall not thereby obtain as many constants as if we suppose Q > 2p — 2 and large enough. Also the Q infinities are at our disposal. We can then presumably dispose of 2Q-p + 1 of the branch points of the new surface. But these are, in number, 2Q + 2p — 2 when the correspondence is reversible. Hence we can dispose of all but 2Q + 2p - 2 - (2Q -p + 1) = 3p - 3 of the branch points of the new surface J.
Ex. 1. The surface associated with the equation
y*=x(l -x] (l-tfx) (1 -XV) (1 -MV) (l-v*x) (1 -p%) is of deficiency 3. It depends on 5 = 2p- 1 parameters, /c2, X2, /u2, v2, p2.
Ex. 2. The surface associated with the equation
y*+y*(x, l\+y(x, !), + (#, 1)4=0,
wherein the coefficients are integral polynomials of the orders specified by the suffixes, is of deficiency 3. Shew that it can be transformed to a form containing only 5 = 2^-1 parametric constants.
* See the Chapters on the geometrical theory and on the inversion of Abelian Integrals. The reason for the exception in case ^ = 0 or 1 will appear most clearly in the Chapter on the self- correspondence of a Riemann surface. But it is a familiar fact that the elliptic functions which can be constructed for a surface of deficiency 1 depend upon one parameter, commonly called the modulus : and the trigonometrical functions involve no such parameter.
t Forsyth, p. 459. The theorems here quoted are considered in detail in Chapter III. of the present book.
£ Cf. Kiemann, Ges. Werke (1876), p. 113. Klein, Ueber Riemann's Theorie (Leipzig, Teubner, 1882), p. 65.
c;
UN I VI.
Of ~ >-.
10 SELF-CORRESPONDENCE. [8
8. But there is a case in which this argument fails. If it be possible to transform the original surface into itself by a rational reversible transforma tion involving r parameters, any r places on the surface are effectively equivalent with, as being transformable into, any other r places. Then the Q poles of the function £ do not effectively supply Q but only Q — r dispose- able constants with which to fix the new surface. So that there are 3/> — 3 + r branch points of the new surface which remain beyond our control. In this case we may say that all the surfaces of the class contain 3p - 3 disposeable parameters beside r parameters which remain indeterminate and serve to represent the possibility of the self-transformation of the surface. It will be shewn in the chapter on self- transformation that the possibility only arises for p = 0 or p = 1, and that the values of r are, in these cases, respectively 3 and 1. We remark as to the case p = 0 that when the fundamental surface has only one sheet it can clearly be transformed into itself by
a transformation involving three constants x— 5 , : and in regard to p = 1,
c% -f d
the case of elliptic functions, that effectively a point represented by the elliptic argument u is equivalent to any other point represented by an argument u + 7. For instance a function of two poles is
and clearly Fa>ft has the same value at u as has Fa+y>p+y at u -f 7 : so that the poles (a, ft) are not, so far as absolute determinations are concerned, effective for the determination of more than one point.
9. The fundamental equation
a0yn + aiyn-l + ...+an = 0,
so far considered as associated with a Riemann surface, may also be regarded as the equation of a plane curve : and it is possible to base our theory on the geometrical notions thus suggested. Without doing this we shall in the following pages make frequent use of them for purposes of illustration. It is therefore proper to remind the reader of some fundamental properties*.
The branch points of the surface correspond to those points of the curve where a line x = constant meets the curve in two or more consecutive points : as for instance when it touches the curve, or passes through a cusp. On the other hand a double point of the curve corresponds to a point on the surface where two sheets just touch without further connexion. Thus the branch place of the surface which corresponds to a cusp is really a different singu larity to that which corresponds to a place where the curve is touched by a
* Cf. Forsyth, Theory of Functions, p. 355 etc. Harkness and Morley, Theory of Functions, p. 273 etc.
9] GEOMETRICAL VIEW. 11
line x = constant, being obtained by the coincidence of an ordinary branch place with such a place of the Riemann surface as corresponds to a double point of the curve.
Properties of either the Riemann surface or a plane curve are, in the simpler cases, immediately transformed. For instance, by Pliicker's formulae for a curve, since the number of tangents from any point is
f-(n-l)n-2£-3/c,
where n is the aggregate order in a; and y, it follows that the number of branch places of the corresponding surface is
w = t + K = (n - 1) n - 2 (8 + K)
= 2n-2 + 2{iO-l)O-2)-S-4
Thus since w = 2n — 2 -j^p, the deficiency of the surface is
£0-1)0- 2)- S-K, namely the number which is ordinarily called the deficiency of the curve.
To the theory of the birational transformation of the surface corresponds a theory of the birational transformation of plane curves. For example, the branch places of the new surface obtained from the surface f(x, y) = 0 by means of equations of the form <£ (x, y} — ty (x, y) = 0, $ (x, y) — 77% (x, ?/) = 0 will arise for those values of £ for which the curve </> (x, y) — jfy (x, y) — 0 touches f(x, y} = 0. The condition this should be so, called the tact inva riant, is known to involve the coefficients of <f> (as, y) — % \Jr (x, y~) = 0, and therefore in particular to involve £, to a degree* n (n — 3) — 28 — 3/c + 2nn, where n' is the order of <£ (x, y) — £i/r (x, y} = 0. Branch places of the new surface also arise corresponding to the cusps of the original curve. The total number is therefore n (n — 3) — 25 — 2* + Znri = *2p — 2'+ 2nn'. Now nri is the number of intersections of the curves f(x, y) = Q and <£ (x, y) — jfy (x, y) = 0, namely it is the number of values of t] arising for any value of £, and is thus the number of sheets of the new surface, which we have previously denoted by v : so that the result is as before.
In these remarks we have assumed that the dependent variable occurs to the order which is the highest aggregate order in x and y together — and we have spoken of this as the order of the curve. And in regarding two curves as intersecting in a number of points equal to the product of their orders we have allowed count of branches of the curve which are entirely at infinity. Some care is necessary in this regard. In speaking of the Riemann surface represented by a given equation it is intended, unless the contrary be stated, that such infinite branches are unrepresented. As an example the curve y- = (x, 1)6 may be cited.
Ex, Prove that if from any point of a curve, ordinary or multiple, or from a point not on the curve, t be the number of tangents which can be drawn other than those touching
* See Salmon, Higher Plane Curves (1879), p. 81.
12 GENERALITY [9
at the point, and K be the number of cusps of the curve — and if v be the number of points other than the point itself in which the curve is intersected by an arbitrary line through the point— -then t + K — 2i/ is independent of the position of the point. If the equation of the variable lines through the point be written u — gv = 0, interpret the result by regarding the curve as giving rise to a Riemann surface whose independent variable fa |*.
10. The geometrical considerations here referred to may however be stated with advantage in a very general manner.
In space of any (k) dimensions let there be a curve — (a one-dimension ality). Let points on this curve be given by the ratios of the k + 1 homo geneous variables xly ... , xk+1. Let u, v be any two rational integral homo geneous functions of these variables of the same order. The locus u — gv = 0 will intersect the curve in a certain number, say v, points — we assume the curve to be such that this is the same for all values of £, and is finite. Let all the possible values of £ be represented by the real points of an infinite plane in the ordinary way. Let w, t be any two other integral functions of the
w
coordinates of the same order. The values of t] = — at the points where
t
u — %v = 0 cuts the curve for any specified value of £ will be v in number. As before it follows thence that 77 satisfies an algebraic equation of order v whose coefficients are one- valued functions of £. Since 77 can only be infinite to a finite order it follows that these coefficients are rational functions of f . Thence we can construct a Riemann surface, associated with this algebraic equation connecting f and 77, such that every point of the curve gives rise to a place of the surface. In all cases in which the converse is true we may regard the curve as a representation of the surface, or conversely.
Thus such curves in space are divisible into sets according to their deficiency. And in connexion with such curves we can construct all the functions with which we deal upon a Riemann surface.
Of these principles sufficient account will be given below (Chapter VI.) : familiar examples are the space cubic, of deficiency zero, and the most general space quartic of deficiency 1 which is representable by elliptic functions.
11. In this chapter we have spoken primarily of the algebraic equation — and of the curve or the Riemann surface as determined thereby. But this is by no means the necessary order. If the Riemann surface be given, the algebraic equation can be determined from it — and in many forms, according to the function selected as dependent variable (y). It is necessary to keep this in view in order fully to appreciate the generality of Riemann's methods. For instance, we may start with a surface in space whose shape is that of an
* The reader who desires to study the geometrical theory referred to may consult : — Cayley, Quart. Journal, vn. ; H. J. S. Smith, Proc. Lond. Math. Soc. vi. ; Noether, Math. Annul. 9 ; Brill, Math. Annal. 16 ; Brill u. Noether, Math. Annul. 7.
11] OF THE THEORY. 13
anchor ring*, and construct upon this surface a set of elliptic functions. Or we may start with the surface on a plane which is exterior to two circles drawn upon the plane, and construct for this surface a set of elliptic functions. Much light is thrown upon the functions occurring in the theory by thus considering them in terms of what are in fact different independent variables. And further gain arises by going a step further. The infinite plane upon which uniform functions of a single variable are represented may be regarded as an infinite sphere ; and such surfaces as that of which the anchor ring above is an example may be regarded as generalizations of that simple case. Now we can treat of branches of a multiform function without the use of a Riemann surface, by supposing the branch points of the function marked on a single infinite plane and suitably connected by barriers, or cuts, across which the independent variable is supposed not to pass. In the same way, for any general Riemann surface, we may consider branches of functions which are not uniform upon that surface, the branches being separated by drawing barriers upon the surface. The properties obtained will obviously generalize the properties of the functions which are uniform upon the surface.
* Forsyth, p. 318 ; Kiemann, Ges. Werke (1876), pp. 89, 415.
[12
CHAPTER II.
THE FUNDAMENTAL FUNCTIONS ON A RTEMANN SURFACE.
12. IN the present chapter the theory of the fundamental functions is based upon certain a priori existence theorems*, originally given by Riemann. At least two other methods might be followed : in Chapters IV. and VI. sufficient indications are given to enable the reader to establish the theory independently upon purely algebraical considerations : from Chapter VI. it will be seen that still another basis is found in a preliminary theory of plane curves. In both these cases the ideas primarily involved are of a very elementary character. Nevertheless it appears that Riemann's descriptive theory is of more than equal power with any other ; and that it offers a generality of conception to which no other theory can lay claim. It is therefore regarded as fundamental throughout the book.
It is assumed that the Theory of Functions of Forsyth will be accessible to readers of the present book ; the aim in the present chapter has been to exclude all matter already contained there. References are given also to the treatise of Harkness and Morley*.
13. Let t be the infinitesimal f at any place of a Riemann surface : if it is a finite place, namely, a place at which the independent variable x is finite, the values of x for all points in the immediate neighbourhood of the place are expressible in the form x = a + tw+1 : if an infinite place, x = t~(w+1>. There exists a function which save for certain additive moduli is one-valued on the whole surface and everywhere finite and continuous, save at the place in question, in the neighbourhood of which it can be expressed in the form
* See for instance : Forsyth, Theory of Functions of a Complex Variable, 1893 ; Harkness and Morley, Treatise on the Theory of Functions, 1893 ; Schwarz, Gesam. math. Abhandlungen, 1890. The best of the early systematic expositions of many of the ideas involved is found in C. Neumann, Vorlesungen ilber Riemann's Theorie, 1884, which the reader is recommended to study. See also Picard, Traite d" Analyse, Tom. n. pp. 273, 42 and 77.
t For the notation see Chapter I. §§ 2, 3.
14] ELEMENTARY NORMAL INTEGRALS. 15
Herein, as throughout, P (t) denotes a series of positive integral powers of t vanishing when t = 0, G, A, ... , Ar^, are constants whose values can be arbitrarily assigned beforehand, and r is a positive integer whose value can be assigned beforehand.
We shall speak of all such functions as integrals of the second kind : but the name will be generally restricted to that * particular function whose behaviour near the place is that of
This function is not entirely unique. We suppose the surface dissected by 2p cutsf, which we shall call period loops; they subserve the purpose of rendering the function one-valued over the whole of the dissected surface. We impose the further condition that the periods of the function for transit across the p loops of the first kind j shall be zero ; then the function is unique save for an additive constant. It can therefore be made to vanish at an arbitrary place. The special function§ so obtained whose infinity is that
of - - is then denoted by Tax> c, c denoting the place where the function vanishes and as the current place. When the infinity is an ordinary place,
at which either sc = a or # = oo , the function is infinite either like ----
x — a
or - x. The periods of T/' * for transit of the period loops of the second kind will be denoted by fl1} ..., flp.
14. Let Oi^/i), (#ay2) be any two places of the surface: and let the infinitesimals be respectively denoted by tlt L, so that in the neighbourhood of these places we have the equations x — xl = £1W]+1, ac — x2 = t.?'*+1. Let a cut be made between the places (a?,^), (#2<y2). There exists a function, here denoted by n*1 c , which (a) is one-valued over the whole dissected surface,
3-1, <<2
(/3) has p periods arising for transit of the period loops of the second kind and has no periods at the period loop of the first kind, (7) is everywhere continuous and finite save near (a^) and (x.,ij.^), where it is infinite re spectively like log£j and -logt,, and, (8), vanishes when the current place denoted by x is the place denoted by c. This function is unique. If the cut between (a?^), (aray2) be not made, the function is only definite apart from an additive integral multiple of 2iri, whose value depends on the
* This particular function is also called an elementary integral of the second kind.
t Those ordinarily called the a, b curves; see Forsyth, p. 354. Harkness and Morley, p. 242, etc.
£ Those called the a cuts. ^,-
§ The fact that the function has no periods at the period loops of the first kind is gene rally denoted by calling the function a normal integral of the second kind.
16
ELEMENTARY NORMAL INTEGRALS.
[14
path by which the variable is supposed to pass from c. It will be called* the integral of the third kind whose infinity is like that of Iog(tift2).
15. Beside these functions there exist also certain integrals of the first kind — in number p. They are everywhere continuous and finite and one- valued on the dissected surface. For transit of the period loops of the first kind, one of them, say Vi, has no periods except for transit of the iih loop, ai. This period is here taken to be 1. The periods of Vi for transit of the period loops of the second kind are here denoted by rtV ..., T;P. We may therefore form the scheme of periods
|
a. |
do |
dp |
frl |
k |
|||
|
•Si |
1 |
0 |
0 |
TU |
T1P |
||
|
v.2 |
0 |
1 |
0 |
T21 |
T2P |
||
|
• |
|||||||
|
VP |
0 |
0 |
1 |
Tfl |
Tpp |
Each of these functions v^ is unique when a zero is given. They will there fore be denoted by v*' °, ..., vpx> c, the zero denoted by c being at our disposal. The periods ry- have certain properties which will be referred to in their proper place : in particular ry- = T^, so that they are certainly not equivalent to more than %p (p + 1) algebraically independent constants. As a fact, in accordance with the previous chapter, when p > 1 they are subject to l)- (3p - 3) = %(p - 2) (p - 3) relations.
