Szego Seminar: An introduction to the Weil Conjectures

Abstract: Let $q$ be a power of a prime. For each positive integer $n$, let $\mathbb{F}_{q^n}$ be the finite field of $q^n$ elements. Let $V$ be a smooth projective variety defined over $\mathbb{F}_q$, which lies in some projective space $\mathbb{P}^N$. An $\mathbb{F}_{q^n}$-point of $V$ is a point on $V$ with $\mathbb{P}^N$-coordinates in $\mathbb{F}_{q^n}$, and we denote the set of such points by $V(\mathbb{F}_{q^n})$. In 1949, Andr\'{e} Weil made a series of conjectures regarding the generating function $Z(V/\mathbb{F}_q; T) := \exp{(\sum_{n = 1}^\infty\frac{\#V(\mathbb{F}_{q^n})}{n}T^n)} \in \mathbb{Q}[[T]]$ and proved by himself the cases of curves and abelian varieties. The full result was gradually developed by Dwork, M. Artin, Grothendieck, and Deligne in the following twenty four years. In particular, the conjectures state that $Z(V/\mathbb{F}_q; T)$ is actually a rational function and even describe it explicitly. In this talk, I will introduce the Weil conjectures and show you a proof of the special case where $V$ is an elliptic curve.