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Lang S. Diophantine Geometry
Lang S.  Diophantine Geometry









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: Diophantine Geometry

: Lang S.

:

Between number theory and geometry there have been several stimulating influences, and this book records of these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the braod scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication... Although in the series of number theory, this volume is on diophantine geometry, the reader will notice that algebraic geometry is present in every chapter. ...The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities.


: en

: /

:

ed2k: ed2k stats

: 1997

: 316

: 20.09.2008

: | | ID
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$C_i$ property      245249
$g_2$ and $g_3$      12 126
$L^2$-degree      173
$T_l(A)$      82 94
$V_l(A)$      83
$X_0(N)$      127131
$Y_0(N)$ and $Y_1(N)$      127
abc conjecture      29 47 49 65
Abelian logarithm      237
Abelian varieties      16 20 2542 68100 101122 158162 181183 220 221 232 236239
Abelian varieties, equations for      26 77 Birch Descent Faltings Finiteness Function Gauss Jacobian l-adic Lang Manins Masser Moduli Mordell Neron Neron Parshin Polarization Rank Raynaud Semisimplicity Semistable Subvarieties Tate Theorem Torsion Trace
Absolute case      12
Absolute conjecture      67
Absolute norm      55
Absolute value      44
Adams type for e      214
Adeles      93
Adjunction formula      168
Admissible metric      165
Affine bounded      207
Affine coordinate ring      4
Affine coordinate ring, coordinates and integral points      217
Affine variety      2
Ahlfors on Nevanlinna theory      199 203
Ahlfors Schwarz lemma      185
Ahlfors Shimizu height      200
Albanese variety      31 32
Albert Brauer Hasse Noether theorem      252253
Algebraic equivalence      30 33
Algebraic families      12 18 2327 28 62 7482 118121 158162 178 187 189 192 221
Algebraic families of abelian varieties      27 28 7482 118121 158162 189
Algebraic families of heights      7682
Algebraic families of pseudo-canonical varieties      24 25
Algebraic families, split      12 24 25 62 78 79 178 192 Function Manin Silverman
Algebraic integers      54
Algebraic point      3
Algebraic point in Vojtas conjecture      50 6367 222225
Algebraic special set      16 182
Algebraically hyperbolic      16 17 179
ample      7 1115 20 22 67 181 198
Ample, anti-canonical class      19 20 67
Ample, canonical class      14 15 18 22 198
Ample, cotangent bundle      181
Ample, vector sheaf      20
Analytic torsion      173 175
Anticanonical class      15 19 20 67 169 245 258262
Anticanonical varieties      15 259262
Arakelov, degree      168
Arakelov, height      169
Arakelov, inequality      151
Arakelov, metric      165
Arakelov, Picard group      167
Arakelov, Shafarevich conjecture in function field case      104
Arakelov, theory      71 163171 228 230 231
Arakelov, volume form      166
Arithmetic Chern character      173
Arithmetic Chow group      174
Arithmetic discriminant      64 171
Arithmetic discriminant and Vojtas inequality      171
Arithmetic Euler characteristic      173
Arithmetic Picard group      167
Arithmetic surface      166
Arithmetic Todd class      173
Arithmetic variety      171
Artin theorem on local specialization      259
Artin Tate      250 253
Artin Winters      150
Artin, conductor      71
Artin, conjectures on $C_i$ fields      246247
Ax theorems, one-parameter subgroups      182
Ax theorems, quasi-algebraic closure      246
Ax Kochen theorem      247
Baily Borel compactification      118
Baker      235 237 239 240
Baker Feldman inequality      235
Basic Hilbert subset      41
Basic isogeny problem      121
Batyrev      20 262
Batyrev Manin conjectures      260262
Batyrev Manin conjectures, relation with Vojta conjecture      261
Beilenson Bloch conjectures      34
Biduality      33
Birational map      5
Birationally equivalent      5
