دانلود کتاب Zeta and L-functions of varieties and motives

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Contents......Page 6
Introduction......Page 10
1.1 A bit of history......Page 15
1.2 Absolute convergence......Page 16
1.3 The Euler product......Page 17
1.4 Formal Dirichlet series......Page 19
1.5 Extension to ℜ(s) > 0; the pole and residue at s = 1......Page 21
1.6 The functional equation......Page 22
1.7 The Riemann hypothesis......Page 24
1.8 Results and approaches......Page 26
1.10 Dedekind zeta functions......Page 27
2.1 A bit of history......Page 29
2.2 Elementary properties of ζ(X, s)......Page 30
2.3 The case of a curve over a finite field: the statement......Page 33
2.4 Strategy of the proof of Theorem 2.7......Page 34
2.6 The Riemann–Roch theorem......Page 35
2.7 Rationality and the functional equation (F.K. Schmidt)......Page 36
2.9 The Riemann hypothesis: Weil’s first proof......Page 38
2.10 First applications......Page 49
2.11 The Lang–Weil theorems......Page 50
3.1 From curves to abelian varieties......Page 53
3.2 The Riemann hypothesis for an abelian variety......Page 61
3.3 The Weil conjectures......Page 63
3.4 Weil cohomologies......Page 66
3.5 Formal properties of a Weil cohomology......Page 69
3.6 Proofs of some of the Weil conjectures......Page 77
3.7 Dwork’s theorem......Page 80
4.1 Dirichlet L-functions......Page 82
4.2 The Dirichlet theorems......Page 85
4.3 First generalisations: Hecke L-functions......Page 93
4.4 Second generalisation: Artin L-functions......Page 103
4.5 The marriage of Artin and Hecke......Page 110
4.6 The constant of the functional equation......Page 111
5.1 “Hasse–Weil” zeta functions......Page 113
5.2 Good reduction......Page 117
5.3 L-functions of l-adic sheaves......Page 119
5.4 The functional equation in characteristic p......Page 129
5.5 The theory of weights......Page 138
5.6 The completed L-function of a smooth projective variety over a global field......Page 142
6.1 The issue......Page 151
6.2 Adequate equivalence relations......Page 153
6.3 The category of correspondences......Page 155
6.4 Pure effective motives......Page 156
6.5 Pure motives......Page 157
6.6 Rigidity......Page 159
6.7 Jannsen’s theorem......Page 160
6.8 Specialisation......Page 161
6.9 Motivic theory of weights (pure case)......Page 163
6.10 Example: Artin motives......Page 166
6.11 Example: h1 of abelian varieties......Page 167
6.12 The zeta function of an endomorphism......Page 168
6.13 The case of a finite base field......Page 170
6.14 The Tate conjecture......Page 173
6.15 Coronidis loco......Page 175
Appendix A Karoubian and monoidal categories......Page 177
Appendix B Triangulated categories, derived categories, and perfect complexes......Page 190
Appendix C List of exercises......Page 204
Bibliography......Page 206
Index......Page 216