From: Robin Chapman
Subject: RE: Elliptic curves over finite field
Date: Wed, 30 May 2001 13:51:53 -0400
Newsgroups: sci.math
Summary: Structure of groups of points of elliptic curves over finite fields
>===== Original Message From azmitamid@REMOVEmy-deja.com (Azmi Tamid) =====
>The possible torsion groups of elliptic curve over the rationals Q are
>known .
>Now if E is elliptic curve defined over the finite field GF(q) is
>there some information about the group in this case ?
The group is isomorphic to Z/nZ x Z/mZ with m | n.
Also m |(q-1) (this comes from the Weil pairing).
Also the order mn satisfies the Hasse-Weil bound
|mn - (q+1)| <= sqrt(q).
------------------------------------------------------------
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"His mind has been corrupted by colours, sounds and shapes."
The League of Gentlemen
==============================================================================
From: Jyrki Lahtonen
Subject: Re: Elliptic curves over finite field
Date: Thu, 31 May 2001 08:35:40 +0300
Newsgroups: sci.math
Robin Chapman wrote:
> Also the order mn satisfies the Hasse-Weil bound
> |mn - (q+1)| <= sqrt(q).
I'm sure that you meant to write
|mn-(q+1)|<=2*sqrt(q)
At least in the case that q is a power of four, there are
curves achieving equality here. So in this sense the bound
is the best possible
--
Jyrki Lahtonen, docent
Department of Mathematics,
University of Turku,
FIN-20014 Turku, Finland
http://users.utu.fi/lahtonen
tel: (02) 333 6014
[quotes of previous message trimmed --djr]