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Published online by Cambridge University Press: 28 April 2020
This last chapter is dedicated to motives and their zeta functions. We consider only an elementary case: that of pure motives of Grothendieck associated to smooth projective varieties over a finite field. One can go much further using the triangulated categories of motives introduced by Voevodsky and developed by Ivorra, Ayoub, and Cisinski–D’eglise, but this would go beyond the scope of the book. Nonetheless, it is explained how this viewpoint considerably clarifies how Weil cohomologies are used to prove rationality and the functional equation, and this theory is applied to prove a somewhat forgotten theorem of Weil: Artin’s conjecture for non-abelian L-functions in positive characteristic.
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