Koblitz N. — Introduction to Elliptic Curves and Modular Forms :: Электронная библиотека попечительского совета мехмата МГУ

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Koblitz N. — Introduction to Elliptic Curves and Modular Forms

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Название: Introduction to Elliptic Curves and Modular Forms

Автор: Koblitz N.

Аннотация:

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. The second edition of this text includes an updated bibliography indicating the latest, dramatic changes in the direction of proving the Birch and Swinnerton conjecture. It also discusses the current state of knowledge of elliptic curves.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1984

Количество страниц: 256

Добавлена в каталог: 09.12.2012

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Предметный указатель

$\delta(z)$       111—112 122 164
$\eta(z)$       78 121 122
$\mathfrak{G}(\lambda)$       107 142 148
$\tau(n)$       122 123 164
Addition law      7 29—35
Affine variety      52
Affine variety, coordinate ring of      55
Algebraic geometry      25—26
Algorithm for congruent number problem      4 5 222
Algorithm, semi-algorithm      5
Automorphy factor      148 152 153 177—178
Bernoulli numbers      110
Bernoulli polynomials      54
Bezout's theorem      32
Birch — Swinnerton — Dyer conjecture      3 46 90—93 218 221 222
Branch points      21 25
Character of group      61
Character, additive      57 62
Character, conductor      67
Character, Dirichlet      62 75
Character, multiplicative      57 62
Character, primitive      62
Character, quadratic      82 176 187—188 191—192
Character, trivial      57
Class number      176 194 218
Class number, relations      194
Coates — Wiles theorem      92 96 221
Coates, J.      92
Commensurable subgroups      165
Complex multiplication      42 50 92 124 143 222
Conductor of character      67
Conductor of elliptic curve      143
Congruence of elliptic curve      53 59
Congruence, principal      99
Congruence, subgroup      99—100
Congruence, zeta-function      51 52
Congruent number      1 3 5 46 70 92 221—222
Congruent Number Problem      1 2 4 221—222
Congruent number problem, generalized      8 123—124 223—224
Coordinate ring      55
Critical value      90 95 193 215 216 217
Cusp      103 106 108 126
Cusp, condition      125—126 180—182
Cusp, form      108 117—118 125 127 155 182
Cusp, irregular      144 182
Cusp, regular      144 174 182
Cyclotomic fields      37
Dedekind, eta-function      78 121 122
Dedekind, zeta-function      56 88 89
Deligne, P.      53 122 164
Diagonal hypersurface      56
Different      89
Dilogarithm      76 78
Diophantus      1
Dirichlet L-series      75 188 190 193
Dirichlet series      80 141
Dirichlet series and modular forms      140—143
Dirichlet theorem (primes in arithmetic progression)      45 142
Discriminant of polynomial      26
Discriminant, modular form      111—112 12 164
Double coset      165 204
Doubly periodic      14
Eigenforms for Hecke operators      163 173—174 201
Eigenforms for Hecke operators, Euler product      163
Eigenforms for Hecke operators, half integer weight      210—211 214
Eigenforms for Hecke operators, normalized      163
Eigenforms for involution of $M_{2}(\Gamma_{0}(4))$       146
Eisenstein series      109—110 123 154 164 174 185
Eisenstein series of half integer weight      186—188 193
Eisenstein series of half integer weight, Euler product      199—201
Eisenstein series of level N      131—134
Eisenstein series, L-function of      146
Eisenstein series, normalized      111 122
Eisenstein, F.      177
Elementary divisor theorem      202
Elliptic curve      9 11
Elliptic curve over finite fields      40—41 43
Elliptic curve, addition law      7 29—35
Elliptic curve, additive degeneracy      36
Elliptic curve, complex multiplication      42 50 92 124 143 222
Elliptic curve, inflection points      13 35 41
Elliptic curve, Legendre form      224
Elliptic curve, multiplicative degeneracy      36
Elliptic curve, points of order      21 36 38—40
Elliptic curve, rank      44 46 51 91
Elliptic curve, torsion subgroup      36 43—44 49—50
Elliptic curve, Weierstrass form      24 26 33 120
Elliptic curve, Weierstrass form, functions      14—16 18 25
Elliptic curve, Weierstrass form, integrals      27—29 217
Elliptic curve, Weierstrass form, point      102 146
Euclid      1
Fermat      2 96
Fermat curve      56
Fermat last theorem      2 5
Fields of division points      37
Fields, cyclotomic      37
Fields, finite      40
Fourier transform      71 83
Fourier transform for finite group      76
Fractional linear transformation      98 102
Functional equation, Dedekind eta-function      78 121
Functional equation, Dedekind zeta-function      88 89
Functional equation, Dirichlet L-series      77—78
Functional equation, Hasse — Weil L-series      81 84 90 91
Functional equation, L-series of modular forms      140—143 216
Functional equation, Riemann zeta-function      73—74
Functional equation, theta-functions      73 76—78 85 88—89 124
Fundamental domain      100 103 105—107 146 231—232
Fundamental parallelogram      14
Galois action on division points      37—38 42 50
Gamma-function      70—71
Gauss lemma (on quadratic residues)      136
Gauss sums      56 62 67—68 188
Gaussian integers      14 41 42 65 165
General linear group      38 98
Genus      53—54
Good reduction      43 90
Grassmannian      55
Greenberg, R.      222
Gross, B.H.      93 222
Hardy, G.H.      177
Hass — Weil L-function      3 61 64 75 79 81 84 90 141
Hass — Weil L-function and modular forms      143
Hasse — Davenport relation      60 62—63 70
Hecke, E.      88 141 142
Hecke, E., character      81
Hecke, E., L-series      81
Hecke, E., operators      155 156 158 167 202
Hecke, E., operators, algebra of      157 210

