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Brünjes L. — Forms of Fermat Equations and Their Zeta Functions

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Название: Forms of Fermat Equations and Their Zeta Functions

Автор: Brünjes L.

Аннотация:

In this volume, an abstract theory of ‘forms’ is developed, thus providing a conceptually satisfying framework for the classification of forms of Fermat equations. The classical results on diagonal forms are extended to the broader class of all forms of Fermat varieties.

The main topic is the study of forms of the Fermat equation over an arbitrary field K. Using Galois descent, all such forms are classified; particularly, a complete and explicit classification of all cubic binary equations is given. If K is a finite field containing the d-th roots of unity, the Galois representation on l-adic cohomology (and so in particular the zeta function) of the hypersurface associated with an arbitrary form of the Fermat equation of degree d is computed.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 238

Добавлена в каталог: 19.06.2008

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Предметный указатель

$\mathbb{Q}_l-G_k$ -representation      see representation
Algebraic cycle      79 85 199
Base change      6 17 19 25
Base change in etale cohomology      88
Base change, conjecture      see Weil conjectures base
Betti number      14
Binary cubic equation      4 8 135—152 162 209
Calibration      80 83
Cardano, formula of      145
Category of (discrete) S-groups      35—74 166 178 179
Category of (discrete) S-sets      34 35 57 62
Category of coefficient extensions      see coefficient extension
Category of discrete -modules      77
Category of discrete S-modules      33
Category of etale sheaves      75
Category of graded anticommutative L-algebras      78
Category of Grothendieck motives      see motif
Category of k-varieties      see variety
Category of morphisms      21
Category of non-singular equations      135
Category of pointed sets      36—74 94 95 100 127 157
Category of S-groups      see S-group
Category, fibre-      20
Category, fibred-      19—22
Category, filtered-      72
Category, pseudo-abelian-      9 165 170 175 222—226
Category, R-linear-      170 175 217—226
CHARACTER      9 109 165 167—170 173 175 177—179 187 188 205 206
Character, cyclotomic-      89
Character, Hecke-      see Hecke character
Character, induced-      206
Character, multiplicative-      186 188 227
Character, unit-      165 167
Characteristic polynomial      87 88 90 211 214 215
Chow group      79
Cocycle      6 37—74 94—98 100—102 107—113 115—119 125 129 159 160 213
Cocycle condition      38—40 43 46 55 61 64 100
Cocycle, trivial-      38 47 100 110
Coefficient extension      5—7 9 15—33 36 91—102 156
Coefficient extension, continuous-      31—32 34 96 97 101 156
Cohomologous      38 40 41 43 46 47 55 56 58—62 64—66 70 71 95 100 110
Cohomology of groups      see group cohomology
Cohomology, etale-      14 75 76 88 103
Cohomology, Forms of Fermat Equations and their Zeta Functions      77—78 80 85—92 102 165 181—211 214
Cohomology, Galois-      108 143—144
Cohomology, group      33 96
Cohomology, l-adic-      2 3 5—7 9 10 14 75 235
Cohomology, long exact sequence in-      45—48 50 53 63 111 159
Cohomology, nonabelian-      6 7 33—74 94
Colimit of pointed sets      72—74
Complete system of idempotents      167—169 171 222—224
Connecting morphism      45—48
Continuous group cohomology      see group cohomology
CYCLE      117 204
Cycle map      79 80 83
Deligne, Pierre      1 2 7 14 88 90 186 191 192
Descent      see Galois descent
Diagonal equation      2—4
Diagonal hypersurface      2
Discrete S-group      see category of (discrete) S-groups
Discrete S-set      see category of (discrete) S-sets
Discriminant      8 135—152
Dual abelian group      see group dual
Dual homomorphism      79 166
Eigenspace      9 10 165
Eigenvalue      209—211 214 215
Elliptic curve      102 103
Etale cohomology      see cohomology etale-
Etale sheaf      see category of etale sheaves
Etale site      75
Euler — Poincare characteristic      13
Exact sequence of discrete S-groups      44—64 107 159
fcs-algebra      65—68 108 116 118 119 121 122 127 138 160 163 213
Fermat equation      1 3 5 105 135 153 181
Fermat equation, twisted form of the-      see form of the Fermat equation
Fermat hypersurface      1 2 5 7 165 181 192
Fermat hypersurface, twisted-      165 208
Fermat's last theorem      1
Fibre category      see category fibre-
Fibre functor      20 21
Fibred category      see category fibred-
Formula      see Lefschetz trace
Formula form      3 6 93—100 102 103
Formula form of the Fermat equation      3—5 7—10 90 91 105—139 150 153—164 213 214
Frobenius, arithmetic-      78 89 103 119 122 183 185 213
Frobenius, geometric-      2 39 78 86 87 90 117 181 185 186 191 192 194 208
Functional equation      see Weil conjectures functional
Galois cohomology      see cohomology
Galois descent      3 4 6 15—32 165
Galois representation      see representation
Galois theory      15

