From Zeta Functions to Cohomology: An Insight into the Weil Conjecture

This thesis provides an introductory exploration of the Weil conjectures, focusing on the deep connections between the zeta functions of smooth projective varieties over finite fields and the concept of Weil cohomology. Building on Andr Weils foundational conjectures, we delve into how these conjectures led to the development of cohomology theories capable of capturing arithmetic information about varieties over finite fields. A key result discussed is the Lefschetz trace formula, a powerful tool in cohomology that allows for the computation of point counts of varieties in terms of the traces of an endomorphism on cohomology groups. Through this study, we aim to outline the theoretical framework behind Weil cohomology and its impact on algebraic geometry.