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
Canonical name	HassesBoundForEllipticCurvesOverFiniteFields
Date of creation	2013-03-22 13:55:41
Last modified on	2013-03-22 13:55:41
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Theorem
Classification	msc 14H52
Synonym	Hasse’s bound
Related topic	LSeriesOfAnEllipticCurve
Related topic	EllipticCurve
Related topic	BadReduction
Related topic	ArithmeticOfEllipticCurves