16. In regard to these enunciations, the reader will notice that the word period here used for that additive constant arising for transit of a period loop — namely, in consequence of a path leading from one edge of the period loop to the opposite edge — would be more properly called the period for circuit of this path than the period for transit of the loop.
The integrals here specified are more precisely called the normal ele mentary integrals of their kinds. The general integral of the first kind is a linear function of Vj , . . . , vp with constant coefficients ; its periods at the first p loops will not have the same simple forms as have those of ^ ... vp. The general integral of the third kind, infinite like C log (t^/t^, G being a constant, is obtained by adding a general integral of the first kind to CHJ x ; similarly for the general integral of the second kind.
The function II*' ° hasf the property expressed by the equation
X, C
* More precisely, the normal elementary integral of the third kind, t Forsyth, p. 453. Harkness and Morley, p. 445.
16] VARYING PARAMETER NEARLY EQUAL TO ARGUMENT. 17
A more general integral of the third kind having the same property is
wherein the arbitrary coefficients satisfy the equations Ay = Aji. The pro perty is usually referred to as the theorem of the interchange of argument (a1) and parameter (a^).
The property allows the consideration of
Il
*1 , 2
as a function of x^ for fixed positions of x, c, x». In this regard a remark should be made :
For an ordinary position of x, the function
is a finite continuous function of ar/ when #/ is in the neighbourhood of x. But if xl be a branch place where w+l sheets wind, and #/, x be two positions in its neighbourhood, the functions of x
IT,' -log (a?/-*), Ux'c -- — 1log(a; ,-x)
*1 , *2 *„ X2 W+l
are respectively finite as x approaches #/ and aclt so that
is not a finite and continuous function of x/ for positions of a-/ up to and including the branch place a?lt
In this case, let the neighbourhood of the branch place be conformally represented upon a simple plane closed area and let £, £/, £ be the represent atives thereon of the places xlt a:/, x. Then the correct statement is that
is a continuous function of ar/ or |/ up to and including the branch place a^. This is in fact the form in which the function n*1''*2 arises in the proof
X, C
of its existence upon which our account is based*. In a similar way the function
-p*. c
regarded as a function of #/, is such that
is a finite continuous function of £' in the immediate neighbourhood of x.
* The reader may consult Neumann, p. 220. B- 2
18 ONE INFINITY AT A BRANCH PLACE. [17
17. It may be desirable to give some simple examples of these integrals. (a) For the surface represented by
y*=x(x-al)...(x-aap + l),
wherein alt ..., a2p + i are a^ finite and different from zero and each other, consider the integral
i (dxfy+ri y+m\ ^ J y \*-k *-&/
(£> "?)) (£i> »?i) being places of the surface other than the branch places, which are
(0, 0),(alt 0), ..., (a2p + 1, 0). It is clearly infinite at these places respectively like log (x - £), - log (x - £ t).
It is not infinite at (£, -r,), (&, -7l); for (y + ?)/(# - £), (y + ih)/(* - &) are finite at these places respectively.
At a place #=00 , where .« = r1, y = ft-f~l (l+P^t}}, t being ±1, and P1(t) a series of positive integral powers of t vanishing for t = 0, we have
and the integral has the form
A being a constant. It is therefore finite. At a place y = 0, for instance where
B being a constant, the integral has the form
C C being a constant, and is finite.
Thus it is an elementary integral of the third kind with infinities at (£, »/), (£1} It may be similarly shewn that the integral
, [dx fy y + r)i\
*j^U~^rJ
is infinite at (|1} j^) like — log(.r- £j) and is not elsewhere infinite except at (0, 0). Near (0, 0), we have x=P,y = Dt [1 +P5 (t2)] and this integral is infinite like
Cdt
It is therefore an elementary integral of the third kind with one infinity at the branch place (0, 0) and the other at (glt rjj).
Consider next the integral
(dx d
where rf = -^. It can easily be seen that it is not infinite save at (£, 17). Writing for the ag
neighbourhood of this place, which is supposed not to be a branch place,
17] ONE INFINITY AT A DOUBLE POINT. 19
the integral becomes
(_dx
](x-
which is equal to
Thus the integral is there infinite like -- ^, and is thus an elementary integral of
x~ £ the second kind.
The elementary integral of the second kind for a branch place, say (0, 0), is a multiple of
»/*.
2 ]xy In fact near # = 0, writing x=tz, y = Dt[l +P(t2)], this integral becomes
which is equal to
as desired.
The integral is clearly not infinite elsewhere.
Example 1. Verify that the integral last considered is the limit of
y~
y L#-f
as the place (£, rf) approaches' indefinitely near to (0, 0).
Example 2. Shew that the general integral of the first kind for the surface is [dx I A A A _n
y 1 P-I •
(/9) We have in the first chapter §§ 2, 3 spoken of a circumstance that can arise, that two sheets of the surface just touch at a point and have no further connexion, and we have said that we regard the points of the sheets as distinct places. Accordingly we may have an integral of the third kind which has its infinities at these two places, or an integral of the third kind having one of its infinities at one of these places. For example, on the surface
/(#» y) = (y- »h#) (y - m??) + (#, y)3 + (x, y\ = 0
where (x, y)3, (.?:, y\ are integral homogeneous polynomials of the degrees indicated by the suffixes, with quite general coefficients, and ml, mz are finite constants, there are at #=0 two such places, at both of which y = 0.
In this case
dx
f'(yY
where f(y) = g- , is a constant multiple of an integral of the third kind with infinities at
these two places (0, 0) ; and
'-mlx + A x2 +Bxy + Cyz dx
2—2
20 EXAMPLES. [17
is a constant multiple of an integral of the third kind, provided A , B, C be so chosen that y — iri]X-\- Ax2 + Bxy + Cy2 vanishes at one of the two places other than (0, 0) at which Lx+My is zero. Its infinities are at (i) the uncompensated zero of Lx + My which is not at (0, 0), (ii) the place (0, 0) at which the expression of y in terms of x is of the form
y = m^x + Px2 + Qx3 + ...
In fact, at a branch place of the surface where x = a + t'2, f'(y) is zero of the first order,
[ dx and dx=2tdt; thus I-^TT-^ is finite at the branch places. At each of the places (0, 0),
f(y] is zero of the first order, Lx + My is zero of the first order and y - m^x -f- A x2 + Exy -f Oy2 is zero at these places to the first and second order respectively. These statements are easy to verify ; they lead immediately to the proof that the integrals have the character enunciated.
The condition given for the choice of A , B, C will not determine them uniquely — the integral will be determined save for an additive term of the form
dx
'f'(yY
where P, Q are undetermined constants. The reader may prove that this is a general integral of the first kind. The constants P, Q may be determined so that the integral of the third kind has no periods at the period loops of the first kind, whose number in this case is two. The reasons that suggest the general form written down will appear in the explanation of the geometrical theory.
(•y) The reader may verify that for the respective cases
^/4 _ — ( /\t fy\ ( M J\\ //v» Ci
the general integrals of the first kind are
fdx , * w ^
I _ (3C 0) (3C C) *
Jy6
'dx . —z(x~c^
I
—a(x- c)2 [A y2 + By (x - c) + C (x - c)2], f
where A, B, C are arbitrary constants.
See an interesting dissertation "de Transformatione aequationis yn = R(x}.." Eugen. Netto (Berlin, Gust. Schade, 1870).
(S) Ex. Prove that if F denote any function everywhere one valued on the Riemann surface and expressible in the neighbourhood of every place in the form
the sum of the coefficients of the logarithmic terms log t of the integral / Fdx, for all places where such a term occurs, is zero.
18] PERIODS OF INTEGRAL OF SECOND KIND. 21
It is supposed that the number of places where negative powers of t occur in the expansion of F is finite, but it is not necessary that the number of negative powers be
finite. The theorem may he obtained by contour integration of I Fdx, and clearly generalizes a property of the integral of the third kind.
18. The value of the integral* jr*'c dv*'° taken round the p closed curves
formed by the two sides of the pairs of period loops (alt b^\ ..., (av, bp\ in such a direction that the interior of the surface is always on the left hand, is equal to the value taken round the sole infinity, namely the place a, in a counter clockwise direction. Round the pair ar, br the value obtained is
flr I dv*'C ,
taken once positively in the direction of the arrow head round what in the figure is the outer side of br. This value is Qr(- a)ir), where a>ir denotes the period of vt for transit of ar, namely, from what in the figure is the inside of the oval ar to the outside.
The relations indicated by the figure for the signs adopted for wir, rir and the periods of T*' ° will be preserved throughout the book.
Since a>ir is zero except when r = i, the sum of these p contour integrals 18 — <>>i ,i^i- Taken in a counter-clockwise direction, round the pole of F***
a '
where
the integral gives
- \ + A + Bt + CP + ...1 \Dv*c + t&va.'c + -...left,
where D denotes . Hence, as wit t = 1,
* Cf. Forsyth, pp. 448, 451. Harkness and Morley, p. 439.
22 ALL INTEGRALS AND RATIONAL FUNCTIONS [IS
This is true whether a be a branch place or a place at infinity (for which, if not a branch place, x = t-1) or an ordinary finite place. In the latter case
. d ( x,
x,c\
v. .
* /
j- dx\
Similarly the reader may prove that the periods of 11^' are
Orv , ...... 0,
In this case it is necessary to enclose x± and xz in a curve winding Wi + 1 times at x1} w2 + 1 times at #2, in order that this curve may be closed.
19. From these results we can shew that the integral of the second kind is derivable by differentiation from the integral of the third kind. Apart from the simplicity thus obtained, the fact is interesting because, as will appear, the analytical expression of an integral of the third kind is of the same general form whether its infinities be branch places or not ; this is not the case for integrals of the second kind.
We can in fact prove the equation
namely, if, to take the most general case, x± be a winding place and #/ a place in its neighbourhood such that #/ = xl + t™ , the equation,
For, let the neighbourhood of the branch place xl be conformally represented upon a simple closed area without branch place, by means of the infinitesimal of x, as explained in the previous chapter. Let £/, & be the representatives of the places #/, #1} and f the representative of a place x which is very near to #!, but is so situate that we may regard #/ as ultimately infinitely closer to #1 than x is.
Then x-x^ = (f - £)w+1,
where C does not vanish for #/ = x,
and E«!/*« = 1°£ (x ~ ^i') + 3*' = l°g (f - £0 + 9 >
where <£' is finite for the specified positions of the places and remains finite when gi is taken infinitely near to £j (§ 16).
X C
Also II ' = n log (« — a,) + d> = log (£ — tj) 4- 9,
Xi. X* nil _i_lC>x ' ' O^3 ;>/ I
19] DERIVABLE FROM INTEGRAL OF THIRD KIND. 23
where <f> is also finite. Therefore
X,', a^ *„ x,
rn*',c - n*'c ~i i
im. -^r? — fc~^'~~ ' = ~~ e — fc
and thus
lim
where \/r is finite.
Now as £/ moves up to £ , for a fixed position of £, we have
i
fc ' _ fc — (T ' _ ™ yt!+l — / ?i <Ti — y^i *i/ ~ •*! >
and rx' e = r!1 ° = :L + «y,
%\ «1 ^ £j
where ^ is finite.
Hence Dtx H*' r - r*' c
is finite when x is near to a;^
Moreover it does not depend on #2. For from the equation
U*'c =I%'X2,
•*J\ j 2/2 •*'» "
we may regard H^' c^ as a function of xl , which is determinate save for an additive constant by the specification of a; and c only. This additive constant, which is determined by the condition that the function vanishes when x^ =xz, is the only part of the function which depends on a?2. It disappears in the differentiation.
Finally, by the determination of the periods previously given, it follows that
has no periods at the 2p period loops. Hence it is a constant, and therefore zero since it vanishes when x = c.
Corollary i.
Hence D,^ = I>tfDt^^D^Dtx^' = D,^'', \ :
of which neither depends on the constant position c.
Corollary ii. The functions
24 PROOF FOR RATIONAL FUNCTIONS. [19
are respectively infinite like
111
tx 2 ' tx 3 ' tx 4 ' —
We shall generally write DXi, D2Xi, ... instead of Dtv , D\v ..... When XT is an ordinary place DXi will therefore mean -=— , etc.
Corollary iii.
By means of the example (8) of § 17 it can now be shewn that the infinite parts of the integral
\Fdx,
J
in which F is any uniform function of position on the undissected surface having only infinities of finite order, are those of a sum of terms consisting of proper constant multiples of integrals of the third kind and differential coefficients of these in regard to the parametric place.
20. One particular case of Cor. iii. of the last Article should be stated. A function which is everywhere one-valued on the undissected surface must be somewhere infinite. As in the case of uniform functions on a single infinite plane (which is the particular case of a Riemann surface for which the deficiency is zero), such functions can be divided into rational and transcendental, according as all their infinities are of finite order and of finite number or not. Transcendental functions which are uniform on the surface will be more particularly considered later. A rational uniform function can be expressed rationally in terms of x and y*. But since the function can be expressed in the neighbourhood of any of its poles in the form
A A A
n _L 1 _j_ 2 _i_ i •"•»». T + ^+">+~r
we can, by subtracting from the function a series of terms of the form
obtain a function nowhere infinite on the surface and having no periods at the first p period loops. Such a function is a constant f. Hence F can also be expressed by means of normal integrals of the second kind only. Since F has no periods at the period loops of the second kind there are for all rational functions certain necessary relations among the coefficients Alt...,Am. These are considered in the next Chapter.
* Forsyth, p. 369. Harkness and Morley, p. 262. t Forsyth, p. 439.
21] SPECIAL RATIONAL FUNCTIONS. 25
21. Of all rational functions there are p whose importance justifies a special mention here ; namely, the functions
dvi dv2 dvp dx ' dx ' dx
In the first place, these cannot be all zero for any ordinary finite place a of the surface. For they are, save for a factor 2?™', the periods of the normal integral F*1 c. If the periods of this integral were zero, it would be a rational uniform function of the first order; in that case the surface would be repre- sentable conformally upon another surface of one sheet*, £= F/-6 being the new independent variable ; and the transformation would be reversible (Chap. I. § 6). Hence the original surface would be of deficiency zero ; in which case the only integral of the first kind is a constant. The functions are all infinite at a branch place a. But it can be shewn as here that the quantities to which they are there proportional, namely J)avly ..., Davp, cannot be all zero. The functions are all zero at infinity, but similarly it can be shewn that the quantities, Dv1} ... , Dvp> cannot be all zero there.
