Hasse’s bound for elliptic curves over finite fields

Let $E$ be an elliptic curve defined over a finite field $\\mathbb{F}_{q}$ with $q=p^{r}$ elements ( $p\\in\\mathbb{Z}$ is a prime). Thefollowing theorem gives a bound of the size of $E(\\mathbb{F}_{q})$ , $N_{q}$ , i.e. the number points of $E$ defined over $\\mathbb{F}_{q}$ .This was first conjectured by Emil Artin (in his thesis!) andproved by Helmut Hasse in the 1930’s.

Theorem 1 (Hasse).

\\mid N_{q}-q-1\\mid\\leq 2\\sqrt{q}

Remark: Let $a_{p}=p+1-N_{p}$ as in the definition of theL-series of an ellitpic curve. Then Hasse’s bound reads:

\\mid a_{p}\\mid\\leq 2\\sqrt{p}

This fact is key for the convergence of the L-series of $E$ .