Prasolov V., Solovyev Y. — Elliptic functions and elliptic integrals :: Электронная библиотека попечительского совета мехмата МГУ

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Prasolov V., Solovyev Y. — Elliptic functions and elliptic integrals

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Название: Elliptic functions and elliptic integrals

Авторы: Prasolov V., Solovyev Y.

Аннотация:

This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 191

Добавлена в каталог: 01.12.2013

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Предметный указатель

$C_{n}$       71
$f_{1}(\tau)$       155
$f_{2}(\tau)$       155
$K_{n}$       72
$r_{E}$       124
$S_{p}$ , Serret's curve      95
$\equiv$ as identically equal to      5
$\eta$ -function, Dedekind's      160
$\eta(\tau)$ , Dedekind's $\eta$ -function      160
$\overset{\frown}{BM}$ the length of $\smile BM$       55
$\wp(z)$ , the Weierstrass function      32
Abel      26 68 140
Abel's theorem      140
Algebraic addition theorem      49
am u, the amplitude      47
Amplitude      47
Automorphisms, trivial      101
Axes, principal      58
Basis of transcendentality      78
bernoulli      68
Bernoulli's lemniscate      25 68
Bernoulli, Nicholas      40
Birch      125
Blow-up      93
Bring      147
Cassini      67
Cassini's ovals      68
Chain      76
cn u      47
Complex multiplication      101
conic      1
Conjugate (diameters)      58
Cubic      1 131
Cubic, nonsingular      16
Curve, algebraic      1
Curve, elliptic associated with Serret's curve      97
Curve, Serret      95
Dedekind function      160
Descartes      103
Diagram, Newton      99
Diameters, conjugate      58
Diophantine equation      103
Diophantus      103
Division of arcs      55
Division of the arc "by half"      56
dn u      47
Domain, fundamental      173
Doubly periodic function      29
Duality, projective      18
Duplication of the arc      56
E( $\mathbb{Q}$ ), the set of rational points on the curve E      115
E( $\phi$ )      43
Eisenstein      68 84
Eisenstein's theorem      85
Element, primitive      139
ellipse      25
Elliptic integral      43 54
Elliptic Jacobi functions      46 47
Equation, cubic      131
Equation, general      140
Equation, modular      166
Equation, quartic      131
Equation, resolving      136
Equation, solvable in radicals      140
Euclid      103
Euler      25 64 68
Euler formula      12
Exponential      50
F( $\phi$ )      43
f( $\tau$ )      155
Fagnano      25 60 68
Fagnano's theorem      56
Fermat      104 118
Fermat primes      68
Field, normal or Galois      140
Form, Legendre      42
Form, Weierstrass      16 42
Formula, Euler      12
Fueter      118
Function, doubly periodic      29
Function, elliptic      29
Function, elliptic Jacobi      31 47
Function, meromorphic      29
Function, Weierstrass      31 32
Fundamental domain      173
Fundamental parallelogram      29
Galois field      140
Gauss      26 68
Generic lines      18
Generic points      18
Genus of the curve      98
Group of substitutions      140
Group, torsion of the elliptic curve      124
Hasse, H.      126
hermite      131
Hesse curve      13
hessian      13
Holzer      109
Inflection point      13
Integral, elliptic      43 54
Involution      89
j( $\tau$ )      169
J-invariant      101
jacobi      26
Jacobi elliptic function      31

L-function of a rational elliptic curve      126
Lagrange's resolvent      135
legendre      26 60 107
Legendre form      42
Legendre's Theorem      41
Leibniz      40 145
Lemniscate      67
Lemniscate, Bernoulli      25 68
Lexicographically ordered monomials      133
Lines, generic      18
Main theorem on symmetric polynomials      133
Mazur      128
Mazur's theorem      128
Meromorphic function      29
Mesiriac      104 118
Mestre      125
Method of secants      104
Modular equation      166
Moebius transformation      159
Mordell      118 119
Mordell's Theorem      119
Multiplicity      93
Newton diagram      99
Normal field      140
Normal form, Weierstrass'      16
Normal forms of a nonsingular cubic      16
Number, congruent      111
Numbers, complex, odd      81
Odd complex numbers      81
Order of a point      114
Order of the elliptic function      31
Ovals, Cassini      68
Pappus's Theorem      5
Parallelogram, fundamental      29
Pascal's theorem      1 4
Picard's big theorem      51
Pluecker, J.      7
Poincare, A.      119
Point, fixed      89
Point, inflection      13
Point, nonsingular of the curve      12
Point, simple      93
Point, singular      93
Point, the order of      114
Points, generic      18
Primitive element      139
Primitive root      137
Principal axes      58
Projective duality      18
Projective plane      9
Quartic      131
Rank of the elliptic curve      124
Rationality field      140
Reduction of the curve modulo p      125
Regiomontanus      103
Resolvents, Lagrange's      135
Resolving equation      136
Resultant      14
Rosen      68 86
Ruffini, Paulo      140
Serret      64
Serret's curve      95
Serret's Theorem      62
Serret, Joseph-Alfred      61
Set, Zorn closed      76
sn u      47
Solution, integer      103
Solution, Rational      103
Steiner, J.      7
Sum of points      1
Swinnerton-Dyer      125
symmetric      133
Symmetric polynomials, elementary      133
Tangent to the curve      12
Theorem, Abel      140
Theorem, algebraic addition      49
Theorem, Eisenstein      85
Theorem, Fagnano      56
Theorem, Legendre      41
Theorem, Mazur      128
Theorem, Mordell      119
Theorem, on symmetric polynomials, main      133
Theorem, Pappus      5
Theorem, Pascal      1 4
Theorem, Picard's big      51
Theorem, Serret      62
Theorem, Vieta      166
Theorem, Weierstrass      50
Theta constants      151
Theta functions      150
Tors E( $\mathbb{Q}$ )      124
Trace      123
Transcendentality, basis of      78
Transformation, fractional linear      159
Transformation, Moebius      159
Transformation, Tschirnhaus      146
Tschirnhaus's transformation      146
Vieta theorem      166
von Tschirnhaus, E.W.      145
Weber      131
Weierstrass form      16 42
Weierstrass function      31 32
Weierstrass' theorem      50
Wiles, A.      126
Zagier      89

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