Thus p linearly independent linear aggregates of these quantities cannot all vanish at the same place. We remark, in connexion with this property, that surfaces exist of all deficiencies such that p - 1 linearly independent linear aggregates of these quantities vanish in an infinite number of sets of two places. Such surfaces are however special, and their equation can be putf into the form
y = w "• /2P + 2 •
We have seen that the statement of the property requires modification at the branch places, and at infinity ; this particularity is however due to the behaviour of the independent variable x. We shall therefore state the pro perty by saying: there is no place at which all the differentials dvlt ..., dvp vanish. A similar phraseology will be adopted in similar cases. For instance, we shall say that each of dvl} dv^, ... , dvp has| 2p — 2 zeros, some of which may occur at infinity.
In the next place, since any general integral of the first kind
must necessarily be finite all over any other surface upon which the original surface is conformally and reversibly represented and therefore must be an integral of the first kind thereon, it follows that the rational function
dx p dx
* I owe this argument to Prof. Klein. + See below, Chap. V.
J See Forsyth, p. 461. Harkness ami Morley, p. 450.
26 INVARIANCE OF THEIR RATIOS. [21
is necessarily transformed with the surface into
dV
where Vi = Vt is an integral of the first kind, not necessarily normal, on the new surface, f being the new independent variable, and M = ~ .
(LOG
Thus, the ratios of the integrands of the first kind are transformed into ratios of integrands of the first kind ; they may be said to be invariant for birational transformation.
This point may be made clearer by an example. The general integral of the first kind for the surface
y- = (as, 1)8 can be shewn to be
'dx ,
y
A, B, C being arbitrary constants.
If then 0! : 0o : 03 denote the ratios of any three linearly independent integrands of the first kind for this surface, we have
for proper values of the constants altbi, ... , c3, and hence
Such a relation will therefore hold for all the surfaces into which the given one can be birationally transformed.
22. It must be remarked that the determination of the normal integrals here described depends upon the way in which the fundamental period loops are drawn. An integral of the first kind which is normal for one set of period loops will be a linear function of the integrals of the first kind which are normal for another set ; and an integral of the second or third kind, which is normal for one set of period loops, will for another set differ from a normal integral by an additive linear function of integrals of the first kind.
27
CHAPTER III.
THE INFINITIES OF RATIONAL UNIFORM FUNCTIONS.
23. IN this chapter and in general we shall use the term rational function to denote a uniform function of position on the surface of which all the infinities are of finite order, their number being finite. We deal first of all Avith the case in which these infinities are all of the first order.
If k places of the surface, say a^, a2 •••«*, be arbitrarily assigned we can always specify a function with p periods having these places as poles, of the first order, and otherwise continuous and uniform ; namely, the function is of the form
where the coefficients /*0, /^ ... /A^ are constants, the zeros of the functions F being left undetermined. Conversely, as remarked in the previous chapter (§ 20), a rational function having a,, ..., a^ as its poles must be of this form. In order that the expression may represent a rational function the periods must all be zero. Writing the periods of F£ in the form fij (a), ...,£lp (a), this requires the equations
(cr.a) + . . . + A**n,- (at) = 0,
for all the p values, i = 1, 2, . . . , p, of i. In what follows we shall for the sake of brevity say that a place c depends upon r places c1} c2, ..., cr when for all values of i, the equations
fli (C) =f&i (C,) + . . . +/A (C,.)
hold for finite values of the coefficients fi,--',fr, these coefficients being independent of i. Hence we may also say :
In order that a rational function should exist having k assigned places as its poles, each simple, one at least of these places must depend upon the others.
24. Taking the k places c^, a2, ..., a* in the order of their suffixes, it may of course happen that several of them depend upon the others, say a,+i, ...,««,-
28 DEPENDENCE OF POLES OF A RATIONAL FUNCTION [24
upon ttx, ..., as, the latter set an . .., as being independent: then we have equations of the form
,+lf
fti (a*) = nt, ! fti (aO + . . . + nk> , Of (a,),
the coefficients in any of the rows here being the same for all the p values of i. In particular, if s be as great as p and alt ... , as be independent, equations of this form will hold for all positions of as+1, ..., ak. For then we have enough disposeable coefficients to satisfy the necessary p equations.
When it does so happen, that a8+1,...,ak depend upon 0,1... at, there exist rational functions, of the form
i —
wherein cr4+1 ... o-^, Xs+1 ... X^ are constants, which are all infinite once in ttj ... as and are, beside, infinite respectively at as+1) ..., a^ ; and the most general function uniform on the dissected surface, which is infinite to the first order at a1, . . . , a^ , being, as remarked, of the form
/*<> + PI r 4-
can be written in the form
4
+ /%|r- ^* -hn*. i It, + ...... +nktS F^-
\_^k
namely, in the form
v0 + z/ir*i + ...... + v, F^ + vg+1Rt+1 + ...... +vkRk.
If this function is to have no periods, the equations
vini(al) + ...... + v.n» (a.) = 0, (i=l, 2, ...,p),
must hold. Since a1} ...,as are independent, such equations can only hold when Vi = 0 = . . . = vg. Thus the most general rational function having k poles of the first order, at a1} ..., a*, is of the form
i/o + vs+lJRg+1 + ...... + vkRk,
and involves k —s+ I linearly entering constants, s being the number of places among alf ... , ak which are independent. These constants will generally be called arbitrary : they are so only under the convention that a function
25] DETERMINES EXPRESSION OF FUNCTION. 29
which has all its poles among a1} ...,ak be reckoned a particular case of a function having each of these as poles ; for it is clear that, for instance, Rk is only infinite at a1( ..., at, ak. The proposition with a slightly altered enuncia tion, given below in § 27 and more particularly dealt with in § 37, is called the Riemann-Roch Theorem, having been first enunciated by Riemann*, and afterwards particularized by Rochf.
25. Take now other places ak+l, ak+2, ... upon the surface in a definite order, and consider the possibility of forming a rational function, which beside simple infinities at alt ..., ak has other simple poles at, say, ak+1, ak+z, ...,ah. By the first Article of the present chapter it follows that the least value of h for which this will be possible will be that for which ah depends on ch ... akak+l ... a/,-i, that is, depends on a^ . . . as ak+l . . . «&_!. This will certainly arise at latest when the number of these places a^ ... as ak+l . . . ah-i is as great as p, namely h — l=k + p — s, and if none of the places ak+l . . . «/,_! depend upon the preceding places ax ... as, it will not arise before: in that case there will be no rational function having for poles the places
ak+j for any value of j from 1 to p — s.
But in order to state the general case, suppose there is a value of j less than or equal to p — s, such that each of the places
ak+j+i ...... ah
depends upon the places
the smallest value of j for which this occurs being taken, so that no one of ak+1 . . . ak+j depends on the places which precede it in the series
Then there exists no rational function with its poles at a, ... ak ak+l ... ak+j, but there exist functions
ia
,s 1 as ~ nk+j+i,k+i 1 ak + l ~ ...... ~ nk+}+i,k+j 1 ak + i J >
whose poles are respectively at
for all values of i from 1 to h — k — j.
* Riemann, Ges. Werke, 1876, p. 101 (§ 5) and p. 118 (§ 14) and p. 120 (§ 16). t Crelle, 64. Cf. also Forsyth, pp. 459, 464. The geometrical significance of the theorem has been much extended by Brill and Noether. (Math. Ann. vii.)
30 STATEMENT OF COMPLETE RESULT. [25
Then the most general rational function with poles at
is in fact
and involves k — s + i + 1 arbitrary constants, namely the same number as that of the places of the set
which depend upon the places that precede them. For such a function must have the form
-1- P-k+j lak
namely,
— s
"i" ^ Ps+r ^ -tts+r T ^s+r, I *• «, T ...... T ^s+r,s *• a, C
r=l L^-«+r ^-s+r
1 T&
AfcffH + nk+}+t,i !«, + ......
H
r"31 . rix n*
t, s 1 a, + Vk+j+t, k+i i ak + 1 + ...... + Kk+j+t, k+j A ak +,
which is of the form
V0 + Vi t\^ + ...... + V8
+ vk+l r^ + 1 + ......
and the p periods of this, each of the form Vi H (ttj) + ...... + i/»ft (a,.) + i>t+1 ft
cannot be zero unless each of vl ... vsvk+i ••• »k+j be zero, for it is part of the hypothesis that none of ak+1 . . . ak+j depend upon preceding places.
26. Proceeding in this way we shall clearly be able to state the following result —
Let there be taken upon the surface, in a definite order, an unlimited number of places al} a2, — Suppose that each of al...a,Q_ is inde pendent of those preceding it, but each of a^,^ ... aQi depends on a, ... a« Suppose that each of an , , a~ ^ . . an is independent of
Ifc—fl Vi + l Vi + 4 Va~9i
those that precede it in the series a, ... an an,....an but each of
VI -9i 9fT* V2~9a'
aQ,-^i ••• % dePends uP°n «i ••• aQl-qiaQl+i ••• aQt.qt' This requires that
26] EXPRESSION OF FUNCTION OF ASSIGNED POLES. 31
Suppose that each of aQ +l . . . aQ _ is independent of those that precede it in the series a, ... a~ a~ , . . . . an «„ , . . . . an , but each of an ..... an
Qi-9i Ci+l Qi-<lt Qs+l Qi-Qs §3-93+1 Qa
depends upon the places of this series. This requires that
Qi-qi + [Q2-q^-Qi] + [Q3-q3-Q2]>p-
Let this enumeration be continued. We shall eventually come to places aQ +i'aQ +2' "• ao - ' eac^ ^dependent of the places preceding, for which the total number of independent places included, that is, of places which do not depend upon those of our series which precede them, is p — so that the equation
will hold. Then every additional place of our series, those, namely, chosen in order from aQ _ +l,aQ _ +2, ... will depend on the preceding places of the
whole series.
This being the case, it follows, using Rf as a notation for a rational function having its poles among al ... a/, that rational functions
do not exist.
The number of these non-existent functions is p.
For all other values off, a rational function Rj exists.
To exhibit the general form of these existing rational functions in the present notation, let m be one of the numbers 1, 2, ..., h; i be one of the numbers 1, 2, ... qtn, and let the dependence of aQ _ . upon the preceding
places arise by p equations of the form
then, denoting P* by F,., there is a rational function
which has its poles at
a' •'• aft-7,' %+!••• aQ.>-<,.>> •— aQn and the general rational function having its poles at
~'a<jm-9m>
32 THERE ARE p GAPS. [26
is of the form
and involves ql + qz + . . . 4- <?m_i + i + 1 arbitrary coefficients.
The result may be summarised by putting down the line of symbols
(&-?, + !),..,&, & + !,... ,&-, + !, ...,(&-?*), (&-?*+!),... with a bar drawn above the indices corresponding to the places which depend upon those preceding them in the series. The bar beginning over Qh — qh + I is then continuous to any length. The total number of indices over which no bar is drawn is p. There exists a rational function Rf, in the notation above, for every index which is beneath a bar.
The proposition here obtained is of a very fundamental character. Sup pose that for our initial algebraic equation or our initial surface, we were able only to shew, algebraically or otherwise, that for an arbitrary place a there exists a function Kxa, discontinuous at a only and there infinite to the first order, this function being one valued save for additive multiples of & periods, and these periods finite and uniquely dependent upon a, then, taking arbitrary places a1} a2, ... upon the surface, in a definite order, and considering func tions of the form
that is, functions having simple poles at al} ..., a#, we could prove, just as above, that there are k values of N for which such functions cannot be one valued ; and obtain the number of arbitrary coefficients in uniform functions of given poles. Namely, the proposition would furnish a definition of the characteristic number k — which is the deficiency, here denoted byp — based upon the properties of the uniform rational functions.
We shall sometimes refer to the proposition as Weierstrasss gap theorem*.
27. When a place a is, in the sense here described, dependent upon places bi} 62, ... , br, it is clear that of the equations
* " Liickensatz." The proposition has been used by Weierstrass, I believe primarily under the form considered below, in which the places ax, a2, ... are consecutive at one place of the surface, as the definition of p. Weierstrass's theory of algebraic functions, preliminary to a theory of Abelian functions, is not considered in the present volume. His lectures are in course of publication. The theorem here referred to is published by Schottky : Conforme Abbildung mehrfach zusammenhangender ebener Flachen, Crelle Bd. 83. A proof, with full reference to Schottky, is given by Noether, Crelle Bd. 97, p. 224.
27] TRANSPOSITION OF THE LINEAR CONDITIONS. 33
A.n, (br) + . . . + Apfip (br) = o
A.fl^a) +... + Apflp(a) = 0
the last is a consequence of those preceding — and conversely that when the last equation is a consequence of the preceding equations the place a depends upon the places b1} b2, ..., br.
Hence the conditions that the linear aggregate
0 (as) = A& (as) + . . . + Apnp (as) should vanish at the places
^"•aQlaQi+l--'aQ3aQ,+l'"aQm-<)a+i'
wherein i$> qm, are equivalent to only
or
linearly independent equations.
If then r + 1 be the number of linearly independent linear aggregates of the form £1 (as), which vanish in the Qm - qm -f i specified places, we have
T + 1 =p - (Qm - ql - ... - qm).
Denoting Qm — qm + i by Q, and the number of constants in the general rational function with poles at the Q specified places, of which constants one is merely additive, by q + 1,
q + 1 = q, + q2 + ... + qm^ + i + 1. We therefore have
Q-q=p-(r + i).
Recalling the values of fl^ar)... Clp(x) and the fact (Chapter II. § 21) that every linear aggregate of them vanishes in just 2p - 2 places, we see that when Q is greater than 2p - 2, T + 1 is necessarily zero.
In the case under consideration in the preceding article the number T + 1 for the function EQ , namely the number of linearly independent linear aggregates ft (as) which vanish in the places
is given, by taking m = h-l and i = qh_, in the formula of the present article, by the equation
r + 1 = p -
= Qh B.
34 POLES AT ONE PLACE. [27
Hence one such linear aggregate vanishes in the places
and therefore
&-?*-! >2p-2
or, the index associated with the last place aQ _ of our series, corresponding to
^h 'k
which a rational function RQ _ does not exist, is not greater than 2p — 1. A
^A ^A
case in which this limit is reached, which also furnishes an example of the theory, is given below § 37, Ex. 2.
28. A limiting case of the problem just discussed is that in which the series of points a1} a.2, ... are all consecutive at one place of the surface.
A rational function which becomes infinite only at a place, a, of the surface, and there like
GI GZ Cf
t P tr '
where any of the constants Glt C2, ... Cr_lt but not Cr, maybe zero, t being the infinitesimal, is said to be there infinite to the rth order. If— A.t- = G[/(i — 1)!, such a function can be expressed in a form
x + XjF* + x2z>ar* + ... + vo^rs
where, in order that the function be one valued on the undissected surface, the p equations
X, flf (a) + \2Da nf (a) + . . . + X^"1 «; (a) = 0
must be satisfied : and conversely these equations give sufficient conditions for the coefficients X1; Xg, ... , X,..
