Hasse’s bound for elliptic curves over finite fields

Let E be an elliptic curveMathworldPlanetmath defined over a finite fieldMathworldPlanetmath 𝔽q with q=pr elements (p is a prime). The following theorem gives a bound of the size of E(𝔽q), Nq, i.e. the number points of E defined over 𝔽q. This was first conjectured by Emil Artin (in his thesis!) and proved by Helmut Hasse in the 1930’s.

Theorem 1 (Hasse).

Remark: Let ap=p+1-Np as in the definition of the L-series of an ellitpic curve. Then Hasse’s bound reads:


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