Serre J.-P. — Lectures on the Mordell-Weil Theorem :: Электронная библиотека попечительского совета мехмата МГУ

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Serre J.-P. — Lectures on the Mordell-Weil Theorem

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Название: Lectures on the Mordell-Weil Theorem

Автор: Serre J.-P.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 3-d

Год издания: 1997

Количество страниц: 218

Добавлена в каталог: 21.06.2006

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      138
, translation on an abelian variety      38
      67—70
      119—120
      67
$\Gamma_0(N)$       68
$\Theta$ , theta divisor      74
$\zeta_K(s)$ , zeta function of a number field K      182 188
abc conjecture      196
Abelian variety      2 31
Abelian variety elliptic curve      see “Elliptic curve”
Abelian variety generalised Mordell conjecture      73—74
Abelian variety Manin — Demjanenko theorem      see “Manin — Demjanenko theorem”
Abelian variety Manin — Mumford conjecture      73—74
Abelian variety Mordell — Weil theorem      see “Mordell — Weil theorem”
Abelian variety normalised heights      see “Height normalised”
Abelian variety Poincare divisor      37 76 78
Abelian variety points of bounded height      53—55
Abelian variety torsion points      43—44 53—54 69
Abelian variety, approximation theorem on abelian varieties      4 95 98—101
Abelian variety, dual abelian variety      36—39 45
Abhyankar’s lemma      116
Absolute values, satisfying a product formula      7—10
Albanese variety Alb(X)      45 62
Antisymmetric divisor class      32
Approachable to within $1/q^\delta$       96
Approximation of real numbers      95—97
Approximation theorem on an abelian variety      98—101
Approximation theorem on an abelian variety effectivity      99—101 106—107
Approximation theorem on an abelian variety Thue — Siegel — Roth theorem      see “Thue — Siegel — Roth theorem”
Artin — Schreier polynomial      105
Baire’s category theorem      61
Baker’s lower bounds for linear forms in logarithms      5 93 100 110—112 189 194 198—199
Baker’s method      5 94 97 108—120
Baker’s method effectivity      5 111 114 115 117
Baker’s method elliptic curves with good reduction outside a given set of places      118—120
Baker’s method on       119—120
Baker’s method on $\mathbf{P}^1$ - $\{0,1,\infty\}$       112—114
Baker’s method on elliptic curves      115 117
Baker’s method on hyperelliptic curves      115—116
Baker’s method on superelliptic curves      117
Baker’s method on Thue curves      114
Baker’s method on X(2)      118
Banach space      29 88
Belyi — Fried — Matzat — Thompson theorem      150
Belyi’s theorem      70—73
Bertini’s theorem      127 130
Birch Swinnerton — Dyer conjecture      189
Blowing up      26—27 157
Bombieri — Davenport large sieve constant      172
Bounded sets of points      81—83
Brauer group      100
Cartan subgroup      194
Cartan subgroup split and non-split      194
Cartan subgroup, normaliser of      194
Cartier divisor      83
Catalan equation      117
Chabauty’s theorem      3 58—62
Chebotarev density theorem      61
Chevalley — Weil theorem      3 50 51 109
Class number of imaginary quadratic fields      188—199
Class number of imaginary quadratic fields, class number 1      188—199
Class number of imaginary quadratic fields, class number 2      199
Classification of finite simple groups      151
Cohen’s theorem      6 177
Complex multiplication      see “Elliptic curve”
Cube, theorem of the      32 34 38 77
Cubic resolvent      124
Curves Chabauty’s theorem      see “Chabauty’s theorem”
Curves elliptic      see “Elliptic curve”
Curves exceptional (Siegel’s theorem)      94—95 110
Curves hyperelliptic      115—116 142
Curves Mordell’s conjecture      see “Mordell’s conjecture”
Curves Mumford’s inequality      see “Mumford’s inequality”
Curves Mumford’s theorem      see “Mumford’s theorem”
Curves superelliptic      117
Curves Tate      91
Curves Thue      114
Cusps, on modular curves      68 71 73 118
Davenport — Halberstam theorem      166—172
Demjanenko — Manin theorem      3 58 62—67
Descent lemma      3 53
Differential form      59 68 90
Dirac $\delta$ -function      166
Dirichlet series      44 181 182
Divisor, on a variety      21
Divisor, on a variety algebraically equivalent to zero      25—26 44—45
Divisor, on a variety ample      22
Divisor, on a variety class group      21
Divisor, on a variety height associated to a      2 22—24
Divisor, on a variety Poincare divisor class      37 76
Divisor, on a variety symmetric and antisymmetric divisor classes      32
Divisor, on a variety, $\Theta$ -divisor, on a jacobian      74
Divisor, on a variety, support of      21
Effectivity effective construction of an elliptic curve $E/\mathbf{Q}$ of rank $\geq$ 9      155—156
Effectivity of Baker’s method      5 111 114 115 117
Effectivity of Chabauty’s theorem      60
Effectivity of Siegel’s theorem      5 100—101 106—107 116
Effectivity of the approximation theorem on an abelian variety      99—101
Effectivity of the Manin — Demjanenko theorem      63
Effectivity of the Mordell — Weil theorem      52 99—100 107
Elliptic curve complex multiplication      44 70 93 119 191—194 196
Elliptic curve conductor      119
Elliptic curve explicit form of Mordell — Weil theorem      56—57
Elliptic curve integral points      97 115 117
Elliptic curve j-invariant      67 118 145—147 158 162 191—193
Elliptic curve local heights      90—93
Elliptic curve normalised heights      40—41 90—93
Elliptic curve of large rank over $\mathbf{Q}$       121 154—162
Elliptic curve Tate module      70 119
Elliptic curve torsion points      69
Elliptic curve with good reduction outside a finite set of places      118—120
Embedding a field finitely generated over $\mathbf{Q}$ in $\mathbf{Q}_p$       61—62
Exceptional curve (Siegel’s theorem)      94—95 110
Exceptional units      104
Fermat curve      110
Fermat quartics (Demjanenko’s theorem)      66—67
Function fields      7—8 19
Function fields heights      11—13
Function fields number of points of bounded height in $\mathbf{P}^n$       19
Functions of degree $\leq$ 2 between abelian groups      32—34 38
Fundamental group      141 150
Galois cohomology      51—52
Galois group       144—145
Galois group       138—144 145
Galois group Belyi — Fried — Matzat — Thompson theorem      150
Galois group finite simple      151
Galois group infinite      147—149
Galois group Noether’s method      147
Galois group Schur’s examples      145
Galois group Shih’s theorem      146—147
Galois group using elliptic curves : , ,       145—147
Galois group with rigid family of rational conjugacy classes      149—150
Galois group, construction of field extensions of $\mathbf{Q}$ with given      6 121
Gel’fond — Linnik — Baker method      197—199
Generalised Riemann Hypothesis (GRH)      188
Goldfeld’s theorem      189
Grassmannian $G_{n,d}$       127
Greenberg’s theorem      83
Gross — Zagier theorem      188 189
Grothendieck — Deligne (Weil conjectures)      184
Hasse’s theorem      156
Height associated to a divisor algebraically equivalent to zero      25—26 44—45
Height associated to a line bundle       2 22—28
Height associated to a morphism $h_\phi(x)$       19—20
Height associated to a Poincare divisor      36—39
Height associated to a torsion divisor      24 46—48
Height behaviour under change of coordinates      13
Height bilinear form associated to $\tilde h_c(x)$       36 41
Height change of height under projection      13—16 19—20