In other words, since Xr cannot be zero because the function is infinite to the ?'th order, the p differential coefficients D^~lCli(a), each of the r— 1th order, must be expressible linearly in terms of those of lower order,
with coefficients which are independent of i. We imagine the p quantities Du~1fli(a), for i = l, 2, ...,p, written in a column, which we call the rth column ; and for the moment we say that the necessary and sufficient con dition for the existence of a rational function, infinite of the rth order at a, and not elsewhere infinite, is that the rth column be a linear function of the preceding columns.
Then as before, considering the columns in succession, they will divide themselves into two categories, those which are linear functions of the pre ceding ones and those which are not so expressible. And, since the number of elements in a column is p, the number of these latter independent columns
30] CORRESPONDING TRANSPOSITION OF LINEAR CONDITIONS. 35
will be just p. Let them be in succession the ^th, &2th, ...,kpt\\. Then there exists no rational function infinite only at a, and there to these orders klt k2, ..., kp, though there are integrals of the second kind infinite to these orders. But if Q be a number different from klt ..., kp, there does exist such a rational function of the Qth order, its most general expression being of the form
xQD(?-ir* + XQ-xA?-2^ + ... + xxr* + x,
namely, the integral of the second kind whose infinity is of order Q is expressible linearly by integrals of the second kind of lower order of infinity, with the addition of a rational function.
If q + 1 be the number of linearly independent coefficients in this function, one being additive, we have an equation
Q-q=P-(r + i),
where p — (r + 1) is the number of the linearly independent equations of the form
\iflf (a) + X2Z)nt-(a) + ... + \QD^fli (a) = 0, (i = 1, 2, ..., p),
from which the others may be linearly derived. As before, r + I is the number of linearly independent linear aggregates of the form
which satisfy the Q conditions
A.D^, (a) + ... + ApDrnp (a) = 0 forr = 0, 1,2, ...,Q-1.
29. In regard to the numbers ^ . . . kp we remark firstly that, unless p = 0, &! = 1 — for if there existed a rational function with only one infinity of the first order, the positive integral powers of this function would furnish rational functions of all other orders with their infinity at this one place, and there would be no gaps (compare the argument Chapter II. § 21); and further that in general they are the numbers 1, 2, 3 ... p, that is to say, there is only a finite number of places on the surface for which a rational function can be formed infinite there to an order less than p + 1 and not otherwise infinite. We shall prove this immediately by finding an upper and a lower limit to the number of such places (§ 31).
30. Some detailed algebraic consequences of this theory will be given in Chapter V. It may be* here remarked, what will be proved in Chapter VI. in considering the geometrical theory, that the zeros of the linear aggregate
It is possible that the reader may find it more convenient to postpone the complete discussion of § 30 until after reading Chapter vi.
3—2
36 ILLUSTRATION FROM THE SUBSEQUENT [30
can be interpreted in general as the intersections of a certain curve, of the form
</> = A^ (x) + ...+ Ap(f>p O) = 0,
wherein ^...^>p are integral polynomials in x and y, with the curve repre sented by the fundamental equation of our Riemann surface. In such interpretation, the condition for the existence of a rational function of order Q with poles only at the place a, is that the fundamental curve be of such character at this place that every curve <£, obtained by giving different values to A1... Ap, which there cuts it in Q— 1 consecutive points, necessarily cuts it in Q consecutive points. As an instance of such property, which seems likely also to make the general theory clearer, we may consider a Riemann surface associated with an equation of the form
f(x, y) = K + (x, y\ + (x, y\ + (x, y)3 + (x, y\ = 0,
wherein (x, y)r is a homogeneous integral polynomial of the rth degree, with quite general coefficients, and K is a constant. Interpreted as a curve, this equation represents a general curve of the fourth degree ; it will appear subsequently that the general integral of the first kind is
dx (A+Bx+Cy),
where f (y) = df/dy, and A , B, 0 are arbitrary constants ; and thence, if we recall the fact that flj (as), ...,£lp(x) are differential coefficients of integrals of the first kind, that the zeros of the aggregate
may be interpreted as the intersections of the quartic with a variable straight line.
Take now a point of inflexion of the quartic as the place a. Not every straight line there intersecting the curve in one point will intersect it in any other consecutive point ; but every straight line there intersecting the curve in two consecutive points will necessarily intersect it there in three consecu tive points. Hence it is possible to form a rational function of the third order whose only infinities are at the place of inflexion ; in fact, if
be the equation of the inflexional tangent, and
be the equation of any line through the fourth point of intersection of the inflexional tangent with the curve, the ratio of the expressions on the left hand side of these equations, namely
Ax + By + 1
30] GEOMETRICAL THEORY. 37
is a general rational function of the desired kind, as is immediately obvious on consideration of the places where it can possibly be infinite. Thus for the inflexional place the orders of two non-existent rational functions are 1, 2. It can be proved that in general there is no function of the fourth order — the gaps at the orders 1, 2, 4 are those indicated by Weierstrass' theorem.
In verification of a result previously enunciated we notice that since Ax + By+ 1 = 0 may be taken to be any definite line through the fourth intersection of the inflexional tangent with the curve, the function contains # + 1 = 2 arbitrary constants. From the form of the integrals of the first kind which we have quoted, it follows that p = 3 ; thus the formula
wherein Q = 3, requires r + 1 = 1 ; now by § 28 r + 1 should be the number of straight lines which can be drawn to have contact of the second order with the curve at the point : this is the case.
If the quartic possess also a point of osculation, a straight line passing through two consecutive points of the curve there will necessarily pass through three consecutive points and also necessarily through four. Hence, for such a place, we can form a rational function of the third order and one of the fourth. In fact, if A<p + B0y + 1=0 be the tangent at the point of osculation and A^x + BJJ + 1 = 0 be any other line through this point, while \£c + fj,y+v = 0 is any other line whatever, these functions are respectively, in their most general forms,
A^x + B$ + 1 Xx + fjuy +j/
' + ** A«x + B0y + 1 ' A^i+B0y + l '
wherein X, p, v are arbitrary constants.
It can be shewn that in general we cannot form a rational function of the fifth order whose only infinity is at the place of osculation. Thus the gaps indicated by Weierstrass's theorem occur at the orders 1, 2, 5. (Cf. the concluding remark of § 34.)
In case, however, the place a be an ordinary point of the quartic, the lowest order of function, whose only infinity is there, is p + 1 = 4 : it will subsequently become clear that a general form of such a function in S'/S, where S = 0 is any conic drawn to intersect the quartic in four con secutive points at a, and S' = 0 is the most general conic drawn through the other four intersections of S with the quartic. S' will in fact be of the form \S + p,T, where T is any definite conic satisfying the conditions for S', and X, /j, are arbitrary constants; the equation Q— q=p — (r + 1) is clearly satisfied by Q = 4, q = 1, p = 3, T + 1 = 0.
The present article is intended only by way of illustration ; the examples given appear to find their proper place here. The reader will possibly
38
FUNCTION OF ORDEK
[30
find it desirable to read them in connexion with the geometrical account given in Chapter VI.
31. Consider now what places of the surfaces are such that we can form a rational function infinite, only there, to an order as low as p.
For such a place, as follows from § 28, the determinant
A =
> (r\ DP"1 O (v\ T}P— ! O ff\
4i v*v> ^ i£2vv> >-L/ ***v~y
must vanish. Assume for the present that none of the minors of A vanish at that place. It is clear by § 28 that A only vanishes at such places as we are considering.
Let v be any integral of the first kind. We can write
/ \ dvi . , „ dv dvi (x) = -j- in the form -=- — , at at av
and similarly put
and so write
where D is the determinant whose rth row is formed with the quantities
dvr '
' dvr
Now -T* is a rational function; and it is infinite only at the zeros of dv,
whose aggregate number is 2p — 2; and -y-0* is a rational function of the
(4>p — 4)th order, its poles being also at the zeros of dv; and a similar state ment can be made in regard to the other rows of D.
Hence D is a rational function whose infinities are of aggregate number
(2p - 2) (1 + 2 + ... +p) = (p - l)p (p + 1), and this is therefore the number of zeros of D.
32] EXISTS ONLY FOR CERTAIN PLACES. 39
Now A can vanish either by the vanishing of the factor D or by the
(dv\%p (.P+I) — I The zeros of the last factor are, however,
dtj
the poles of D. Hence the aggregate number of zeros of A is (p — 1) p (p + 1). We shall see immediately that these zeros do not necessarily occur at as many as (p — I)p (p + 1) distinct places of the surface.
In order that a rational function should exist of order less than p, its infinity being entirely at one place, say of order p — r, it would be necessary that the r determinants formed from the matrix obtained by omitting the last r rows of A should all vanish at that place. We can, as in the case of A, shew that each of these minors will vanish only at a finite number of places. It is therefore to be expected that in general these minors will not have common zeros ; that is, that the surface will need to be one whose 3/; — 3 moduli are connected in some special way.
Moreover it is not in general true that a rational function of order p + 1 exists for a place for which a function of order p exists, these functions not being elsewhere infinite. For then we could simultaneously satisfy the two sets of p equations
(a) + \DCli (a) + ...... + Xp^DP-2^ (a) + \pDf~1 Ot- (a) = 0,
(a) + ^Dtli (a) + ...... + ffv-lDP-afli (a) + /v,., jDPflf (a) = 0,
namely, A and -7- would both be zero at such a place. The condition that at
this be so would require that a certain function of the moduli of the surface — what we may call an absolute invariant — should be zero.
Therefore when of the p gaps required by Weierstrass's theorem, p — 1 occur for the orders 1, 2, ..., p — 1, the other will in general occur for the order p+l. The reader will see that there is no such reason why, when a function of order p exists, a function of order p + 2 or higher order should not exist.
32. The reader who has followed the example of § 30 will recall that the number of inflexions of a non-singular plane quartic* is 24 which is equal to the value of (p - 1) p (p + 1) when p = 3. The condition that the quartic possess a point of osculation is that a certain invariant should vanish^.
When the curve has a double point, there are only two integrals of the first kind J, and p is equal to two. Thus in accordance with the theory above, there should be (p — 1) p (p + 1) = 6 places for which we can form functions
* Salmon, Higher Plane Curves (1879), p. 213.
t The equation can be written so as to involve only 5 = 3/> - 3 - 1 parametric constants (Chap. V. p. 98, Exs. 1, 2).
+ Their forms are given Chapter II. § 17 /3. lleasons are given in Chapter VI. The reader may compare Forsyth, p. 395.
40 EXAMPLES. [32
of the second order infinite only at one of these places. In fact six tangents can be drawn to the curve from the double point : if A^c + B9y = 0 be the equation of one of these and \ (Ax + By) + fi(A0x + B0y) = 0 be the equation of any line through the double point, the ratio
fc Ax + By XA^ + B0y + f*
represents a function of second order infinite only at the point of contact of
For the point of contact of one of these tangents the^) gaps occur for the orders 1 and 3.
The quartic with a double point can be biratioually related to a surface expressed by an equation of the form
£ being the function above. The reader should compare the theory in Chapter I. and the section on the hyperelliptic case, Chapter V. below.
33. Ex. For the surface represented by the equation
where the brackets indicate general integral polynomials of the order of the suffixes, p is equal to 4, and the general integral of the first kind is
r
where f(y ) = + . Prove that at the (p - 1 ) p (p + 1 ) = 60 places for which rational functions of the 4th order exist, infinite only at these places, the following equations are satisfied
2/7</-3(/'/y)2=o,
2f 3 s3/ ff2 83/,3i
>* + 6 a^ap fvfx ~ ap J* J
Where y'=' etc-'^=' etc>
Explain how to express these functions of the fourth order. Enumerate all the zeros of the second differential expression here given.
Ex. 2. In general, the corresponding places are obtained by forming the differential equation of the pth order of all adjoint <f> curves. In a certain sense A is a differential invariant, for all reversible rational transformations. (See Chapter VI.)
* Here the number of integrands of the integrals of the first kind, which are of the form (Lx + My)lf'(y) (cf. Chapter III. § 28), which vanish in two consecutive points at the point of contact of Avx + H0y = 0, is clearly 1, or T + 1 = 1 : hence the formula Q - q -p - (r + 1) is verified by Q = 2, q = l, p = 2, so that the form of function of the second order given in the text is the most general possible.
34]
CONSIDERATION OF THESE EXCEPTIONAL PLACES,
41
34. We pass now to consider whether the (p — 1) p (p + 1) zeros of A will in general fall at separate places*.
Consider the determinant
V = 0 fl (x} fl (x)
wherein flj^ ((?)=* J5f Of (£•), and k1} ..., kp are the orders of non-existent rational functions for a place £, in ascending order of magnitude, (A^ = 1) ; and let its value be denoted by
so
that ur = I (i>r (x) dtx is an integral of the first kind.
Then wr(x) vanishes at % to the (kr — l)th order. For <w,. (x) is the determinant
v.-r-i
now the (kr — l)th differential coefficient of this determinant (in regard to the infinitesimal at x) has at £ a value which is in fact the minor of the element (1, 1) of V, save for sign. That this minor does not vanish is part of the definition of the numbers k1} k2, ..., kp. But all differential coeffi cients of Vr of lower than the (kr — l)th order do vanish at £: some, because for a; = £ they are determinants having the first row identical with one of the following rows, this being the case for the differential coefficients of orders ^ — 1, &2 — 1, ... ; others, because when /* is not one of the numbers fcj, k2, ..., kp, D^ifli^) is a linear function of those of Dk'~1Cli(^), Lb~* {li(g), ... for which p is greater than klf k.2, ... , the coefficients of the linear functions being independent of i. This proves the proposition.
It is clear that the &rth differential coefficient of Vr may also vanish at £. In particular Wi(x) does not vanish at £: a result in accordance with a remark previously made (Chapter II. § 21), that there is no place at which the differentials of all the integrals of the first kind can vanish.
* The results in §§ 34, 35, 36 are given by Hunvitz, Math. Annul. 41, p. 409. They will be useful subsequently.
42 AND OF THE NON-EXISTING-ORDERS. [34
An important corollary is that the highest order for which no rational function exists, infinite only at the place £, is less than 2p. For wp (x) vanishes only 2p — 2 times, namely, kp — 1 < 2p — 2.
35. We can now prove that if k2 > 2, the sum of the orders k\, &2> ••• , kp is less than p2. For if there be a rational function of order in, infinite only at £, and r be one of the non-existent orders* ^ ... kp, r — m is also one of these non-existent orders — otherwise the product of the existent rational function of order i — m with the function of order m would be an existent function of order r. The powers of the function of order m are existent functions, hence none of kl . . . kp are divisible by in.
