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Zeta and L-Functions of Varieties and Motives [Minkštas viršelis]

  • Formatas: Paperback / softback, 214 pages, aukštis x plotis x storis: 226x152x13 mm, weight: 330 g, Worked examples or Exercises
  • Serija: London Mathematical Society Lecture Note Series
  • Išleidimo metai: 07-May-2020
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108703399
  • ISBN-13: 9781108703390
Kitos knygos pagal šią temą:
Zeta and L-Functions of Varieties and MotivesDidesnis paveikslėlis
  • Formatas: Paperback / softback, 214 pages, aukštis x plotis x storis: 226x152x13 mm, weight: 330 g, Worked examples or Exercises
  • Serija: London Mathematical Society Lecture Note Series
  • Išleidimo metai: 07-May-2020
  • Leidėjas: Cambridge University Press
  • ISBN-10: 1108703399
  • ISBN-13: 9781108703390
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781108703390.html
Raktažodžiai:
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.

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.

Recenzijos

'The book will be of interest to both young mathematicians and physicists as well as experienced scholars.' Nikolaj M. Glazunov, zbMATH Open

Daugiau informacijos

Discover how zeta and L-functions have shaped the development of major parts of mathematics over the past two centuries.
Introduction 1(5)
1 The Riemann Zeta Function
6(14)
1.1 A bit of history
6(1)
1.2 Absolute convergence
7(1)
1.3 The Euler product
8(2)
1.4 Formal Dirichlet series
10(2)
1.5 Extension to R(s) > 0; the pole and residue at s = 1
12(1)
1.6 The functional equation
13(2)
1.7 The Riemann hypothesis
15(2)
1.8 Results and approaches
17(1)
1.9 The prime number theorem
18(1)
1.10 Dedekind zeta functions
18(2)
2 The Zeta Function Of A Z-Scheme Of Finite Type
20(24)
2.1 A bit of history
20(1)
2.2 Elementary properties of ξ (X, s)
21(3)
2.3 The case of a curve over a finite field: the statement
24(1)
2.4 Strategy of the proof of Theorem 2.7
25(1)
2.5 Review of divisors
26(1)
2.6 The Riemann-Roch theorem
26(1)
2.7 Rationality and the functional equation (F.K. Schmidt)
27(2)
2.8 The Riemann hypothesis: reduction to (2.4.1) (Hasse, Schmidt, Weil)
29(1)
2.9 The Riemann hypothesis: Weil's first proof
29(11)
2.10 First applications
40(1)
2.11 The Lang--Weil theorems
41(3)
3 The Weil Conjectures
44(29)
3.1 From curves to abelian varieties
44(8)
3.2 The Riemann hypothesis for an abelian variety
52(2)
3.3 The Weil conjectures
54(3)
3.4 Weil cohomologies
57(3)
3.5 Formal properties of a Weil cohomology
60(8)
3.6 Proofs of some of the Weil conjectures
68(3)
3.7 Dwork's theorem
71(2)
4 L-Functions From Number Theory
73(31)
4.1 Dirichlet L-functions
73(3)
4.2 The Dirichlet theorems
76(8)
4.3 First generalisations: Hecke L-functions
84(10)
4.4 Second generalisation: Artin L-functions
94(7)
4.5 The marriage of Artin and Hecke
101(1)
4.6 The constant of the functional equation
102(2)
5 L-Functions From Geometry
104(38)
5.1 "Hasse--Weil" zeta functions
104(4)
5.2 Good reduction
108(2)
5.3 L-functions of l-adic sheaves
110(10)
5.4 The functional equation in characteristic p
120(9)
5.5 The theory of weights
129(4)
5.6 The completed L-function of a smooth projective variety over a global field
133(9)
6 Motives
142(26)
6.1 The issue
142(2)
6.2 Adequate equivalence relations
144(2)
6.3 The category of correspondences
146(1)
6.4 Pure effective motives
147(1)
6.5 Pure motives
148(2)
6.6 Rigidity
150(1)
6.7 Jannsen's theorem
151(1)
6.8 Specialisation
152(2)
6.9 Motivic theory of weights (pure case)
154(3)
6.10 Example: Artin motives
157(1)
6.11 Example: h1 of abelian varieties
158(1)
6.12 The zeta function of an endomorphism
159(2)
6.13 The case of a finite base field
161(3)
6.14 The Tate conjecture
164(2)
6.15 Coronidis loco
166(2)
Appendix A Karoubian and monoidal categories 168(13)
Appendix B Triangulated categories, derived categories, and perfect complexes 181(14)
Appendix C List of exercises 195(2)
Bibliography 197(10)
Index 207
Bruno Kahn is Directeur de recherche at CNRS. He has written around 100 research papers in areas including algebraic and arithmetic geometry, algebraic K-theory and the theory of motives.