# 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). The following 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!) and proved 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 the L-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 $E$.

Title Hasse’s bound for elliptic curves over finite fields HassesBoundForEllipticCurvesOverFiniteFields 2013-03-22 13:55:41 2013-03-22 13:55:41 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 14H52 Hasse’s bound LSeriesOfAnEllipticCurve EllipticCurve BadReduction ArithmeticOfEllipticCurves