Let Ti be the greatest of the non-existent orders k^ ... kp which is con gruent to i(< m) for the modulus m : then, by the remark just made,
TI, Ti — in, Ti — 2m, ... , m + i, i
are all non-existent orders — and all congruent to i for the modulus m. Since i'i occurs among ki...kp, all these also occur. Take i in turn equal to
1, 2, ... 771-1.
Then, the number of non-existent orders being p,
p =
so that T! + r2 + . . . + rm_^ = mp — \ m (m — 1)
= \ m(2p — m+ 1).
Now the sum of the non-existent orders is
m-l
2 [ri + (n - m) + (n - 2m) + ... + i] , which is equal to
+ lm (m - 1) - TL (in - 1) (2m - 1), and, since Sr^ = ^ m (%p — m + 1), this is equal to
~ Sr, [r, - (2p - 1)] + I [4p» - (m - Vf\ + ^ (m - 1) (m + 1), or ^-r2-l-r-m-lm-2.
* i.e. orders of rational functions, infinite only at £, which do not exist: and similarly in what follows.
36]
LEAST NUMBER OF THESE EXCEPTIONAL PLACES.
Since, by the corollary of the preceding article, 2p — 1 is not less than riy this is less than p- unless m is 1 or 2. Now m cannot be equal to 1 ; and if it is 2 then also k2 > 2. Hence the statement made at the beginning of the present Article is justified.
When there is a rational function of order 2, it is easy to prove that there are places for which L\ ... kp are the numbers 1, 3, 5, ... , 2p — 1, whose sum* is^>2. An example is furnished by § 32 above.
Ex. For the surface
for which p = 3, there is, at #=ao , only one place, and the non-existent orders are 1, 2, 5 : whose sum is p* — l.
36. We have in § 34 defined p integrals of the first kind I wl(x)dtx, ... , I wp(x)dtx
by means of a place £. Since the differential coefficients of these vanish at £ to essentially different orders, these integrals cannot be connected by a homo geneous linear equation with constant coefficients. Hence a linear function of them with parametric constant coefficients is a general integral of the first kind. Therefore each of ^(x) ... O^ (x) is expressible linearly in terms of o>! (x) ... wp (x) in a form
&i 0) = Cn$r(» + . . . + Cip&p (x},
where the coefficients are independent of x. Thus the determinant A (§ 31), which vanishes at places for which functions of order less than p + 1 exist, is equal to
>i(x) , ...... , <op(x)
xwl(x) , ...... , DXG>P(X)
where C is the determinant of the coefficients c,-j. It follows from the result of § 34 that the determinant here multiplied by C vanishes at £ to the order
Thus, the determinant A vanishes at any one of its zeros to an order equal to the sum of the non-existent orders for the place diminished by %p(p + l).
For example, it vanishes at a place where the non-existent orders are 1, 2, ... , p- 1, p + 1 to an order $p(p-~L)+p + l-^p(p + l) or to the first order. We have already remarked that such places are those which most usually occur.
* Cf. Burkhardt, Math. Annal. 32, p. 388, and the section iu Chapter V., below, on the hyper- elliptic case.
44 RIEMANN-ROCH THEOREM. [36
Hence, since fa -f . . . + kp ^ p2, A vanishes at one of its zeros to an order
Further, if r be the number of distinct places where A vanishes, and mly m2, . .., mr be the orders of multiplicity of zero at these places, it follows, from
and raa + ... + mr < r %p(p — 1),
that r > 2p + 2, or
there are at least 2p + 2 distinct places for which functions of less order than p + 1, infinite only thereat, exist] this lower limit to the number of distinct places is only reached when there are places for which functions of the second order exist.
Ex. For the surface given by
p is equal to 3 ; there are 12 = 2^ + 6 distinct places where A vanishes.
37. We have called attention to the number of arbitrary constants con tained in the most general rational function having simple poles in distinct places (§ 27) and to the number in the most general function infinite at a single place to prescribed order (§ 28) : in this enumeration some of the con stants may be multipliers of functions not actually becoming infinite in the most general way allowed them, that is, either of functions which are not really infinite at all the distinct places or of functions whose order of infinity is not so high as the prescribed order.
It will be convenient to state here the general result, the deduction of which follows immediately from the expression of the function in terms of integrals of the second kind : —
Let tt1; a.,, ... be any finite number of places on the surface, the infinitesi mals at these places being denoted by tl} t.2, .... The most general rational function whose expansion at the place di involves the terms
JL JL _L
&' W t*<' '"
— whose number is finite, = Q» say, — and no other negative powers, involves q + 1 linearly entering arbitrary constants, of which one is additive, q being given by the formula
Q-q=P-(r + i),
where Q is the sum of the numbers Q{, and r + 1 is the number of linearly independent linear aggregates of the form
fl(a;) = A A (a;) + ... + ApQ,p(x\
37] EXAMPLE. 45
which satisfy the sets of Qi relations, whose total number is Q, given by A, DV-1 f^ (at) + A,D^ 02 (ai) + . . . + ApD*--1 np (tti) = 0, Ail>-1 nt (a;) + 4a J> -i n2 (a<) + . . . + ^I> -1 Op (a,-) = 0,
As before, this general function will as a rule be an aggregate of functions of which not every one is as fully infinite as is allowed, and it is clear from the present chapter that in the absence of further information in regard to the places a1} a.2, ... it may quite well happen that not one of these functions is as fully infinite as desired, the conditions analogous to those stated in §§ 23, 28 not being satisfied. See Example 2 below.
The equation Q — q=p — (r + l) will be referred to as the Riemann-Roch Theorem.
Ex. 1. For a rational function having only simple poles or, more gene rally, such that the numbers X;, /^t-, vi, ... for any pole are the numbers 1, 2, 3, ... Qit
if Q > 2p — 2, r + 1 is zero, since fl (x) has only an aggregate number 2p — 2 of zeros : the function involves Q — p + 1 constants,
if Q = 2p — 2, r + 1 cannot be greater than 1 ; for the ratio of two of the aggregates £l(x) then vanishing at the poles, being expressible in a form
dV
_™ , where V, W are integrals of the first kind, would be a rational function
a w
without poles, namely a constant ; then the linear aggregates fl (#) would be identical : thus the function involves Q — p + I or Q — p + 2, constants, namely p — 1 or p constants,
if Q= 2p — 3, T+ 1 cannot be greater than 1, since the ratio of two of the aggregates H (x) then vanishing at the poles would be a rational function of the first order and therefore p be equal to unity — in which case 2p — 3 is negative : thus the function involves p — 2 or p — 1 constants,
if Q = 2p — 4, and T + 1 be greater than unity, the ratio of two of the vanishing aggregates fl (#) would be a rational function of the second order : we have already several times referred to this possibility as indicative that the surface is of a special character — called hyperelliptic — and depends in fact only on 2p — 1 independent moduli. In general such a function would involve p — 3 constants.
Ex. 2. Let V be an integral of the first kind and a be an arbitrary definite place which is not among the 2p — 2 zeros of dV. We can form a rational function infinite to the first order at the 2p — 2 zeros of dV and to the second order at a; the general form of such a function would contain 2j9 — 2 + 2— p + I =p + 1 arbitrary constants. But there exists no rational function infinite to the first order at the zeros of dV and to the first order at
46 IMPORTANT EXAMPLE. [37
the place a. Such a function would indeed by the Riemann-Roch theorem here stated, contain 2p — 2 + 1— p -{• l=p arbitrary constants : but the coeffi cients of these constants are in fact infinite only at the zeros of d V. For when the places a1} ... , 0^-2 are all zeros of an aggregate of the form
AA(a;) + ...+Apnp(ac), the conditions that the periods of an expression
be all zero, namely the equations
Xj nt (aO + . . . + \2p_2 fli (oap-a) + fjLfli (a) = 0, (i = 1, 2, . . . , p), lead to
p, [AfMa) + ... + Apflp(d)] = 0,
and therefore to /JL — 0.
Thus the function in question will be a linear aggregate of p functions whose poles are among the places a1} ... , a^-s- As a matter of fact, if W be a general integral of the first kind, expressible therefore in the form
2F2 + ...+\PVP,
dW
wherein V2, ..., Vp are integrals of the first kind, v^ involves the right
a v
number of constants and is the function sought.
In this case the place a does not, in the sense of § 23, depend upon the places a1} ... , 0^-2 j ^ne symbol suggested in § 26 for the places a1} ... , a^-a, a, ... is
1,2,3, .. .,^-1,^+1, ...,2p-2, 2^-1, 2^,2^ + 1,....
It may be shewn quite similarly that there is no rational function having simple poles in a1} a2, ... , a2p_2 and infinite besides at a like the single
term — , t being the infinitesimal at the place a.
v
Ex. 3. The most general rational function R which has the value c at each of Q given distinct places, R — c being zero of the first order at each of these places, is obviously derivable by the remark that l/(R — c) is infinite at these places.
38]
CHAPTER IV.
SPECIFICATION OF A GENERAL FORM OF RIEMANN'S INTEGRALS.
38. IN the present chapter the problem of expressing the Riemann integrals is reduced to the determination of certain fundamental rational functions, called integral functions. The existence of these functions, and their principal properties, is obtained from the descriptive point of view natural to the Riemann theory.
It appears that these integral functions are intimately related to certain functions, the differential-coefficients of the integrals of the first kind, of which the ratios have been shewn (Chapter II. § 21) to be invariant for birational transformations of the surface. It will appear, further, in the next chapter, that when these integral functions are given, or,, more pre cisely, when the equations which express their products, of pairs of them, in terms of themselves, are given, we can deduce a form of equation to re present the Riemann surface ; thus these functions may be regarded as anterior to any special form of fundamental equation.
Conversely, when the surface is given by a particular form of fundamental equation, the calculation of the algebraic forms of the integral functions may be a problem of some length. A method by which it can be carried out is given in Chapter V. (§§ 72 ff.). Compare § 50 of the present chapter.
It is convenient to explain beforehand the nature of the difficulty from which the theory contained in §§ 38 — 44 of this chapter has arisen. Let the equation associated with a given Riemann surface be written
wherein A, A1,..., An are integral polynomials in x. An integral function is one whose poles all lie at the places .r=o> of the surface; in this chapter the integral functions considered are all rational functions. If y be an integral function, the rational symmetric functions of the n values of y corresponding to any value of .r, whose values, given by the equation, are -AJA, Ay/A, -A^A, etc., will not become infinite
48 RATIONAL FUNCTIONS WHOSE POLES [38
for any finite value of x, and will, therefore, be integral polynomials in x. Thus when y is an integral function, the polynomial A divides all the other polynomials Alt A 2, ...... , An. Conversely, when A divides these other polynomials, the form of the
equation shews that y cannot become infinite for any finite value of x, and is therefore an integral function.
When y is not an integral function, we can always find an integral polynomial in x, say /3, vanishing to such an order at each of the finite poles of y, that /3y is an integral function. Then also, of course, |32/2, /33y 3, . . . are integral functions: though it often happens that there is a polynomial /32 of less order than /32, such that /32y2 *s an integral function, and similarly an integral polynomial #3 of less order than /33, such that ft3y* is an integral function ; and similarly for higher powers of y.
In particular, if in the equation given we put Ay = rj, the equation becomes
r,n + AlT)n-l + A2Ar,n-2 + ... + AnAn-l = 0, and T) is an integral function.
Suppose that y is an integral function. Then any rational integral polynomial in x and y is, clearly, also an integral function. But it does not follow, conversely, though it is sometimes true, that every integral rational function can be written as an integral polynomial in x and y. For instance on the surface associated with the equation
f + Bfx + Cyx* + Dtf -E(f- A-2) = 0 ,
the three values of y at the places .r = 0 may be expressed by series of positive integral powers of x of the respective forms
, y— -
Thus, the rational function (/• — Ey^x is not infinite when #=0. Since y is an integral function, the function cannot be infinite for any other finite value of x. Hence (y2 - Ey}jx is an integral function. And it is not possible, with the help of the equation of the surface, to write the function as an integral polynomial in x and y. For such a polynomial could, by the equation of the surface, be reduced to the form of an integral polynomial in x and y of the second order in y ; and, in order that such a polynomial should be equal to (y^-Ey^lx, the original equation would need to be reducible.
Ex. Find the rational relation connecting x with the function 77 = (#2 — Ey}jx ; and thus shew that 17 is an integral function.
39. We concern ourselves first of all with a method of expressing all rational functions whose poles are only at the places where x has the same finite value. For this value, say a, of x there may be several branch places : the most general case is when there are k places specified by such equations as
x - a = £ri+1, • • • , x- a = tkwk+\
The orders of infinity, in these places, of the functions considered, will be specified by integral negative powers of tlf . .., tk respectively. Let F be such a function. Let o- + 1 be the least positive integer such that (x — aY+lF is finite at every place x = a. We call <r + 1 the dimension of F. Let f(xt y) = 0 be the equation of the surface. In order that there may be any branch places at x = a, it is necessary that df/dy should be zero for this value
39] ARISE FOR THE SAME VALUE OF X. 49
of x. Since this is only true for a finite number of values of x, we shall suppose that the value of x considered is one for which there are no branch places.
We prove that there are rational functions h1} ..., hn^ infinite only at the n places x = a, such that every rational function whose infinities occur only at these n places can be expressed in the form
(— > l] +(— > l) h+. ..+(-?— , l] hn. ..(A),
\ao -a J\ \x-a' A, \x - a' Jx^
in such a way that no term occurs in this expression which is of higher dimension than the function to be expressed : namely, if a + 1 be the dimen sion of the function to be expressed and o-; + 1 the dimension of hi, the function can be expressed in such a way that no one of the integers
X, AX + al + 1, . . . , A,^ + a-n^ + 1
is greater than cr + 1. We may refer to this characteristic as the condition of dimensions. It is clear conversely that every expression of the form (A) will be a rational function infinite only for x = a.
Let the sheets of the surface at x = a be considered in some definite order. A rational function which is infinite only at these n places may be denoted by a symbol (R1} R2> ... , Rn), where R1} R2, ... , Rn are the orders of infinity in the various sheets. We may call Rlf R2, ... , Rn the indices of the function. Since the surface is unbranched at x — a, it is possible to find a
certain polynomial in - - , involving only positive integral powers of this
SO ^~ CL
1 \ 72
quantity, the highest power being [- -) , such that the function
\x — a i
l), = (£,$,, ...,,SU,0)say ......... (i),
'•"
— a. is not infinite in the nth sheet at a; = a.
