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









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: Lectures on the Mordell-Weil Theorem

: Serre J.-P.

: en

: /

:

ed2k: ed2k stats

: 3-d

: 1997

: 218

: 21.06.2006

: | | ID
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$Gal_T$      138
$T_a$, translation on an abelian variety      38
$X_0(N)$      6770
$X_1(N)$      119120
$Y_0(N)$      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      7374
Abelian variety Manin Demjanenko theorem      see Manin Demjanenko theorem
Abelian variety Manin Mumford conjecture      7374
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      5355
Abelian variety torsion points      4344 5354 69
Abelian variety, approximation theorem on abelian varieties      4 95 98101
Abelian variety, dual abelian variety      3639 45
Abhyankars lemma      116
Absolute values, satisfying a product formula      710
Albanese variety Alb(X)      45 62
Antisymmetric divisor class      32
Approachable to within $1/q^\delta$      96
Approximation of real numbers      9597
Approximation theorem on an abelian variety      98101
Approximation theorem on an abelian variety effectivity      99101 106107
Approximation theorem on an abelian variety Thue Siegel Roth theorem      see Thue Siegel Roth theorem
Artin Schreier polynomial      105
Baires category theorem      61
Bakers lower bounds for linear forms in logarithms      5 93 100 110112 189 194 198199
Bakers method      5 94 97 108120
Bakers method effectivity      5 111 114 115 117
Bakers method elliptic curves with good reduction outside a given set of places      118120
Bakers method on $X_1(n)$      119120
Bakers method on $\mathbf{P}^1$ - $\{0,1,\infty\}$      112114
Bakers method on elliptic curves      115 117
Bakers method on hyperelliptic curves      115116
Bakers method on superelliptic curves      117
Bakers method on Thue curves      114
Bakers method on X(2)      118
Banach space      29 88
Belyi Fried Matzat Thompson theorem      150
Belyis theorem      7073
Bertinis theorem      127 130
Birch Swinnerton Dyer conjecture      189
Blowing up      2627 157
Bombieri Davenport large sieve constant      172
Bounded sets of points      8183
Brauer group      100
Cartan subgroup      194
Cartan subgroup split and non-split      194
Cartan subgroup, normaliser of      194
Cartier divisor      83
Catalan equation      117
Chabautys theorem      3 5862
Chebotarev density theorem      61
Chevalley Weil theorem      3 50 51 109
Class number of imaginary quadratic fields      188199
Class number of imaginary quadratic fields, class number 1      188199
Class number of imaginary quadratic fields, class number 2      199
Classification of finite simple groups      151
Cohens theorem      6 177
Complex multiplication      see Elliptic curve
Cube, theorem of the      32 34 38 77
Cubic resolvent      124
Curves Chabautys theorem      see Chabautys theorem
Curves elliptic      see Elliptic curve
Curves exceptional (Siegels theorem)      9495 110
Curves hyperelliptic      115116 142
Curves Mordells conjecture      see Mordells conjecture
Curves Mumfords inequality      see Mumfords inequality
Curves Mumfords theorem      see Mumfords theorem
Curves superelliptic      117
Curves Tate      91
Curves Thue      114
Cusps, on modular curves      68 71 73 118
Davenport Halberstam theorem      166172
Demjanenko Manin theorem      3 58 6267
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      2526 4445
Divisor, on a variety ample      22
Divisor, on a variety class group      21
Divisor, on a variety height associated to a      2 2224
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      155156
Effectivity of Bakers method      5 111 114 115 117
Effectivity of Chabautys theorem      60
Effectivity of Siegels theorem      5 100101 106107 116
Effectivity of the approximation theorem on an abelian variety      99101
Effectivity of the Manin Demjanenko theorem      63
Effectivity of the Mordell Weil theorem      52 99100 107
Elliptic curve complex multiplication      44 70 93 119 191194 196
Elliptic curve conductor      119
Elliptic curve explicit form of Mordell Weil theorem      5657
Elliptic curve integral points      97 115 117
Elliptic curve j-invariant      67 118 145147 158 162 191193
Elliptic curve local heights      9093
Elliptic curve normalised heights      4041 9093
Elliptic curve of large rank over $\mathbf{Q}$      121 154162
Elliptic curve Tate module      70 119
Elliptic curve torsion points      69
Elliptic curve with good reduction outside a finite set of places      118120
Embedding a field finitely generated over $\mathbf{Q}$ in $\mathbf{Q}_p$      6162
