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
2 - The zeta function of a Z-scheme of finite type
Published online by Cambridge University Press: 28 April 2020
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
Summary
This chapter introduces zeta functions of Z-schemes of finite type. It is essentially dedicated to the proof of the Riemann hypothesis for curves over a finite field. An idea of Weil’s proof of the Castelnuovo–Severi inequality is included, and the easy case of curves of genus 1 (due to Hasse) is given. For the general case, the proofs of Mattuck–Tate and Grothendieck, which rely on an a priori weaker inequality, are given; the two inequalities are compared and it is shown that we can recover the first one using the second and the additivity of the numerically trivial divisors.
- Type
- Chapter
- Information
- Zeta and L-Functions of Varieties and Motives , pp. 20 - 43Publisher: Cambridge University PressPrint publication year: 2020