The Weil conjectures

Abstract

In discussing the question of rational points on algebraic curves, we are usually concerned with ℚ. André Weil looked instead at curves over finite fields; assembling the counts into a function, he discovered that it always had some surprising properties. His conjectures, posed in 1949 and since proven, have been the source of much development in algebraic geometry. In this thesis we introduce the zeta function of a variety (named after the Riemann zeta function for reasons which we explain), present the Weil conjectures, and show how they can be used to simplify the process of counting points on a curve. We also present the proof of the conjectures for the special case of elliptic curves.