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1 results

  • 2 - The zeta function of a Z-scheme of finite type

  • Bruno Kahn
  • Book:
    Zeta and L-Functions of Varieties and Motives
    Published online:
    28 April 2020
    Print publication:
    07 May 2020, pp 20-43

  • 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.