Height elementary properties      10—16
Height for function fields      11—13
Height for number fields      11
Height logarithmic height h(x)      2 11
Height non-degeneracy of normalised heights      41—43
Height normalised height $\tilde h(x)$ , $\tilde h_c(x)$       2 30—31 35—43 63 134
Height Northcott’s finiteness theorem      16—17
Height of a rational point H(x)      2—6
Height on $\mathbf{P}^n$       10—13
Height positivity      24—25
Height quadraticity of normalised heights on abelian varieties      35—41
Height relation between quadratic and linear parts of $\tilde h_c(x)$       38—39
Height Schanuel’s theorem      17—19
Height, functoriality of      23
Height, local height      see “Local heights”
Hensel’s Lemma      62
Hermite’s finiteness theorem      3 49—52 109
Hermitian structure on a line bundle      84—85
Hilbertian field      129—130 137—138
Hilbertian field, finitely generated extensions of hilbertian fields are hilbertian      130
Hilbert’s irreducibility theorem      5—6 121 130 149
Hilbert’s irreducibility theorem $\mathbf{Q}$ is hilbertian      130—132
Hilbert’s irreducibility theorem Neron’s specialisation theorem      see “Neron’s specialisation theorem”
Hilbert’s irreducibility theorem relation with integral points      135
Hilbert’s irreducibility theorem specialisation of Galois groups      122—126 137—138
Hilbert’s irreducibility theorem thin sets      see “Thin sets”
Hilbert’s irreducibility theorem, hilbertian field      129—130 137—138
Honda’s conjecture      162
Hurwitz’ genus formula      142—143
Hyperelliptic curve      115—116
Inertia subgroup      140 143 150
Integral points      4—6 94—95 97 100 102—106
Integral points Baker’s method      5 94 97 108—120
Integral points behaviour under morphisms      108—110 114
Integral points elliptic curves with good reduction outside a finite set of places      118—120
Integral points exceptional curves      94—95 110
Integral points of bounded height      6 115 117 177—178
Integral points on       119—120
Integral points on $\mathbf{P}_1$ - $\{0,1,\infty\}$       5 102—104 112—114
Integral points on elliptic curves      97 115 117
Integral points on hyperelliptic curves      5 115—116
Integral points on modular curves and the class number 1 problem      194—199
Integral points on superelliptic curves      117
Integral points on Thue curves      114
Integral points on X(2)      118
Integral points quasi-integral sets      94—95
Integral points S-integral points      94 104—105
Integral points Siegel’s theorem (see also “Siegel’s theorem”)      95 102—104
Intersection multiplicities      85—86
Jacobian of a curve      1—2 5 46 58—59 66 73 98 101 134
Jacobian of a curve $\Theta$ divisor      74
Jacobian of a curve is principally polarised      46 76
Jacobian of a curve Poincare divisor class      76
Kissing number      79
Kronecker’s limit formula      194 199
Kubert — Lang      115 119—120 197
Kummer theory      55—56
L-function      188 189 198—199
Lang — Weil theorem      6 62 184
Large point      79 80
Large sieve inequality      6 163—164
Large sieve inequality Bombieri — Davenport bound      172
Large sieve inequality Davenport — Halberstam theorem      166—172
Large sieve inequality improved inequalities      170—172
Large sieve inequality Selberg’s bound      171—172
Legendre’s equation of an elliptic curve      118
Lenstra’s example      147
Lie group      60 149
Line bundle, generalities      20—22
Line bundle, generalities ample      22
Line bundle, generalities generated by its global sections      21—22
Line bundle, generalities the Picard group Pic(X)      20—22
Linear torus      73
Local heights      83—93
Local heights as intersection multipicities      85—86
Local heights case of an abelian variety with good reduction      89
Local heights in terms of theta functions      88—89
Local heights normalisation on abelian varieties      87—89
Local heights relation with global heights      89—90
