The first étale cohomology group of an elliptic curve

In this thesis we state the Weil conjectures and prove them in the case of elliptic curves. Then we construct the étale cohomology groups of a scheme as the right derived functors of the functor of global sections in the category of abelian étale sheaves. We conclude by showing that the first étale cohomology group of an elliptic curve is the dual of the Tate module.