Consider then all rational functions, infinite only at x = a, of which the nth index is zero. It is in general possible to construct a rational function having prescribed values for the (n - 1) other indices, provided their sum be p + 1. When this is not possible a function can be constructed* whose indices have a less sum than p + 1, none of them being greater than the prescribed values. Starting with a set of indices (p + 1, 0, ... , 0), consider how far the first index can be reduced by increasing the 2nd, 3rd, ... , (n - l)th indices. In constructing the successive functions with smaller first index, it will be necessary, in the most general case, to increase some of the 2nd, 3rd, ..., (n — l)th indices, and there will be a certain arbitrariness as to the way in which this shall be done. But if we consider only those functions of which the sum of the indices is less than p + 2, there will be only a finite number
* The proof is given in the preceding Chapter, (§§ 24, 28). B. 4
50 SPECIFICATION [39
possible for which the first index has a given value. There will therefore only be a finite number of functions of the kind considered*, for which the further condition is satisfied that the first index is the least possible such that it is not less than any of the others. Let this least value be r1} and suppose there are ^ functions satisfying this condition. Call them the reduced functions of the first class — and in general let any function whose nth index is zero be said to be of the first class when its first index is greater or not less than its other indices. In the same way reckon as functions of the second class all those (with nth index zero) whose second index is greater than the first index and greater than or equal to the following indices. Let the functions whose second index has the least value consistently with this condition be called the reduced functions of the second class ; let their number be k2 and their second index be r2. In general, reckon to the ith class (i < n) all those functions, with nth index zero, whose t'th index is greater than the preceding indices and not less than the succeeding indices. Let there be ki reduced functions of this class, with iih index equal to i\. Clearly none of the integers t\, ... , rn_j are zero.
Let now (^ ... s;_! r{Si+1 ... sn_i 0),
where r{ >slt ... , i\ > st-_,, n > si+l, ... , n > sn-i,
be any definite one of the ki reduced functions of the iih class. Make a similar selection from the reduced functions of every class. And let
($! . . . $£_! R{ Si+l . . . Sn-i 0) be any function of the iih class other than a reduced function, so that
Ri > Si, ... , Ri> Si-i, Ri > Si+1 , . . . , Ri > Sn-i- Then by choice of a proper constant coefficient X we can write
(& . . . £<_! Ri Si+i . . . Sn_! 0) - X (x - a)~(Ri~Ti) (sl . . . «;_! n si+l . . . sn_! 0)
in the form
(^...T^R/T^-.-Tn^Ri-ri) (ii),
where R{ < Ri', 2\ may be as great as the greater of S1} Ri — (n - s^, but is certainly less than Ri] and similarly T2, ... , T^ are certainly less than Rt', while T{+1 may be as great as the greater of $f+1, Ri — (rt — si+l), and is there fore not greater than R^, and similarly Ti+2, ... , Tn^ are certainly not greater than Ri.
* Functions which have the same indices are here regarded as identical. Of course the general function with given indices may involve a certain number of arbitrary constants. By the function of given indices is here meant any one such, chosen at pleasure, which really becomes infinite in the specified way.
39]
OF A FUNDAMENTAL SYSTEM.
51
Further, if
, 1
be a suitable polynomial of order Ri — r\ in 1
\.x — a (x — a)~l, we can write
\iv — a / tii-Vi
/<y a/ -p>> a' &' r\\ (\\\\
— (/o i ... io i—i £L i ij i+i ... ij n_i \}) V111^'
where R"i may be as great as the greater of R'{, R{ — rit but is certainly less than Ri; S\ may be as great as the greater of 1\, Ri — r{, but is certainly less than Ri; and similarly $'2, ..., $';_! are certainly less than R^; while S'i+l may be as great as the greater of 7\-+1, Ri—ri, and is certainly not greater than RI\ and similarly S'i+2, ... , S'n-\ are certainly not greater than Rt.
Hence there are two possibilities.
(1) Either (S\ . . . £'f_i R"i S'i+1 . . . £'n_i 0) is still of the ith class, namely, R"i > Slf ... , R"i > S'i^ , R"i > S'i+1 , . . . , R"t > £'„_, ,
and in this case the greatest value occurring among its indices (R"i) is less than the greatest value occurring in the indices of (Si... Si-i Ri Si+1 . . . $n_j 0).
(2) Or it is a function of another class, for which the greatest value occurring among its indices may be smaller than or as great as Rt (though not greater) ; but when this greatest value is Ri, it is not reached by any of the first i indices.
If then, using a term already employed, the greatest value occurring among the indices of any function (Ri, ..., Rn) be called the dimension of the function, we can group the possibilities differently and say, either
(S\ . . . S'i^i R"i S'i+i . . . S'n-i 0) is of lower dimension than
(Si ... Si-i Ri Si+1 . . . Sn-i 0),
or it is of the same dimension and then belongs to a more advanced class, that is, to an (i + &)th class where k > 0.
In the same way if (^ ... ^ r{ ti+i . . . tn-i 0) be any reduced function of the t'th class other than (^ ... st-_i rf si+1 . . . sn^ 0), we can, by choice of a suitable constant coefficient p,, write
(t-L ... ti—iTi t-+ ... t — 0) — /x (s ... s-_ r-s- s 0)
where r'i<ri, t\... £';_i may be respectively as great as the greater of the pairs (ti, s^ ... (^_j, Si_i) but are each certainly less than rit while similarly no one of t'i+1, ... , t'n-i is greater than rt.
The function (t\ ... t'^i r{ tft+1 ... £'n_i 0) cannot be of the ith class, since no function of the tth class has its tth index less than rt : and though the greatest value reached among its indices may be as great as rt (and not greater), the number of indices reaching this value will be at least one less
4—2
52 EXAMPLE [39
than for (s1 . . . s;_j rt si+1 . . . sn^ 0). Namely (t\ . . . JV-i r'i t'i+l . . . t'n^ 0) is certainly of more advanced class than (si . . . £;_! Vi Si+1 . . . sn^ 0), and not of higher dimension than this.
Denote now by hlt ... , hn^ the selected reduced functions of the 1st, 2nd, ..., (n — l)th classes. Then, having regard to the equations given by (ii), (iii), (iv), we can make the statement,
Any function (Sl... $;_j Rt Si+l ... $n_j 0) can be expressed as a sum of (I) an integral polynomial in (x — a)~l, (2) one ofhly ... , An_j multiplied by such a polynomial, (3) a function F which is either of lower dimension than the function to be expressed or is of more advanced class.
In particular when the function to be expressed is of the (n — l)th class the new function F will necessarily be of lower dimension than the function to be expressed.
Hence by continuing the process as far as may be needful, every function
f=(S1... Si-! Ri Si+1 . . . Sn-i 0) can be expressed in the form
(— , l] + (— , l) h*. .. + (—, l] hn^+F,, (v)
Ve-a A U-a' AI \ac-a' An_,
where F^ is of lower dimension thany!
Applying this statement and recalling that there are lower limits to the dimensions of existent functions of the various classes, namely, those of the &! + . . . + kn-! reduced functions, and noticing that the reduction formula (v) can be applied to these reduced functions, we can, therefore, put every func tion f=(S1... Si-! Ri Si+l . . . Sn-i 0) into a form
f— , l) + (— , l) hl+... + (— , l) hn-!.
\at-a J\ \x-a /\l \x-a /x^
Now it is to be noticed that in the equations (ii), (iii), (iv), upon which this result is based, no terms are introduced which are of higher dimension than the function which it is desired to express : and that the same remark is applicable to equation (i).
Hence every function (R1} ... , Rn) can be written in the form (A) in such a way that the condition of dimensions is satisfied.
40. In order to give an immediate example of the theory we may take the case of a surface of four sheets, and assume that the places x = a are such that no rational function exists, infinite only there, whose aggregate order of infinity is less than p + 1. In that case the specification of the reduced functions is an easy arithmetical problem. The reduced functions of the first class are (m1} w2, m3, 0), where mx is to be as small as possible without being smaller than m2 or w3 : by the hypothesis we may take
Wj + m3 + m3 = p + I.
40]
OF THE FUNDAMENTAL SYSTEM.
53
Those of the second class require m2 as small as possible subject to ml + w2 + ra3 = p + 1, m2>ml, ra2 > w3 :
those of the third class require w3 greater than m1 and w2 but otherwise as small as possible subject to n^ + m2 + ws = p + 1. We therefore immediately obtain the reduced functions given in the 2nd, 3rd and 4th columns of the following table. The dimension of any function of the t'th class being denoted by <Ti + 1, the values of <rt- are given in the fifth column, and the sum ar1 + a~2 + 0-3 in the sixth. The reason for the insertion of this value will appear in the next Article.
|
P |
Reduced functions of the first class |
Reduced functions of the second class |
Reduced functions of the third class |
°"l> ff-2> ff3 |
ff1 + ff.2 + ffx |
|
= 3H-1 |
(M, M, M, 0) |
(M-2, M+l, M + l, 0) (M-l, M+1,M, 0) (M, M+l, M-l, 0) |
(M-l, M, M+l, 0) |
M-1,M, M |
3M-1 |
|
= 3N-2 |
(N,N,N-1,0) (N,N-1,N,0) |
(N-l, N, N, 0) |
(N-I,N-1,N + 1,0) |
N-1,N-1,N |
3^-2 |
|
= 3P |
(P+l, P, P, 0) (P + 1,P + 1,P-1,0) (P+1,P-1,P + 1,0) |
(P-l, P + l, P + l, 0) (P, P + l, P, 0) |
(P, P, P + l, 0) |
P, P, P |
3P |
Here the reduced functions of the various classes are written down in random order. Denoting those first written by h1} h2, h3, we may exemplify the way in which the others are expressible by them in two cases.
(a) When p = 3M — 1, we have, /* being such a constant as in equa tion (iv) above (§ 39),
(M, M + 1, M- 1, 0) -fi(M- 2, M + 1, M + 1, 0)= {M, M, M + 1, 0},
the right hand denoting a function whose orders of infinity in the various sheets are not higher than the indices given. If the order in the third sheet be less than M + 1, the right hand must be a function of the first class and therefore the order in the third sheet must be M. In that case, since a general function of aggregate order p + 1 contains two arbitrary constants, we have an expression of the form
(M, M + 1, M - 1, 0) = fjih, + Ah, + B, for suitable values of the constants A, B.
If however there be no such reduction, we can choose a constant \ so
that
{M, M, M + I, 0} - \(M- 1, M, M+l, 0) = {M, M, M, 0} = A'h, -f
54 SUM OF DIMENSIONS OF FUNDAMENTAL SYSTEM
and thus obtain on the whole
(M,M+1,M- 1, 0) = fJis + \h3 + A'h, + B', for suitable values of the constants A', B'. (b) When p = 3P we obtain
(P + 1, P + 1, P - 1, 0) - \k, + A(P,P + 1,P,0) + B = \h1 + A \fiht + Ch3 + D}+B.
Ex. 1. Shew for a surface of three sheets that we have the table
[40
|
p |
*Ii h2 |
o-i. 0-2 |
(Tj + O-2 |
|
odd |
/p + l p + i \ /p-i ^ + 3 \ |
p-1 p + l |
|
|
V 2 ' 2 ' ) \ 2 ' 2 ' ) (p + 2 p \ fp p + 2 \ |
2 ' 2 |
||
|
\ 2 ' 2' / \2' 2 ' / |
2' 2 |
* |
2£c. 2. Shew, for a surface of n sheets, that if the places x = a be such that it is impossible to construct a rational function, infinite only there, whose aggregate order of infinity is less than _£> + !, a set of reduced functions is given by
kP.Jtr+l'-(k,..Jk,t-l,..^k-l,<)\(k-ltkt.JLtk-l+.^-l,Q) ...... (k-l,...,k- !,£,...£,(>)
/<r + 2.. A-i = (£- 1, ...,k-l,k + l, k, ...k, G)(k-I, ..., £-1, k,k + l, k, ...A, 0) ......
(k-l, ...,k-l,k, ...k,k+l, 0)
wherein p + I = (n—l)k — r (r<»— 1) and, in the first row, there are r numbers ^ — 1 in each symbol, and, in the second row, there are r+l numbers k—\ in each symbol. In each case k, ...k denotes a set of numbers all equal to k and £—1, ..., £—1 denotes a set of numbers all equal to k — 1.
The values of crj, ..., cr,. + 1 are each k — l, those of o-,. + 2» •••> «"n-i are each ^- Hence 0-J+. .. +o> + ! + o-r + 2+. . .+<rn_1 = (r + !)(&- l) + (n-r- 2) A = (n-l)*-r- 1 =^?.
^!r. 3. Shew that the resulting set of reduced functions is effectively independent of the order in which the sheets are supposed to be arranged at x=a.
41. For the case where rational functions exist, infinite only at the places x = a, whose aggregate order of infinity is less than ^ + 1, the specification of their indices is a matter of greater complexity.
But we can at once prove that the property already exemplified and expressed by the equation o-1 + ... + <rn_^ = p, or by the statement that the sum of the dimensions of the reduced functions is p + n — 1, is true in all cases.
For consider a rational function which is infinite to the rth order in each sheet at x — a and not elsewhere : if r be taken great enough, such a function necessarily exists and is an aggregate of nr — p + 1 terms, one of these being an additive constant (Chapter III. § 37). By what has been proved, such a function can be expressed in the form
- a
,
x - a
,,_,
42] EXPRESSED BY THE DEFICIENCY OF THE SURFACE. 55
where the dimensions of the several terms, namely the numbers
X, Xj + (T} + 1 , . . . , Xn_! + 0"n-l + 1 ,
are not greater than the dimension, r, of the function.
Conversely*, the most general expression of this form in which X^X^ ..., Xn_! attain the upper limits prescribed by these conditions, is a function of the desired kind.
But such general expression contains
(X + 1) + (Xj + 1) + ... + (Xn-, + 1),
that is (r + 1) + (r - O + . . . + (r - cr,^),
or nr — (a1+...+ o^) + 1
arbitrary constants.
Since this must be equal to nr —p+ 1 the result enunciated is proved.
The result is of considerable interest — when the forms of the functions hl...hn-l are determined algebraically, we obtain the deficiency of the surface by finding the sum of the dimensions of //x. . ,hn _ l . It is clear that a proof of the value of this sum can be obtained by considerations already adopted to prove Weierstrass's gap theorem. That theorem and the present result are in fact, here, both deduced from the same fact, namely, that the number of periods of a normal integral of the second kind is p.
42. Consider now the places x = oo : let the character of the surface be specified by k equations
_—fWi + l — fWk+l
— »l » • •• i — "k k >
X X
there being k branch places. A rational function g which is infinite only at these places will be called an integral function. If its orders of infinity at these places be respectively rlt r.2,..., rk and G [n-/(Wi+l)J be the least positive integer greater than or equal to ^/(w; + 1), and p + 1 denote the greatest of the k integers thus obtained, then it is clear that p + 1 is the least positive integer such that or*^1' g is finite at every place x = oo . We shall call p + 1 the dimension of g.
