Trying to understand Deligne’s proof of the Weil conjectures

These notes are an attempt to convey some of the ideas, if not the substance or the details, of the proof of the Weil conjectures by P. Deligne [De1], as far as I understand them, which is to say somewhat superficially – but after all J.-P. Serre (see [Ser]) himself acknowledged that he didn’t check everything. . . What makes this possible is that this proof still contains some crucial steps which are beautiful in themselves and can be stated and even (almost) proved independently of the rest. The context of the proof has to be accepted as given, by analogy with more elementary cases already known (elliptic curves, for instance). Although it is possible to present various motivations for the introduction of the étale cohomology which is the main instrument, getting beyond hand waving is the matter of very serious work, and the only reasonable hope is that the analogies will carry enough weight. What I wish to emphasize is how much the classical study of manifolds was a guiding principle throughout the history of this wonderful episode of mathematical invention – until the Riemann hypothesis itself, that is, when Deligne found something completely different. . .