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]