Baker H.F. — Abel's Theorem and the Allied Theory Including The Theory of the Theta Functions :: Электронная библиотека попечительского совета мехмата МГУ

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Baker H.F. — Abel's Theorem and the Allied Theory Including The Theory of the Theta Functions

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Название: Abel's Theorem and the Allied Theory Including The Theory of the Theta Functions

Автор: Baker H.F.

Аннотация:

Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincaré, Clebsch, Jacobi and other outstanding mathematicians of the last century, was mainly an analytical theory. In our century the methods and ideas of topology, commutative algebra and Grothendieck's schemes enriched it and seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This classic book, written in 1897, covers the whole of algebraic geometry and associated theories. Baker discusses the subject in terms of transcendental functions, and theta functions in particular. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1897

Количество страниц: 684

Добавлена в каталог: 25.03.2006

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Предметный указатель

Periodicity of a (1,1) correspondence      650
Periods of Riemann’s integrals      16 21
Periods of Riemann’s integrals, for integrals of second kind      323
Periods of Riemann’s integrals, general theory of systems of periods      ff. 579 ff.
Periods of Riemann’s integrals, general transformation      536 538
Periods of Riemann’s integrals, linear transformation of periods      532
Periods of Riemann’s integrals, of degenerate integrals      657
Periods of Riemann’s integrals, of factorial integrals      404
Periods of Riemann’s integrals, Riemann’s and Weierstrass’s relations for the periods of integrals of the first kind, and of associated integrals of the second kind      197 285 581 587
Periods of Riemann’s integrals, rule for half-periods on a hyperelliptie surface      297
Picard      14 165 594 659
Picard’s theorem (Weierstrass’s)      658
Pick      360 430 639
Pl $\ddot{u}$ cker      124 165
Pl $\ddot{u}$ cker’s equations, for curves in space      166
Pl $\ddot{u}$ cker’s equations, generalised form of      123 124
Places, after linear transformation      562
Places, branch places      7 9 46 74 122 297 569
Places, determination of, for a Riemann surface with assigned period Places, loops      567
Places, for a hyperelliptie surface      297 563
Places, of a Riemann surface      1 2 3
Places, the places $m_{1}$ , ..., $m_{p}$       255
Places, their geometrical interpretation      265 266
Places, where a rational fnnetion is infinite, to order less than       38 41 90 653
Poincar $\acute{e}$       239 372 373 439 486 594 654 659 663
Poles      see Inflnities.
Polynomial, $\phi$ -polynomials      141
Polynomial, adjoint      121 128
Polynomial, algebraic treatment of      120
Polynomial, expression of rational functions, and algebraic integrals by means of adjoint polynomials      156 see Curves.
Polynomial, grade of      120
Polynomial, transformation of fundamental equation by $\phi$ -polyno- mials      142 154
Positive direction of period loop      529
Primary and associated systems of factorial fnnctions      397
Prime function (or form)      see Function.
Pringsheim      445
Product expression of uniform transcendental function with single essential singularity      205
Prym      2 296 342 392 477 486 511 566 600 627
Quartic. Double tangents of plane quartic curve      381 390 647
Quotients of theta functions      310 311 390 516
Radical function      see Function.
Rational function, of order 1, algebraic expression of, by adjoint polynomials      125 ff. 156
Rational function, of order 1, expressed by Riemann’s integrals      24 212
Rational function, of order 1, expressed by Weier- strase’s function      176
Rational function, of order 1, fundamental integral rational functions, algebraic determination of      105 ff.
Rational function, of order 1, infinities of, Riemann- Roch theorem, Weierstrass’s gap theorem      27 ff.
Rational function, of order 1, integral function      47 ff. 55 91 ff.
Rational function, of order 1, is an uniform function on the Riemann surface whose only infinities are poles      27
Rational function, of order 1, of order       38 137
Rational function, of order 1, of the second order      80 ff.
Rational function, of order 1, only exists when       8
Rational function, of order 1, special      25 137
Rational function, of order 1, Weierstrass’s fundamental      171 175 177 ff. 182
Reciprocal sets of zeros of adjoint polynomials      134
Residual sets of places      135
Residue, fundamental residue theorem      232 189 20
Reversible transformation      see Birational.
Richelot      221 230 529 600 668
Riemann      1 2 6 9 13 45 47 77 113 115 246 248 255 296 343 397 409 628
Riemann and Weierstrass’s period relations      197 285 581 587
Riemann — Roch theorem      44 133
Riemann — Roch theorem, for factorial functions      405
Right side of period loop      529
Ritter      360 373 392 429 439 442
Roch      29
Rosanes      666
Rosenhain      221 222 246 311 459 600 607 662
Row and column      see Matrices
Salmon      5 6 7 11 39 117 124 136 144 165 267 383 389
Schepp      239
Schl $\ddot{a}$ fli      666
Schottky      32 101 283 296 340 343 345 360 371 372 373 387 448 461 469 544 628
Schottky — Klein prime form and function      360 430 433
Schubert      665
Schwarz      14 654
Scott      473
Sequence, theorem of      114 161 165
Sequent sets of places      135
Sets of places on a Riemann surface or algebraic curve      135 see Special.
Sign of equivalence and congruence      236 256 261 264 487
smith      12 600 639 666 674 675
Sohnke      689
Special correspondences on a Riemann surface      648
Special rational functions      25 62 137
Special sets of zeros of adjoint polynomials      134 147