Exceptional curve (Siegels theorem)      9495 110
Exceptional units      104
Fermat curve      110
Fermat quartics (Demjanenkos theorem)      6667
Function fields      78 19
Function fields heights      1113
Function fields number of points of bounded height in $\mathbf{P}^n$      19
Functions of degree $\leq$ 2 between abelian groups      3234 38
Fundamental group      141 150
Galois cohomology      5152
Galois group $A_n$      144145
Galois group $S_n$      138144 145
Galois group Belyi Fried Matzat Thompson theorem      150
Galois group finite simple      151
Galois group infinite      147149
Galois group Noethers method      147
Galois group Schurs examples      145
Galois group Shihs theorem      146147
Galois group using elliptic curves : $GL_2(Z/nZ)$, $PSL_2(F_p)$, $SL_3(F_2)$      145147
Galois group with rigid family of rational conjugacy classes      149150
Galois group, construction of field extensions of $\mathbf{Q}$ with given      6 121
Gelfond Linnik Baker method      197199
Generalised Riemann Hypothesis (GRH)      188
Goldfelds theorem      189
Grassmannian $G_{n,d}$      127
Greenbergs theorem      83
Gross Zagier theorem      188 189
Grothendieck Deligne (Weil conjectures)      184
Hasses theorem      156
Height associated to a divisor algebraically equivalent to zero      2526 4445
Height associated to a line bundle $h_c(x)$      2 2228
Height associated to a morphism $h_\phi(x)$      1920
Height associated to a Poincare divisor      3639
Height associated to a torsion divisor      24 4648
Height behaviour under change of coordinates      13
Height bilinear form $B_c(x,y)$ associated to $\tilde h_c(x)$      36 41
Height change of height under projection      1316 1920
Height elementary properties      1016
Height for function fields      1113
Height for number fields      11
Height logarithmic height h(x)      2 11
Height non-degeneracy of normalised heights      4143
Height normalised height $\tilde h(x)$, $\tilde h_c(x)$      2 3031 3543 63 134
Height Northcotts finiteness theorem      1617
Height of a rational point H(x)      26
Height on $\mathbf{P}^n$      1013
Height positivity      2425
Height quadraticity of normalised heights on abelian varieties      3541
Height relation between quadratic and linear parts of $\tilde h_c(x)$      3839
Height Schanuels theorem      1719
Height, functoriality of      23
Height, local height      see Local heights
Hensels Lemma      62
Hermites finiteness theorem      3 4952 109
Hermitian structure on a line bundle      8485
Hilbertian field      129130 137138
Hilbertian field, finitely generated extensions of hilbertian fields are hilbertian      130
Hilberts irreducibility theorem      56 121 130 149
Hilberts irreducibility theorem $\mathbf{Q}$ is hilbertian      130132
Hilberts irreducibility theorem Nerons specialisation theorem      see Nerons specialisation theorem
Hilberts irreducibility theorem relation with integral points      135
Hilberts irreducibility theorem specialisation of Galois groups      122126 137138
Hilberts irreducibility theorem thin sets      see Thin sets
Hilberts irreducibility theorem, hilbertian field      129130 137138
Hondas conjecture      162
Hurwitz genus formula      142143
Hyperelliptic curve      115116
Inertia subgroup      140 143 150
Integral points      46 9495 97 100 102106
Integral points Bakers method      5 94 97 108120
Integral points behaviour under morphisms      108110 114
Integral points elliptic curves with good reduction outside a finite set of places      118120
Integral points exceptional curves      9495 110
Integral points of bounded height      6 115 117 177178
Integral points on $X_1(n)$      119120
Integral points on $\mathbf{P}_1$ - $\{0,1,\infty\}$      5 102104 112114
Integral points on elliptic curves      97 115 117
Integral points on hyperelliptic curves      5 115116
Integral points on modular curves and the class number 1 problem      194199
Integral points on superelliptic curves      117
Integral points on Thue curves      114
Integral points on X(2)      118
Integral points quasi-integral sets      9495
Integral points S-integral points      94 104105
Integral points Siegels theorem (see also Siegels theorem)      95 102104
Intersection multiplicities      8586
Jacobian of a curve      12 5 46 5859 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
Kroneckers limit formula      194 199
Kubert Lang      115 119120 197
Kummer theory      5556
L-function      188 189 198199
Lang Weil theorem      6 62 184
Large point      79 80
Large sieve inequality      6 163164
Large sieve inequality Bombieri Davenport bound      172