Gixed point      85 86 197—199
Gouvea, Fernando Q.      2
Grothendieck motif      see motif
Grothendieck spectral sequence      76
Grothendieck topology      75 77
Grothendieck, Alexander      11 14 21
Group of automorphisms      3 4 6 8 33 94 96 105—107 128 153
Group of fractional ideals      187
Group, additive-      88 188
Group, cohomology      39
Group, dual abelian-      165 166 172 182
Group, multiplicative-      118 138 188
Group, opposite-      116
Group, symmetric-      3 49 52 66 67 172 207
Group, topological-      6 31 33 34 68 69
Group, twisted-      96 107
Hecke character      2 187
Hodge conjecture      1
Idempotent      167—169 171 177 218 222—224
Infinity type      187
Inflation      69—74 143 145 146
Intersection cycle      199
Intersection number      13 85 198 199
Inverse image      20 21
Inverse image, functor      20
Jacobi sum      2 10 186—191 196 227
Krull-topology      31 65 89 96 104
Kuenneth components      83
Kuenneth formula      79 82 83
l-adic cohomology      see cohomology l-adic-
Lefschetz operator      80
Lefschetz theorem, strong-      80 192
Lefschetz theorem, weak-      80
Lefschetz trace formula      75 80 85 197—200
Long exact cohomology sequence      see cohomology long
Module of definition      187
Moebius transformation      136
Morphism of $\mathcal{F}$ -descent      21—23
Morphism of coefficient extensions      28—31 99—104
Morphism, cartesian-      20 21
Motif      10 165 174
Motivic      10 165
Nonabelian cohomology      see cohomology nonabelian-
Nonsingular binary cubic equation      see binary cubic equation
Pell's equation      3
Permutation      3 120 204—207
Permutation matrix      105 156
Poincare duality      78 82 88
Pointed set      see category of pointed sets
Projection formula      81
Projection functor      20
Projector      170 173 218—226
R-linear category      see category
R-linear-rationality      see Weil conjectures rationality
Representation of a finite group-      168
Representation, -      16
Representation, $\mathbb{Q}_l-G_k$ -      2 5—7 10 102 103 175 181 185 186 191 201
Representation, induced-      206
Restriction      69—71 74 143—147
Restriction, -inflation-sequence      68—71 143 145
Riemann hypothesis      see Weil conjectures Riemann
Riemann sphere      136
Rupprecht, Christopher      7
S-group      see category of (discrete)
S-groups, abelian-      39
S-groups, finite abelian-      166 169 174
S-groups, twisted-      see twisted (discrete) S-group
S-set      see category of (discrete) S-sets
Semidirect product      9 48—50 52 63 129 165—179
Serre, Jean-Pierre      6 33 108
Shioda, Tetsuji      1 10 192 201
Subgroup, closed normal-      70
Symmetric group      see group symmetric-
Tate conjecture      1
Transposition      10 173 196 197 204
Trivial class      46 47 60 71 74 93 96 99 100 121 122 146 147
Trivial cocycle      38 47 100 110
Trivial form      93 94
Twisted (discrete) S-group      54 56—64
Variety      1 6 11—14 17—25 31 76—78 102
Variety over a finite field      4 6 11—14 90 194 201
Weil cohomology theory      6 75 78—86
Weil Conjectures      1 2 4 6 11 13 14 88
Weil conjectures, base change      14 88
Weil conjectures, functional equation      13 88
Weil conjectures, rationality      13 88
Weil conjectures, Riemann hypothesis      14 88
Weil, Andre      1 2 13 189 192
Wiles, Andrew      1
Wreath product      3 6 9 48 52 105 106 119 153 155 165
Yui, Noriko      2
Zeta function      2—4 6 7 10 12—14 85 87 90 91 213—221

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