Of such integral functions there are n — 1 which we consider particularly, namely, using the notation of the previous paragraph, the functions
(x - a)^+l hlt ,(x- a)°n-i+1 hn^ ,
which by the definitions of a-1} , o-n_! are all finite at the places x = a,
and are therefore infinite only for x = oo . Denote (x — a)0^"1"1 hi by </;. If hi do not vanish at every place x = oo , it is clear that the dimension of <ft is
* It is clear that this statement could not be made if any of the indices of the function to be expressed were less than the dimension of the function. For instance in the final equation of § 40 (a), unless /t, X, A' be specially chosen, the right hand represents a function with its third index equal to
56 PARTICULAR CASE OF INTEGRAL FUNCTIONS. [42
o-j + 1. If however hi do so vanish, the dim'ension of gi may conceivably be less than o-^ + l; denote it by pi 4 1, so that pi < a-^ Then x~(?i+v gi} and therefore also (x — a)~(pi+l]gi) = (x — aYi~i>ihi, is finite at all places #=oo : hence (# — a)'Y~pi /^ is a function which only becomes infinite at the places x = a. But, in the phraseology of § 39, it is clearly a function of the same class as hi, it does not become infinite in the nth sheet at x = a, and is of less dimension than hi if a^ > p^ That such a function should exist is contrary to the definition of hi. Hence, in fact, o\- = p^. The reader will see that the same result is proved independently in the course of the present paragraph.
Let now F denote any integral function of dimension p 4 1. Then #-(P+I) F [s finite at all places x = oo : and therefore so also is (x — a)~(p+1} F. This latter function is one of those which are infinite only at places x = a ; if F do not vanish at all places x=a, the dimension cr + 1 of (x — a)~(p+1) F will be p + 1 : in general we shall have a- < p.
By § 39 we can write
x-a /Al \x-a
where cr -f 1 > Xi + o-^ + 1,
and therefore, a fortiori,
p + 1 > \ + <Ti + 1 > \i + pi + 1.
Hence we can also write
F= (1, a; - a)x O - a)'-* + (!,«- a)Al (* - a)"-A^' & 4 ......
4 (1, a? - a)An-x (« - a)"-A»-r%-i ^^j, or say
^=(1,^4(1,^,0! 4 ...... + (l,«U-i0n-i, ............ (B)
where /Ai4pi 4 1 =/) -cr. + p^4 1 =p + 1 -(^ - pf) <p + 1,
namely, there is no term on the right whose dimension is greater than that of F (and each of /-i, p,lt ...... , fin_1 is a positive integer).
Hence the equation (B) is entirely analogous to the equation (A) obtained previously for the expression of functions which are infinite only at places x = a. The set (1, glf ...... , gn-i) will be called a fundamental set
for the expression of rational integral functions*.
It can be proved precisely as in the previous Article that p1 4 p2 4 ......
4 pn-\ = P- For this purpose it is only necessary to consider a function
* The idea, derived from arithmetic, of making the integral functions the basis of the theory of all algebraic functions has been utilised by Dedekind and Weber, Theor. d. alg. Funct. e. Verdnd. Crelle, t. 92. Kronecker, U. die Discrim. alg. Fctnen. Crelle, t. 91. Kronecker, Grundziige e. arith. Theor. d. algebr. Grossen, Crelle, t. 92 (1882).
43] GENERAL PROPERTIES OF FUNDAMENTAL SYSTEMS. 57
which is infinite at the places #=oc respectively to orders r (Wj + 1), ..., r (wk + 1). And the equations Sp = Scr = p, taken with <7f > pit suffice to shew that a-i = pt. It can also be shewn that from the set gl . . . gn^ we can conversely deduce a fundamental set 1, (x — 6)~(pi+1) <ft, ...,(x — b)~lpn-rl} gn-i for the expression of functions infinite only at places x=b; these have the same dimensions as 1, (ft, ..., gn-i*-
43. Having thus established the existence of fundamental systems for integral rational functions, it is proper to refer to some characteristic pro perties of all such systems.
(a) If Gl ... Gn-: be any set of rational integral functions such that every rational integral function can be expressed in the form
(x, l\ + (x, l\ £x+ ...... + 0, 1)AB_1 Gn-, ............... (C),
there can exist no relations of the form
(X) iv + (*, IV, 0i + ...... + (x, i V^ £„_! = o.
For if k such relations hold, independent of one another, k of the functions (TJ ... 6rn_i can be expressed linearly, with coefficients which are rational in x, in terms of the other n — 1 — k. Hence also {3$, (32y2,. . . , (3n-i-k yn~l~k, @n-kyn~k> which are integral functions when &,...,$„_* are proper poly nomials in x, can be expressed linearly in terms of the n— 1— k linearly independent functions occurring among Gi...Gn-i, with coefficients which are rational in x. By elimination of these n — 1 — k functions we therefore obtain an equation
A + A,y + ...... + An_kyn-k = 0,
whose coefficients A, Al} ...... , An-k are rational in x. Such an equation is
inconsistent with the hypothesis that the fundamental equation of the surface is irreducible.
(6) Consider two places of the Riemann surface at which the inde pendent variable, x, has the same value : suppose, first of all, that there are no branch places for this value of x. Let X, \lt ...... , \n-i be constants.
Then the linear function
A. + Xj GI + ...... + \i-i Gn-i
cannot have the same value at these two places for all values of \,
For this would require that each of G1} ...... , Gn-\ has the same value
at these two places. Denote these values by a1} ...... , an_i respectively.
We can choose coefficients filt ...... , /zn_! such that the function
* The dimension of an integral function is employed by Hensel, Crelle, t. 105, 109, 111 ; Acta Math. t. 18. The account here given is mainly suggested by Hensel's papers. For surfaces of three sheets see also Baur, Math. Aniuil. t. 43 and Math. Annal. i. 46.
58 GENERAL PROPERTIES OF [43
which clearly vanishes at each of the two places in question, vanishes also at the other n — 2 places arising for the same value of x. Denoting the value of x by c, it follows, since there are no branch places for a; = c, that the function
[l*i(Gi ~ ai) + ...... + Pn-i(Gn-i - a«_i)]/0 - c)
is not infinite at any of the places x = c. It is therefore an integral rational function.
Now this is impossible. For then the function could be expressed in the form
(x, 1)A + (x, 1)^ G, + ...... + (a?, !)„ GW_a ,
and it is contrary to what is proved under (a) that two expressions of these forms should be equal to one another.
Hence the hypothesis that the function
A + A! GI + ...... + Xn_j 6rn_]
can have the same value in each of two places at which x has the same value, is disproved.
If there be a branch place at x = c, at which two sheets wind, and no other branch place for this value of x, it can be proved in a similar way, that a linear function of the form
cannot vanish to the second order at the branch place, for all values of A!, ...... , A7l_i namely, not all of G1} ...... , Gn-L can vanish to the second
order at the branch place. For then we could similarly find an integral function expressible in the form
...... + pn-i £»-i)/0 - c).
More generally, whatever be the order of the branch place considered, at x = c, and whatever other branch places may be present for x = c, it is always true that, if all of Gly ...... , Gn-i vanish at the same place A of
the Riemann surface, they cannot all vanish at another place for which x has the same value; and if A be a branch place, they cannot all vanish at A t() the second order.
Ex. 1. Denoting the function
by K, and its values in the n sheets for the same value of x by K(l\ /if <-),..., K(n\ we have shewn that, for a particular value of x, we can always choose X, X1)t.., Xn_1( so that the equation K(l) = KW is not verified. Prove, similarly, that we can always choose X, A!,..., Xn_x so that an equation of the form
) = 0, where m1,..., mlc_1, mk are given constants whose sum is zero, is not verified.
43]
FUNDAMENTAL SYSTEMS.
59
Ex. 2. Let x = ylt...,yic be k distinct given values of x: then it is possible to choose coefficients X, A!,..., p, Mi)"-) finite in number, such that the values of the function
at the places x=y1, shall be all different, and also the values of the function, at the places x=yz, shall be all different, and, also, the values of the function, for each of the places #=y3,..., yt, shall be all different.
(c) If 1, HI, H2, , Hn-i be another fundamental set of integral
functions, with the same property as 1, Glt , Gn-\, we shall have
linear equations of the form 1 = 1
where a;, j is an integral polynomial in x.
Now in fact the determinant For if I
j \ is a constant (i= 1, 2, ..., n — 1 ; denote the value of Hi, for a general value
j = 1, 2, ..., n — 1).
of x, in the rth sheet of the surface, we clearly have the identity
1, 1,
,1
|
10 ,0 |
1, 1,. |
1 |
|
' '• |
. £,<"> |
|
|
/-» (i\ ri (2) w n— i > "n— 1 >••••> |
Gn-i(n} |
ff (1) ff (2) ff («)
JJ n -i > •" n— i i ) L± 11—1
If we form the square of this equation, the general term of the square of
the left hand determinant, being of the form H^H^ + + Hi{n)Hj(n}, will
be a rational function of x which is infinite only for infinite values of x ; it is therefore an integral polynomial in x. We shall therefore have a result which we write in the form
TB_1) = V« . A (1, <?,, G,, ....... Gn^\
aitj \. A (1, H1} ...... , Hn^) may be called the
A (1, H,t ...... ,
where V is the determinant discriminant of 1 , H^ , ...... , Hn^.
If /3 be such an integral polynomial in x that fty, = 77, say, is an integral function, an equation of similar form exists when 1, tj, if, ...... , ijn~l are
written instead of 1, Hl} ...... , Hn^. Since then A (1, 77, rf, ...... , V1"1) does
not vanish for all values of x it follows that A (1, G1} G.2 ....... , G^n-i) does
not vanish for all values of x. (Cf. (a), of this Article.)
But because 1, Hlt H«, ...... , Hn_± are equally a set in terms of which all
integral functions are similarly expressible, it follows that A (1,H1, ...... ,Hn_^)
does not vanish for all values of x, and that
A (1, Glt ...... , G_1) = Vi2 A (1, H,, ...... , #„.,),
where V! is an integral function rationally expressible by x only.
60 FUNDAMENTAL SYSTEMS. [43
Hence V2 . Vt2 = 1 : thus each ofV and Vl is an absolute constant.
Hence also the discriminants A (1, Glt , Gn_^) of all sets in terms of
which integral functions are thus integrally expressible, are identical, save for a constant factor.
Let A denote their common value and 771,..., rjn denote any n integral functions whatever ; then if A fa, i)2, ..., rjn) denote the determinant which is the square of the determinant whose (s, r)th element is T/'J1, we can prove, as here, that there exists an equation of the form
A (%,%,..., *») = JfsAt
wherein M is an integral polynomial in x. The function A (77!, rj2,..., r)n) is called the discriminant of the set 77!, tj2,..., rjn. Since this is divisible by A, it follows, if, for shortness, we speak of 1, Hl,..., #„_,, equally with 77^ i}2>-"> *7n> as a set of n integral functions, that A is the highest divisor common to the discriminants of all sets of n integral functions.
(d) The sets (1, GI, , Gn-i), (1, H1} , Hn^) are not supposed
subject to the condition that, in the expression of an integral function in terms of them, no term shall occur of higher dimension than the function to
be expressed. If (1, gl} , gn-i) be a fundamental system for which this
condition is satisfied, the equation which expresses Gi in terms of 1, (ft,
g.2, , gn-i will not contain any of these latter which are of higher
dimension than that of G* Let the sets G1 , , Gn-! , g1 , , gn^ be each
arranged in the ascending order of their dimensions. Then the equations
which express Gly G2, , Gk in terms of gl, , gn_l must contain at least
k of the latter functions ; for if they contained any less number it would be possible, by eliminating those of the latter functions which occur, to obtain an equation connecting G1} , Gk of the form
(as, !)* + (*, l)Al 0,+ + (x, l\ 0* = 0;
this is contrary to what is proved under (a).
Hence the dimension of g^ is not greater than the dimension of Gk '•
hence the sum of the dimensions of Glf G2> , Gn-i is not less than the
sum of the dimensions of g1} g2, , gn-i- Hence, the least value which is
possible for the sum of the dimensions of a fundamental set (1, G1} , Gn-J
is that which is the sum of the dimensions for the set (1, <ft, , gn-i), namely,
the least value is p + n — 1.
We have given in the last Chapter a definition of p founded on Weierstrass's gap theorem : in the property that the sum of the dimensions of (ft,..., gn--i is p + n — 1 we have, as already remarked, another definition, founded on the properties of integral rational functions.
Ex. 1. Prove that if (1, glt ..., gn^v\ (1, hlt ..., hn_l) be two fundamental sets both having the property that, in the expression of integral functions in terms of them, no terms
44] THE COMPLEMENTARY FUNCTIONS. 61
occur of higher dimension than the function to be expressed, the dimensions of the individual functions of one set are the same as those of the individual functions of the other set, taken in proper order.
Ex. 2. Prove, for the surface
y«_
that the function
rt
satisfies the equation
rf - Crj2 + a2br) - «22ai = 0 > and that
A (1, y, rj) = bW + lSa^bc - 27<Va22 - 4a1c3 - 4a263,
A(l, y, /) = a12 A(l, y, ij) A(l, 77, ^2) = «22A(1, y, r,) A (y, y\ r,) = a*<* A(l, y, ij). In general 1, y, rj are a fundamental set for integral functions, in this case.
44. Let now (1, glt g.,, ...... , gn-\) be any set of integral functions in
terms of which any integral function can be expressed in the form
(x, 1)M + (x, 1 V, <7i + ...... + O, 1 ^ <7n_i ,
and let the sum of the dimensions of g1} ...... , #H_X be p + n — 1.
There will exist integral polynomials in x, (3lt /32, ...... ,/37l_i, such that
ftiy1 is an integral function: expressing this by glt ...... , gn-i in the form
above and solving for g^ ...... , gn-i we obtain* expressions of which the
most general form is
_ /*i, n-i 9i
where /*;,«_!, ...... > Pi,i, f*>i, Di are integral polynomials in x. Denote this
expression by gi (y, x}.
Let the equation of the surface, arranged so as to be an integral polynomial in x and y, be written
f(y,x) = Q«yn + Qiyn-1+ ...... + Qn-i y + Qn = o,
and let ^ (y, x) denote the polynomial
Qo y*+ &3T1 + ...... + Q,--i y + Qt-,
so that ^0 (y, #) is Q0.
Let ^>0', 0i', ...... , ^'n_! be quantities determined by equating powers of y
in the identity
* Since JT,, ..., <;n_j are linearly independent.
62 ALTERNATIVE DEFINITIONS OF [44
in other words, if the equations expressing 1, y, y2, , yn-1 in terms of
1-1,
iin~l — n 4- n « 4- -I-/7 n
y ~ "'n— l T W'n— i, i </i r T <*n— i, n— l J/n— u
where the coefficient G^J is an integral polynomial in x divided by /:?;, then
r O /x^1 — 1 V«7 * s 1 s\,^ 2 \,7 > / r • • • • • • i **"n — 1 /^Q
So that if we write n being the matrix of the transformation, we have
where %/ = %; (y', #), and H represents a transformation whose rows are the columns of H, its columns being the rows of D.
