Zeta and L-Functions of Varieties and Motives (London Mathematical Society Lecture Note Series, Series Number 462)

"The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta and L-functions sit at the meeting point of all these theories and have played a profound role in shaping the evolution of number theory. This book presents a big picture of zeta and L-functions and the complex theories surrounding them, combining standard material with results and perspectives that are not made explicit elsewhere in the literature. Particular attention is paid to the development of the ideas surrounding zeta and L-functions, using quotes from original sources and comments throughout the book, pointing the reader towards the relevant history. Based on an advanced course given at Jussieu in 2013, it is an ideal introduction for graduate students and researchers to this fascinating story." -- Prové de l'editor 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 This book is an account of how zeta and L-functions have helped shape number theory, combining standard and less standard material, some of which cannot be found elsewhere in the literature. Particular attention is paid to the development of ideas: quotes from original sources and comments are used throughout the book, pointing the reader towards the relevant history. Based on an advanced course at Jussieu in 2013, it is an ideal introduction to this story for graduate students and researchers. --back cover. Zeta and L-functions have played a major part in the development of number theory. This book for graduate students and researchers presents a big picture of some key results and surrounding theory, whilst taking the reader on a journey through the history of their development Discover How Zeta And L-functions Have Shaped The Development Of Major Parts Of Mathematics Over The Past Two Centuries.