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I know that the Weil's proof of the Weil conjectures for curves and abelian varieties is made under the lenguage of his "Foundation of algebraic geometry", however in "Polarizations and Grothendieck's Standard conjectures" Milne says:

In examining Weil’s proofs (Weil 1948) of the Riemann hypothesis for curves and abelian varieties over finite fields, Grothendieck was led to state two “standard” conjectures (Grothendieck 1969), which imply the Riemann hypothesis for all smooth projective varieties over a finite field, essentially by Weil’s original argument.

Is there a proof of the Weil conjectures for curves and abelian varieties in modern (mathematical)language such that follows the Weil's original argument?

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    $\begingroup$ See Milne's 1986 article on Abelian Varieties, 19.1. $\endgroup$
    – anon
    Commented Aug 4, 2014 at 16:24

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http://jmilne.org/math/xnotes/pRH.html might be of interest to you (proof of the Riemann Hypothesis for curves on p. 10 and for Abelian varieties on p. 22) and http://jmilne.org/math/articles/1986b.pdf (proof of the Riemann Hypothesis for Abelian varieties).

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Kleiman's paper "Algebraic Cycles and the Weil Conjectures", in the book "Dix Exposes Sur La Cohomologie des Schemas" is, I think, what you're looking for.

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