Zeta and L-Functions of Varieties and Motives

Discover how zeta and L-functions have shaped the development of major parts of mathematics over the past two centuries.

Bruno Kahn (Author)

9781108703390, Cambridge University Press

Paperback / softback, published 7 May 2020

214 pages
22.6 x 15.2 x 1.3 cm, 0.33 kg

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.

Introduction
1. The Riemann zeta function
2. The zeta function of a Z-scheme of finite type
3. The Weil Conjectures
4. L-functions from number theory
5. L-functions from geometry
6. Motives
Appendix A. Karoubian and monoidal categories
Appendix B. Triangulated categories, derived categories, and perfect complexes
Appendix C. List of exercises
Bibliography
Index.

Subject Areas: History of mathematics [PBX], Geometry [PBM], Number theory [PBH]