Book contents
- Frontmatter
- Contents
- 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
4 - L-functions from number theory
Published online by Cambridge University Press: 28 April 2020
- Frontmatter
- Contents
- 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
Summary
This chapter returns to more elementary mathematics, introducing Dirichlet, Hecke, and Artin L-functions. A proof of Dirichlet’s theorem on arithmetic progressions is given, by the method expounded by Serre; it would however be a shame to omit Dirichlet’s original method, which gave additional information and anticipated the analytic class number formulae. The two main generalisations of Dirichlet’s L-functions are then introduced: those of Hecke and Artin. Hecke’s main theorem is stated without proof: existence of an analytic continuation and a functional equation, and it is then explained how Artin and Brauer derived the same results for non-abelian L-functions.
- Type
- Chapter
- Information
- Zeta and L-Functions of Varieties and Motives , pp. 73 - 103Publisher: Cambridge University PressPrint publication year: 2020