Birch      148 248
Birch Swinnerton Dyer conjecture      34 91 92 9496 98 136 137 139140
Birch Swinnerton Dyer conjecture, Coates Wiles      136
Birch Swinnerton Dyer conjecture, Gross Zagier      139142
Bismut Vasserot      174 232
Bloch (Andre) conjecture      182
Bloch (Spencer) conjectures      34
Bogomolov      20 151
Bombieri simplification of Vojtas proof      233
Borels theorem      183
Bosch      259
Bounded degree      56 63 64 204 222 223
Bounded degree, denominator      217
Bounded degree, generated group      240243
Bounded degree, height      see Height upper
Brauer group      250258
Brauer group, birational invariant      252
Brauer group, exact sequence      255
Brauer group, unramified      252
Brauer Grothendieck group      250 252
Brauer Grothendieck group, specialization      252
Breen      80
Brody hyperbolic      178 179 183 184 225 226
Brody Green hypersurface      22 181 186
Brodys theorem      179
Brownawell-Masser      66
Brumer McGuinness on average rank      28
Bryuno      214
Canonical bundle      185
Canonical class      11 14 20 21 119 146 151 152 197
Canonical class, in higher dimension      14
Canonical class, inequalities      151 152
Canonical class, on a curve      11
Canonical class, on moduli space      119
Canonical class, on projective space      14 197
Canonical class, relative      146
Canonical class, zero      1921 23
Canonical height      63 64 67 146 169
Canonical height in Nevanlinna theory      202
Canonical metric      165 166
Canonical sheaf      119 145 146 169 170 228 259
Canonical sheaf of an imbedding      145 146 169
Canonical variety      15
Carlsson Griffiths on Nevanlinna theory      201
Cartan Nevanlinna height      194
Cartans theorem      197198
Cartier divisor      6 14
Cartier divisor, divisor class group      6
Cassel Guy example      254
Cassels Tate pairing      89
CC inequalities      151 152
Chabauty      36 38
Chai on cubical sheaves      80 82
Chai Faltings      118 120 122
Chais theorem of the kernel      158
Characteristic polynomial      85
Chatelet Weil group      87 88
Chatelet, Brauer variety      256
Chatelet, surface      22 256 257
Chern form      184 202
Cherry on Nevanlinna theory      204 223
Chevalley Weil theorem      224
Chevalleys theorem on quasi-algebraic closure      246
Chow group      7 33 174
Chow group in higher codimension      34
Chow trace      26 (see also Lang Neron Trace
Chow Lang      69 103
Circle method      248
Class field theory      105
Clemens curves      21 34
Co Lie determinant      116 118
Coates Wiles theorem      136
Cocycle      153
Colemans account of Manins method      153161
Colliot Thelene      249258
Colliot Thelene, conjecture      254
Compact case of Vojta conjecture      67
Complete intersection      4
Complex multiplication      29 39 136
Complex torus      125
Complex torus, isomorphism classes      126
Complexity of divisor      203
Conductor      51 71 9799
Conductor of elliptic curve      9799 (see also Modular elliptic curves)
Congruent numbers      135
Connected component      31 33
Connected component, Neron model      70 7981 98
Connection      154
Conormal sheaf      145 169
Constant field      12
Counting function      193 202 211
Counting, algebraic integers      57
Counting, points      58 61 73 260262
Counting, units      57
Coverings, applied to diophantine approximations      219 222
Coverings, descent and heights      37 85 160
Coverings, Nevanlinna theory      204
Coverings, Parshin construction      104106
Coverings, ramified      104105 224 225
Coverings, torsion points      39
Coverings, X_0(N)$ by $X_1(N)$      128
Cubic forms      22 247 250
Cubical sheaves      80 81
Curvature      185 186
Cuspidal group      128
Cuspidal group, ideal      129
Cusps      127
Cycles      31 32 254
Cyclotomic character      133
Cyclotomic extensions      29 247
Cyclotomic units      141
d(F) or d(P)      55
Davenport      48 49 247
de Franchis theorem      13 24
De Rham cohomology      153160
