# 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).

$$\mid {N}_{q}-q-1\mid \le 2\sqrt{q}$$ |

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:

$$\mid {a}_{p}\mid \le 2\sqrt{p}$$ |

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 |