The aim of this book is to explain the development of the statements and proofs of the Weil conjectures from the analogous results for the Riemann zeta function, at the same time describing the transformation of classical algebraic number theory and algebraic geometry into the language of schemes. In order to motivate the cohomological aspects and in particular the idea of an e"tale map, a certain amount of classical algebraic geometry (over the field of complex numbers) and associated complex manifold theory is included, not only to provide an insight to the more abstract methods, but also to be used in the various comparison techniques.
All these methods and results are contained in existing literature, which to the non-expert may seem both inpenetrable and disjointed. This text is designed to be read by any mathematician who is conversant with the basics of commutative algebra, as in Atiyah fc Macdonald (17), field and Galois theory, as in Winter (217), and 'who has some familiarity with manifolds and algebraic topology (see for example Matsushima (138) and Spanier (183))and categories as in Mitchell (141).
It would be impossible to cover the contents of this book with full details in a small number of pages, and so footnotes have been included to give explicit references to the relevant
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