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Notes by Davide Lombardo from a 2019 course by Tamás Szamuely devoted to Deligne's proof of the Riemann Hypothesis for smooth projective varieties over finite fields. Katz's recent proof of the hypersurface case and Esnault's spectacular application of the integrality theorem are also discussed. The course concludes with a brief survey of the main results of 'Weil II'.