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Published online by Cambridge University Press: 28 April 2020
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.
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