Large sieve inequality Davenport Halberstam theorem      166172
Large sieve inequality improved inequalities      170172
Large sieve inequality Selbergs bound      171172
Legendres equation of an elliptic curve      118
Lenstras example      147
Lie group      60 149
Line bundle, generalities      2022
Line bundle, generalities ample      22
Line bundle, generalities generated by its global sections      2122
Line bundle, generalities the Picard group Pic(X)      2022
Linear torus      73
Local heights      8393
Local heights as intersection multipicities      8586
Local heights case of an abelian variety with good reduction      89
Local heights in terms of theta functions      8889
Local heights normalisation on abelian varieties      8789
Local heights relation with global heights      8990
Local heights Tate normalisation on elliptic curves      9093
Locally compact field      81
Logarithmic height      2 11
Lower bounds for linear forms in logarithms      5 94 110112
Manin Demjanenko theorem      58 6266 154
Manin Demjanenko theorem application to Fermat quartics $ax^4+by^4+cz^4=0$      6667
Manin Demjanenko theorem application to modular curves      6769
Manin Demjanenko theorem application to the Tate module of an elliptic curve      7374
Manin Drinfeld theorem      73 197
Manin Mumford conjecture      76
Mazurs theorem      69
Modular curves $X_0(N)$      6770 146
Modular curves $X_1(N)$      119120
Modular curves $Y_0(N)$      67
Modular curves associated to Cartan subgroups and their normalisers      194197
Modular curves Belyis theorem      7073
Modular curves cusps      69 71 73 118 195197
Modular curves Manin Drinfeld theorem      73
Modular curves Manins theorem      6769
Modular curves X(N)      118 193
Mordell Weil theorem      1 34 52 58 99 100
Mordell Weil theorem classical descent      53
Mordell Weil theorem effectivity      52 99101 107
Mordell Weil theorem explicit form      5557
Mordell Weil theorem for finitely generated ground fields      52
Mordell Weil theorem weak Mordell Weil theorem      3 5152 99
Mordell-Weil group      4344 5153
Mordell-Weil group rank      3 5859 69
Mordell-Weil group torsion subgroup      4344 54 69
Mordell-Weil group, generators for      99
Mordells conjecture      1 34 5859 105 133
Mordells conjecture Chabautys theorem      see Chabautys theorem
Mordells conjecture generalised      7374
Mordells conjecture in characteristic p > 0      80 105
Mordells conjecture Manin Demjanenko theorem      see Manin Demjanenko theorem
Mordells conjecture Manin Mumford conjecture      see Manin Mumford conjecture
Mordells conjecture Mumfords theorem      see Mumfords theorem
Multiplicative function      182
Mumfords inequality      77
Mumfords lemma      23
Mumfords theorem      34 58 7480 105 133
Mumfords theorem optimal for function fields      80
Nakai Moishezon criterion      26 27
Neron model of an abelian variety      89
Neron Severi group NS(X)      25 3132 47 62 64 157
Neron Tate normalisation      2 2931 35
Nerons estimate      5355
Nerons normalisation of local heights      8789
Nerons specialisation theorem for abelian varieties      59 152154
Nerons specialisation theorem for abelian varieties Silvermans theorem      154
Nerons specialisation theorem for abelian varieties specialisation of extensions of abelian varieties      154
Nerons specialisation theorem for abelian varieties specialisation of non-commutative groups      154
Noethers method      147
Non-degeneracy of normalised heights      4243 (see also Height
Normalised heights      2 3031 3448
Normalised heights non-degeneracy      4244
Normalised heights on elliptic curves      3941
Normalised heights positivity      41
Normalised heights quadraticity      3536
Normalised heights, normalised local heights      8789 9093
Northcotts finiteness theorem      1617 44 46 53
Northcotts finiteness theorem quantitative form (Schanuels theorem)      1719 132
NS(X), Neron Severi group      26 3132 47 62 64 157
Number of integral points of bounded height on a thin set      6 134136 163 177
Number of integral points of bounded height on an affine variety      177178
Number of rational points of bounded height on $\mathbf{P}_2$ blown up at a point      2628
Number of rational points of bounded height on $\mathbf{P}_n$      2 1719
Number of rational points of bounded height on a curve      56 80
Number of rational points of bounded height on a thin set      56 132134 178
Number of rational points of bounded height on an abelian variety      5355
Number of rational points of bounded height on an algebraic variety      178
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