Special transformation of a theta function      ff. 639 660
Stahl      288 301 392 430 486 502
Stickelberger      666
Stolz      2
Strength of assigned zeros, as determinators of a polynomial      133
Supplementary transformations of a theta function      552
Sylvester      136
System, G $\ddot{o}$ pel      see Characteristics.
Syzygetic characteristics      487 542
Taber      668
Tables of Characteristics      303 305
Tangents, double, of a plane curve, by the principle of correspondence      644 646
Theta functions, addition theorem for hyperelliptic      882 337
Theta functions, algebraic expression for hyperelliptic      435
Theta functions, algebraic expression of first logarithmic derivatives of      288 290 320
Theta functions, algebraic expression of quotients of      310 311 390 426
Theta functions, algebraic expression of second logarithmic derivatives of      293 324 329 333
Theta functions, convergence of      247
Theta functions, determination of, from periodicity      444
Theta functions, General theta function of first order      283 444
Theta functions, General theta function of first order, addition theorems for      457 472 481 513 521
Theta functions, General theta function of first order, expression of Jacobian functions by means of      594
Theta functions, General theta function of first order, G $\ddot{o}$ pel relation for, in case       see G $\ddot{o}$ pel;
Theta functions, General theta function of first order, period relations      285 197 581 587
Theta functions, General theta function of first order, second logarithmic derivatives of      516
Theta functions, hyperelliptic      296 ff.
Theta functions, identical vanishing of      258 271 276 303
Theta functions, number of odd and even      446
Theta functions, period properties of      249
Theta functions, Riemann’s functions not the most general      248 628
Theta functions, Riemann’s theta functions      246 ff.
Theta functions, solution of inversion problem by means of      275 324 390 426 ff.
Theta functions, Theta functions of second and higher order      448
Theta functions, Theta functions of second and higher order, every theta functions of same order, periods, and characteristic, connected by a homogeneous polynomial relation      453
Theta functions, Theta functions of second and higher order, expression of, number of linearly independent      452 463
Theta functions, Theta functions of second and higher order, of order 2, of special kind      509 510
Theta functions, Transformation of theta functions      see Transformation;
Theta functions, Transformation of theta functions, characteristics of theta functions      see Characteristics;
Theta functions, Transformation of theta functions, complex multiplication of theta functions      629 ff. 639 660
Theta functions, Transformation of theta functions, hyperelliptic case      433
Theta functions, Transformation of theta functions, particnlar cases      430 ff.
Theta functions, Transformation of theta functions, theta functions Theta functions, Transformation of theta functions, expressed by factorial functions and simpler theta fnnetions      426
Theta functions, zeros of      252 255 258 567
Thomae      288 296 318 533 556 600
Thompson      436
toeplitz      383
Transcendental uniform function      200
Transcendental uniform function, application of Laurent’s theorem when the function is integral      444
Transcendental uniform function, expressed in prime factors      205
Transcendental uniform function, Mittag- Leffler’s theorem for      202
Transformation of characteristics      see Characteristics.
Transformation of periods      528 534 539 551 553 555 559 568
Transformation of the algebraic equation (or Riemann surface)      3 143 145 151 152 654 655
Transformation of theta functions      535
Transformation of theta functions for any odd order, general theorem      614
Transformation of theta functions for hyperelliptic case      568
Transformation of theta functions of second order      603 617
Transformation of theta functions, composition of      551
Transformation of theta functions, for any odd order, constants in      620 622
Transformation of theta functions, linear transformation      539
Transformation of theta functions, linear transformation constants in      554 559
Transformation of theta functions, special transformations      629 660
Transformation of theta functions, supplementary transformations      552
Transformation of theta functions, when coefficients not integers      625
uniform      see Rational and
Valentin      101
Valentiner      124 165 665
Vanishing of theta function      253 258 271 276 303
Variables, homogeneous      118 429 441
Voss      137
Weber      8 56 270 272 37 387 392 430 460 486 538 538 559 600 615 620 639 674
Weierstrass      32 93 99 101 177 195 197 205 231 239 242 246 301 311 317 326 389 443 474 486 525 570 571 572 573 577 579 594 628 653 654 658 666
Weierstrass’s gap theorem      32 34 93 174
Weierstrass’s gap theorem, canonical surface (or equation)      90 ff. 93
Weierstrass’s gap theorem, fundamental rational fnnction      171 175 177 178 182 189
Weierstrass’s gap theorem, period relations      197 ff. 285 581 587
Weierstrass’s gap theorem, rule for characteristics of hyperelliptic theta functions      569
Weierstrass’s gap theorem, special places which are the poles of rational functions of order less than       34 ff.
Weierstrass’s gap theorem, theorem for degenerate integrals      658
White      165
Wiltheiss      842 600 629 660 662 674
Wirtinger      340 486 578 628
Zeros, generalised zeros of a polynomial      121
Zeros, zeros of Riemann theta function      252
Zeta function      287 292 320
Zeuthen      647

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