But if (Q) denote the substitution
Qn-2, Qn-3, , Qo, 0
ft, ft, 0,
Q0, 0,
we have
Hence, changing y' to y in fa' and writing therefore fa for fa', we may write Either this, or the original definition, which is equivalent to
y'-y = %o y"-1 + y71-2 %i (y', «) + + y %n-2 (y', <*) + x«-i (y, *) (F),
may be used as the definition of the forms fa, fa, , <£n_j.
The latter form will now be further changed for the purposes of an immediate application : let ylf , yn denote the values of y corresponding
44]
THE COMPLEMENTARY FUNCTIONS.
63
to any general value of x for which the values of y are distinct. Denote fc (Vr, *), ffi (yr, *), by fc<", <7*(r)> etc.
Then putting in (F) in turn y = tf = y1 and y' = ylly = y», we obtain
= 2' 3>
Hence if, with arbitrary constant coefficients cfl, c1} , Cn_i, we write
Co<£o(1> + C^1" + + C,^ ^ = </>(1),
we have
' c0 Cj cn_! ^>(1) = 0,
1 n I1) rt I1' /
1 .<7i ' yn-i J
or
/'(*
1
1 ^1(1)
^n-i
1 ^
0 r 0
1 9^
Cji— i
ffn-i
(n)
.(G);
and we shall find this form very convenient: it clearly takes an inde terminate form for some values of x.
If we put all of d, ...... , Cn-i, = 0 except cr, and put cr = 1, and multiply
both sides of this equation by the determinant which occurs on the left hand, the right hand becomes
where, if sijj = gi^ #/> + g in the determinant
+ ...... +^"1' g}™, Sitj means the minor of sitj
$1 *1, 1 ^1, 2
1, n— i
Sn— i Sn—i, i *n— i, 2 *n— i, n— i
Since this is true for every sheet, we therefore have
<f>r _ Sr + Srt ij(/i+ + Sr> n-i ffn-i
"
^aA !_ 3A 1 ^A_
64 INVERSE DETERMINATION [44
and therefore, also
The equation (H) has the remarkable property that it determines the functions ,,( . from the functions gt with a knowledge of these latter only.
J \<y x
But we can also express g1} ...... , gn-i so that they are determined from
y , y , ...... , -FTJ\ , with a knowledge of these only.
For let these latter be denoted by 70, 71, ...... , yn-i' and, in analogy with
« the definition of sr, i, let a-ft f = "2 <yr{s} 7t(s).
s=l
Then from equation (H)
n I T 1
S 7r(S) #<«> = X >SU + Sr, i «i, i + ...... + Sr, n-i Si, n-i
«=i ^ L
= 0 or 1 according as z =}= r or t = r.
Therefore, also, by equation (H),
+i ...... •
s=i
1
so that equation (H) may be written
Jr = <Tr, o + °V, 1 9\ + ...... + °V, n-i ^n-i-
If then Sr, i denote the minor of ov, < in the determinant of the quantities <rr ti — which determinant we may call V (y0, <y1} ...... ,7*1-1) — we have, in
analogy with (H),
gr=^ (Sr 7o + Sril 7!+ ...... + Srin_!7»_i) ............... (K)*.
Of course V = -^ and 2r> i = -r- s.r> t-, and equation (K) is the same as (H'). Ex. 1. Verify that if the integral functions ffi, ..., gn-i have the forms
wherein Z)15 ...,!>„_! are integral polynomials in x, then <£0, ..., 0n_! are given by
* The equations (H) and (K) are given by Hensel. In his papers they arise immediately from the method whereby the forms of >t , y2 , ...... are found.
45] EXPRESSION OF INTEGRAL OF THE THIRD KIND.
Ex. 2. Prove from the expressions here obtained that
65
and infer that 2 (dv/d.v)i = 0,
8=1
v being any integral of the first kind.
45. We are now in a position to express the Riemann integrals.
Let P£ £ be a general integral of the third kind, infinite only at the places xlt scz. Writing, in the neighbourhood of xl, x — xl = tlWt+l, dP/dx will (§§ 14, 16) be infinite like
namely, like
dP y
thus (x — #1) ^ is finite at the place x1 and is there equal to
Similarly (x — xz) -,-- is finite at #2 and there equal to
w2+ 1'
Assume now, first of all, for the sake of simplicity, that at neither x = x± nor x — x% are there any branch places ; let the finite branch places be at
At any one of these where, say, x = a + tw+1, dPjdx is infinite like 1 d
(w + l)f dP .
-V +...],
and therefore (x — a) -=- is zero to the first order at the place. 7 dx
Hence, if a = (x — aa) (x — a,). . .
be the integral polynomial which vanishes at all the finite branch places of the surface, and g be any integral function whatever, the function
K.a.g.(»-^)(,-^
is a rational function which is finite for all finite values of x and vanishes at every finite branch place.
Therefore the sum of the values of K in the n sheets, for any value of x, being a symmetrical function of the values of K belonging to that value of x, is a rational function of x only, which is finite for finite values of x and is therefore an integral polynomial in x. Since it vanishes for all the values of
66 EXPRESSION OF INTEGRAL OF [45
x which make the polynomial a zero, it is divisible by a, and may be written in the form aJ.
Let the polynomial J be written in the form
Xx (x - a;2) - X2 (x - X-L) + (x - x^ (x - x2) H,
wherein 7^ and X2 are constants and H is an integral polynomial in x. This is uniquely possible. Let H be of degree ^ - 1 in x ; denote it by (x, \Y~\
Then, on the whole,
(g = - - - -- + (.. I)-'.
—
Multiply this equation by a; — a^ and consider the case when x = xl} there being by hypothesis no branch place at as = xt. Thus we obtain the value of Xj ; namely, it is the value of g at the place x^ This we denote by g(xly y^. Similarly X,, is g (ara, y2). Further, at an infinite place where as = t-(w+l),
dP = tw+^ dP dx w + 1 dt
so that x^dPjdx is finite at all places x = oc . Hence if p + 1 be the dimen sion of the integral function g, and we write
a-P-i (^ _ tfj) ^p-1 (a; - x.2) we can infer, since p cannot be negative, that yu, is at most equal to p.
Hence, taking g in turn equal to 1, glt ..., gn-i, the dimensions of these functions being denoted by 0, r, + 1, ... , rn-, + 1, we have the equations
/ V
dP\ dP\
. . + = -
1 (dx), yi \dx)n x-x, x-
(-}
\dx/i
where r\, ... , r'7l_] are positive integers not greater than Tlf ... , TW_I respectively.
Let these equations be solved for (-5-) : then in accordance with equa-
\dxj-i
tions (G) on page 63 we have, after removal of the suffix,
45] THE THIRD AND FIRST KIND. 67
f (y) = (x, IV''-1 <f>, + (x. 1)T'*~J <k + . . . + (x, IVVi"1 <f»rt_i
dx
+
vU ^^ *^\
where </>i stands for <£; (a;, y).
This, by the method of deduction, is the most general form which dP/dx can have ; the coefficients in the polynomials (x, I)1"'*"1 are in number, at most,
T! + T2 + ... +TH_!,
or p ; and no other element of the expression is undetermined. Now the most general form of dP/dx is known to be
1 dx p dx \dx J '
wherein f ~^- 1 is any special form of -y- having the necessary character, and
\i , ..., \p are arbitrary constants. Hence, by comparison of these forms, we can infer the two results —
(i) The most general form of integral of the first kind is f dx ,_j
J f(y} X''J ' 0n-i(^, y)J,
wherein r'i < T; and the coefficients in (x, I)7'"1 are arbitrary :
(ii) A special and actual form of integral of the third kind logarithmically infinite at the two finite, ordinary, places (xly y^, (x», 7/2), namely like log [(x — x1)/(x — x2)], and elsewhere finite, is
f i 77 I I nr o"
J I \y J I tv t*/j
0o (iC> y} + 0i (x, y} gl (x2, y2) + . . . + 0n_a (x, y) gn-\ (&-2> 2/2)!
r _ y
A ^2 J
or
fx dx /"*• , d r^>0 (x, y) + 0! (x, y) gl (£, tj) + ... + 0n_i (x, y) gn-\ (j£, ri}~\
In the actual way in which we have arranged the algebraic proof of this result we have only considered values of the current variable x for which the n sheets of the surface are distinct : the reader may verify that the result is valid for all values of x, and can be deduced by means of the definitions of the forms </>„, ..., <£n_j, which have been given, other than the equation (G).
Ex. Apply the method to obtain the form of the general integral of the first kind only.
5—2
68 DEDUCTION OF INTEGRAL OF SECOND KIND. [45
We shall find it convenient sometimes to use a single symbol for the expression
<f>0 (as, y) + (/>! (x,
and may denote it by (#, £). Then the result proved is that an elementary integral of the third kind is given by
em
Px' = \ dx \(x, #1) — (x, a?.,)"].
xltx.2 Jc LV '
This integral can be rendered normal, that is, chosen so that its periods at the p period loops of the first kind are zero, by the addition of a suitable linear aggregate of the p integrals of the first kind.
Now it can be shewn, as in Chapter II. § 19, that if Ex' c denote an elemen tary integral of the second kind, the function of (x, y) given by the differ ence
,„. ..... «(,dj: ^n;-*?'. M**pt« --••.!«
wherein D% denotes a differentiation, is not infinite at (£, •»?). It follows from the form of P*' °x , here, that this function does not depend upon (x2, y«). Hence it is nowhere infinite, as a function of (x, y}. Therefore, if not inde pendent of (x, y), it is an aggregate of integrals of the first kind. Thus we infer that one form of an elementary integral of the second kind, which is once algebraically infinite at an ordinary place (£, •»;), like — (as — ^)~1, is given by
dx_ d^ ftp (x, y) + 0! (as, y) gl ( £ ??)+... + <f>n-i (
The direct deduction of the integral of the second kind when the infinity is at a branch place, which is given below, § 47, will furnish another proof of this result.
46. We proceed to obtain the form of an integral of the third kind when one or both of its infinities (xly yj, (<KZ, y») are at finite branch places ; and when there may be other branch places for x = xl or x = x2.
As before, let a be the integral polynomial vanishing at all the finite branch places. The function
ga (x — ajj) (x — #2) dP[dx
will vanish at all the places x = xl : and though it may vanish at some of these to more than the first order, it will vanish at (x1} y^) only to as high order as (x — x^}. Hence the sum of the values of this function in the several sheets for the same value of x is of the form aJ, where J is a polynomial in x which does not vanish, in general, for x = x± or x = x.^.
46] INTEGRAL OF THIRD KIND WHEN INFINITIES ARE BRANCH PLACES. 69
Hence as before (§ 45) we can write
/ dP\ I dP\ Xx X.,
Iff T- 1+ ••• + U7 j ~ 1= - +(x,iyt-1.
\ ax /i \ tW /n x — xl x — x»
Multiply this equation by x — xl and consider the limiting form of the resulting equation as (x, y) approaches to (x1, y^) : let w + 1 be the number of sheets which wind at this place. Recalling that the limiting value of (x — x^dPjdx is l/(w+l), we see that w+I terms of the left hand, corre sponding to the w+ 1 sheets at the discontinuity of the integral, will take a form
where e is a (?y + l)th root of unity. The limit of this when t = 0 is 9(xi> y\)l(w + 1); the corresponding terms of the left will therefore have 9(xi>y\) as limit. The other terms of the left hand will vanish.
Hence Xj = g(xlt y^), X2 = ^(^2, y2). The determination of the upper limit for p and the rest of the deduction proceed exactly as before. Thus,
The expression already given for an integral of the third kind holds ivhether (%i> yi), (#2, y-) be branch places or ordinary places.
If we denote the form of integral of the third kind thus determined by •P^ * > the zero c being assigned arbitrarily, it follows, as in § 45, above, that an elementary integral of the second kind, which is infinite at a branch place #!, is given by
Now if we write t for tXl and #/ =xl + tw+1, the coefficient of dxff'(y) in the integrand of the form here given for Px'fc is
Xi , Xi
<t>« + 01 • (ffi + tg,' +...)+... + (/>n-i • (gn-i + tg'n-i + • • •)
x - a? -
wherein ^>0, ..., </>„_, are functions of a-, y, and ^, .... r/,^, #/, f//, ... are written for 5r](^1) y,), ... , gn_, (Xl, y,), Dg^x,, y}), Dg,(xl} y,), ... , respectively, D denoting a differentiation in regard to t. Hence the ultimate form is
70 EXAMPLES. [46
That is, introducing £, tj, instead of xly ylt an elementary integral of the second kind, infinite at a finite branch place (f, rj), is given by
da; 0! (as, y) g( (£, rf) + . . . + <£n_! (x, y) #'„_, (£, 77)
/'(y) f-f
where </i (£, 77), ... are the differential coefficients in regard to the infini tesimal at the place. It has been shewn in (6) § 43 that these differential coefficients cannot be all zero.
Sufficient indications for forming the integrals when the infinities are at infinite places of the surface are given in the examples below (1, 2, 3, ...); in fact, by a linear transformation of the independent variable of the surface we are able to treat places at infinity as finite places.
Ex. 1. Shew that an integral of the third kind with infinities at (xly y^, (x.2t #2) can also be written in the form
(a?, y) ffr (xl , ?/i) X2 - * 00 (x, y ) + 2X2T»- <ftr (x,
_
./'(y) #-#1 ^-^2
wherein X1 = (^-a)/(*'1-a), \2 = (x-a)/(x.2-a}, T,. + I is the dimension of gr, and a is any arbitrary finite quantity.
It can in fact be immediately verified that the difference between this form and that previously given is an integral of the first kind. Or the result may be obtained by con sidering the surface with an independent variable £ = (x — a)~l and using the forms of § 39 of this chapter for the fundamental set for functions infinite only at places x — a. The corresponding forms of the functions <j> are then obtainable by equations (H) § 44.
Ex. 2. Obtain, as in the previous and present Articles, corresponding forms for inte grals of the second kind.
Ex. 3. Obtain the forms for integrals of the third and second kinds which have an infinity at a place x= QO .
It is only necessary to find the limits of the results in Examples 1 and 2 as (x1, y-^) approaches the prescribed place at infinity. It is clearly convenient to take a = 0.
Ex. 4. For a surface of the form
y* = x(x-a1) ...... (#-02P + i),
wherein a1} ..., a2p + 1 are finite and different from zero and from each other, we may* take the fundamental set (1, g^) to be (1, y\ and so obtain (00, </>1)=:(fy, 1). Assuming this, obtain the forms of all the integrals, for