Hecke, E., operators, Euler product      158 160
Hecke, E., operators, Hermitian      168 172
Hecke, E., operators, in half integer weight      168 201 206—207 210
Hecke, E., operators, on q-expansion      161 163 207
Hecke, E., operators, trace      175
Hecke, E., operators, via double cosets      167 168 202
Heegner, K.      92—93
Homogeneous polynomial      10 12 52
Hurwitz, A.      194
Hypergeometric series      29
Irregular cusp      144 182
Isotropy subgroup      102
J-invariant      105 119—120 123—124
Jacobi, K.      112
Jacobi, K., forms      194
Jacobi, K., sums      56 57 61
Jacobi, K., triple product      219
Kohnen plus-subspace      213—214
Kohnen — Shimura isomorphism      201 213—216
Kohnen — Zagier theorem      216
Kohnen, W.      214
Kronecker, L.      194
L-function of modular form      140—143 216
L-function, Dirichlet      75 77—78 188 190 193
L-function, Hasse — Weil      3 61 64 75 79 81 84 90 141
Lattice      14 21 89 153
Lattice, dual      89
Legendre form of elliptic curve      224
Legendre symbol      60 65 82 136 147 178 187—188
Level      99
Line at infinity      10—11
Linear group, general      38 98
Linear group, special      98
Liouville's theorem      15
Mellin transform      71 84 85 139
Mellin transform, inverse      142
Modular curve      91
Modular form      96—97 108—109 117—118 125 127 155
Modular form and Dirichlet series      140—143
Modular form of half integer weight      3 176 178 182
Modular form with character      127 136—139 183
Modular form, Euler product      163
Modular form, weight one      165
Modular function      108 119 125 127 155 182
Modular group $SL_{2}(\mathbb{Z})$       99
Modular point      153
Moebius function      188
Mordell theorem      43—44
Mordell — Weil theorem      44
Multiplicity One      173—174 214
New form      174
Niwa, S.      212 214 215
Pentagonal number theorem      123
Petersson scalar product      168 169—170 216
Points at infinity      10 11 13 103
Points at infinity on elliptic curve      11 12
Poisson summation      72 83
Projective completion of curve      11
Projective line      11 12 98 105
Projective plane      10
Projective plane, dual plane      12
Projective space      11
Projective variety      52
Pythagoras      1
Pythagorean triple      1—2 3 5 8
Pythagorean triple, primitive      5 6—7
q-expansion      104 109 125 126
Quadratic character      82 176 187—188 191—192
Quadratic form      176—177 221
Quadratic reciprocity      66 82 153
Quadratic residue symbol      60 65 82 136 147 178 187—188
Ramanujan conjecture      122 164
Ramanujan — Petersson conjecture      164
Ramanujan, $\tau(n)$       122 123 164
Ramanujan, S.      122
Ramification index      144
Rank of elliptic curve      44 46 51 91
Reduction mod p      43
Regular cusp      144 174 182
Representation      137 165
Representation, theory      215
Residue, field      43 55—56
Residue, theorem      15 30 116
Riemann sphere      10 21 98 105 119 120
Riemann surface      53 54 143
Riemann zeta-function      27 51 64 70 73
Root number      70 84 92 97
Shimura map      200—201 213—215
Shimura theorem      212—213 215
Shimura, G.      177 215
Shintani, T.      215
Smooth curve      9 13 52—53
Special linear group      98
Tangency, order of      13
Taniyama — Weil conjecture      91 143
Taniyama, Y.      91 143
Tate — Shafarevich group      218 222
Tate, J.      88
Theta-functions      72 76—78 84 88—89 96—97 124 176—177
Theta-functions, Hecke transformation formula      148
Torsion subgroup      36—37 43—44
Torus      14 15 21 24
Tunnell theorem      1 3 4 6 193 217 219—222
Tunnell, J.      1 200 217 219
Twisting      81 127 142 176
Type of finite abelian group      40
Upper half-plane      99
Variety, affine      52
Variety, projective      52
Waldspurger theorem      193 200 215 220
Waldspurger, J.-L.      215
Weierstrass $\mathfrak{P}$ -function      16—18 21 134
Weierstrass $\mathfrak{P}$ -function, derivatives of      17 18 20 21 22
Weierstrass $\mathfrak{P}$ -function, differential equation of      21 22 24
Weierstrass form of elliptic curve      24 26 33 120
Weight      108 153—154
Weight of polynomials      183
Weight, half integer      176
Weight, one      165
Weil Conjectures      53—54 91 122 139 164
Weil parametrization      91
Weil theorem      142—143 215
Weil — Taniyama conjecture      91
Weil, A.      44 91 141 142 143
Wiles, A.      92
Zagier, D.      93 194 222
Zeta-function, congruence      51 52
Zeta-function, Dedekind      56 88 89
Zeta-function, partial      132
Zeta-function, Riemann      27 51 64 70 73

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