Weil conjectures for elliptic curves

Abstract

In this paper we studied Weil conjectures, especially the Riemann hypothesis part. It contains Hasse's proof for elliptic curves, two proofs for smooth projective curves by Weil and Bombieri respectively. And also we briefy introduce the etale cohomology theory and its connection with Weil conjectures, calculate the etale cohomology groups of Elliptic curves, and prove the Weil conjectures again.