Local heights Tate normalisation on elliptic curves      90—93
Locally compact field      81
Logarithmic height      2 11
Lower bounds for linear forms in logarithms      5 94 110—112
Manin — Demjanenko theorem      58 62—66 154
Manin — Demjanenko theorem application to Fermat quartics       66—67
Manin — Demjanenko theorem application to modular curves      67—69
Manin — Demjanenko theorem application to the Tate module of an elliptic curve      73—74
Manin — Drinfeld theorem      73 197
Manin — Mumford conjecture      76
Mazur’s theorem      69
Modular curves       67—70 146
Modular curves       119—120
Modular curves       67
Modular curves associated to Cartan subgroups and their normalisers      194—197
Modular curves Belyi’s theorem      70—73
Modular curves cusps      69 71 73 118 195—197
Modular curves Manin — Drinfeld theorem      73
Modular curves Manin’s theorem      67—69
Modular curves X(N)      118 193
Mordell — Weil theorem      1 3—4 52 58 99 100
Mordell — Weil theorem classical descent      53
Mordell — Weil theorem effectivity      52 99—101 107
Mordell — Weil theorem explicit form      55—57
Mordell — Weil theorem for finitely generated ground fields      52
Mordell — Weil theorem weak Mordell — Weil theorem      3 51—52 99
Mordell-Weil group      43—44 51—53
Mordell-Weil group rank      3 58—59 69
Mordell-Weil group torsion subgroup      43—44 54 69
Mordell-Weil group, generators for      99
Mordell’s conjecture      1 3—4 58—59 105 133
Mordell’s conjecture Chabauty’s theorem      see “Chabauty’s theorem”
Mordell’s conjecture generalised      73—74
Mordell’s conjecture in characteristic p > 0      80 105
Mordell’s conjecture Manin — Demjanenko theorem      see “Manin — Demjanenko theorem”
Mordell’s conjecture Manin — Mumford conjecture      see “Manin — Mumford conjecture”
Mordell’s conjecture Mumford’s theorem      see “Mumford’s theorem”
Multiplicative function      182
Mumford’s inequality      77
Mumford’s lemma      23
Mumford’s theorem      3—4 58 74—80 105 133
Mumford’s theorem optimal for function fields      80
Nakai — Moishezon criterion      26 27
Neron model of an abelian variety      89
Neron — Severi group NS(X)      25 31—32 47 62 64 157
Neron — Tate normalisation      2 29—31 35
Neron’s estimate      53—55
Neron’s normalisation of local heights      87—89
Neron’s specialisation theorem for abelian varieties      59 152—154
Neron’s specialisation theorem for abelian varieties Silverman’s theorem      154
Neron’s specialisation theorem for abelian varieties specialisation of extensions of abelian varieties      154
Neron’s specialisation theorem for abelian varieties specialisation of non-commutative groups      154
Noether’s method      147
Non-degeneracy of normalised heights      42—43 (see also “Height” “
Normalised heights      2 30—31 34—48
Normalised heights non-degeneracy      42—44
Normalised heights on elliptic curves      39—41
Normalised heights positivity      41
Normalised heights quadraticity      35—36
Normalised heights, normalised local heights      87—89 90—93
Northcott’s finiteness theorem      16—17 44 46 53
Northcott’s finiteness theorem quantitative form (Schanuel’s theorem)      17—19 132
NS(X), Neron — Severi group      26 31—32 47 62 64 157
Number of integral points of bounded height on a thin set      6 134—136 163 177
Number of integral points of bounded height on an affine variety      177—178
Number of rational points of bounded height on $\mathbf{P}_2$ blown up at a point      26—28
Number of rational points of bounded height on $\mathbf{P}_n$       2 17—19
Number of rational points of bounded height on a curve      5—6 80
Number of rational points of bounded height on a thin set      5—6 132—134 178
Number of rational points of bounded height on an abelian variety      53—55
Number of rational points of bounded height on an algebraic variety      178

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