Weil Conjectures Exposition

TL;DR

This paper provides a comprehensive exposition of the Weil conjectures, including Deligne's proof of the eigenvalue conjecture, using étale cohomology and Lefschetz theory, with detailed explanations and discussions of consequences.

Contribution

It offers a detailed, accessible account of the proof of the Weil conjectures, emphasizing the role of étale cohomology and Lefschetz theory, and clarifies complex aspects for better understanding.

Findings

Proof of the Weil conjectures completed

Verification of eigenvalues of Frobenius endomorphism

Discussion of consequences of the conjectures

Abstract

In this paper we provide a full account of the Weil conjectures including Deligne's proof of the conjecture about the eigenvalues of the Frobenius endomorphism. Section 1 is an introduction into the subject. Our exposition heavily relies on the Etale Cohomology theory of Grothendieck so I included an overview in Section 2. Once one verifies (or takes for granted) the results therein, proofs of most of the Weil conjectures are straightforward as we show in Section 3. Sections 4-8 constitute the proof of the remaining conjecture. The exposition is mostly similar to that of Deligne in [7] though I tried to provide more details whenever necessary. Following Deligne, I included an overview of Lefschetz theory (that is crucial for the proof) in Section 6. Section 9 contains a (somewhat random and far from complete) account of the consequences. Numerous references are mentioned throughout the paper as well as briefly discussed in Subsection 1.4.