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 6 Introduction 10 1 The Riemann zeta function 15 1.1 A bit of history 15 1.2 Absolute convergence 16 1.3 The Euler product 17 1.4 Formal Dirichlet series 19 1.5 Extension to R(s) > 0; the pole and residue at s = 1 21 1.6 The functional equation 22 1.7 The Riemann hypothesis 24 1.8 Results and approaches 26 1.9 The prime number theorem 27 1.10 Dedekind zeta functions 27 2 The zeta function of a Z-scheme of finite type 29 2.1 A bit of history 29 2.2 Elementary properties of ζ(X, s) 30 2.3 The case of a curve over a finite field: the statement 33 2.4 Strategy of the proof of Theorem 2.7 34 2.5 Review of divisors 35 2.6 The Riemann–Roch theorem 35 2.7 Rationality and the functional equation (F.K. Schmidt) 36 2.8 The Riemann hypothesis: reduction to (2.4.1) (Hasse, Schmidt, Weil) 38 2.9 The Riemann hypothesis: Weil’s first proof 38 2.10 First applications 49 2.11 The Lang–Weil theorems 50 3 The Weil conjectures 53 3.1 From curves to abelian varieties 53 3.2 The Riemann hypothesis for an abelian variety 61 3.3 The Weil conjectures 63 3.4 Weil cohomologies 66 3.5 Formal properties of a Weil cohomology 69 3.6 Proofs of some of the Weil conjectures 77 3.7 Dwork’s theorem 80 4 L-functions from number theory 82 4.1 Dirichlet L-functions 82 4.2 The Dirichlet theorems 85 4.3 First generalisations: Hecke L-functions 93 4.4 Second generalisation: Artin L-functions 103 4.5 The marriage of Artin and Hecke 110 4.6 The constant of the functional equation 111 5 L-functions from geometry 113 5.1 “Hasse–Weil” zeta functions 113 5.2 Good reduction 117 5.3 L-functions of l-adic sheaves 119 5.4 The functional equation in characteristic p 129 5.5 The theory of weights 138 5.6 The completed L-function of a smooth projective variety over a global field 142 6 Motives 151 6.1 The issue 151 6.2 Adequate equivalence relations 153 6.3 The category of correspondences 155 6.4 Pure effective motives 156 6.5 Pure motives 157 6.6 Rigidity 159 6.7 Jannsen’s theorem 160 6.8 Specialisation 161 6.9 Motivic theory of weights (pure case) 163 6.10 Example: Artin motives 166 6.11 Example: h1 of abelian varieties 167 6.12 The zeta function of an endomorphism 168 6.13 The case of a finite base field 170 6.14 The Tate conjecture 173 6.15 Coronidis loco 175 Appendix A Karoubian and monoidal categories 177 Appendix B Triangulated categories, derived categories, and perfect complexes 190 Appendix C List of exercises 204 Bibliography 206 Index 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.