Decomposition group      83 88 112
Degree of canonical sheaf      152
Degree of divisor on a curve      9
Degree of hypersurface      4
Degree of isogeny      35
Degree of line sheaf      144
Degree of metrized line sheaf      116
Degree of polarization      35
Degree on a singular curve      144
Degree with respect to Riemann form      238 (see also Isogeny Polarization)
Degree, Arakelov      168
Deligne on Faltings proof      120
Deligne on L-function      92
Deligne Mumford      150
Deligne Serre representation      132
Demjanenko Manin on split function field case      79
Demjanenko, estimate of Neron height      72
Demjanenko, fibering of Fermat surface      23
Descent      85 90 91 160 219
Descent in coverings      160 219
Descent with Selmer groups      90
Determinant of vector sheaf      144
Diagonal hyperplane      183
Different      120
Differential form      10
Differentials      115
Differentials of first kind      11 136
DIMENSION      4
Diophantine approximation on toruses      233243
Diophantine approximation to numbers      213216
Dirichlet box principle      234
Discriminant      50 55 69
Discriminant in coverings      224
Discriminant of elliptic curve      69 96
Distance      177
Division group      37 38 161 Raynaud
Divisor      6
Divisor class group      6 33
Divisor classes and heights      5861 194196
Divisor classes and Neron functions      213
Dobrowolski      243
Dolbeault operator      172
Dual variety      33
Dysons lemma      229 231 232
Effective divisor      6
Effective divisor class      7
Eigenform      131 132
Eigenvalues of Frobenius      85 91 131
Eigenvalues of Laplacian      173 175
Eisenstein ideal      129
Eisenstein ideal, quotient      129
Eisenstein ideal, series      260
Elkies example of integral points      50
Elkies on Fermat hypersurface      23
Elliptic curve      12 21 23 2527 49 50 96 132 135 139142 162
Elliptic curve, conductor      97 98
Elliptic curve, diophantine approximation      235 236
Elliptic curve, fibers of family      21 23
Elliptic curve, Frey      132 135
Elliptic curve, height of generators      99 100
Elliptic curve, integral points      50
Elliptic curve, isomorphism      96
Elliptic curve, L-function      9798 137142
Elliptic curve, minimal discriminant      97
Elliptic curve, modular      130142
Elliptic curve, periods      93 95 97 125 236
Elliptic curve, rank      28 42 92 139142
Elliptic curve, rank one over the rationals      137142
Elliptic curve, Tate curve      162 (see also Torsion points)
Error function in Nevanlinna theory      203 224
Error terms in second main theorem      199204 224
Error terms in second main theorem, Lang conjecture      200
Esnault Viehweg inequality      152
Esnault Viehweg inequality on Roth theorem      229
Euler characteristic      172
Euler characteristic, arithmetic      173
Euler on Fermat      22
Euler product for a variety      91
Exact sequence, de Rham cohomology      156
Exact sequence, group cohomology      88 250 255
Exact sequence, Lang Tate      87
Exact sequence, Selmer and Shafarevich Tate group      8890
Exceptional set in Vojta conjecture      67
Exceptional set in Vojta conjecture in Schmidt Vojta theorem      215 222
Exponential on Lie groups      236
Faltings, canonical height      119
Faltings, finiteness of l-adic representations      114
Faltings, finiteness of rational points      12 18 36 230 232
Faltings, formula for the degree      120
Faltings, height      74 117123 238
Faltings, inequality on abelian varieties      220 237
Faltings, integral points on abelian varieties      220 221
Faltings, positivity of canonical sheaf      170
Faltings, semisimplicity and Tate conjecture      111115
Faltings, stable height      117 238
Faltings, subvariety of abelian variety      36
Fano variety      260
Fermat      11 22 23 48 62 64 132 135 181 225
Fermat, Brody Green perturbation      181
Fermat, curve      11
Fermat, Euler      22
1 2 3 4
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