Date: Fri, 10 Dec 1993 19:50:17 -0600 From: David Rusin To: jbuddenh@papaya.wustl.edu Subject: Re: Families of elliptic curves of positive rank In article <2eahva$n6i@wuecl.wustl.edu> you write: >Many families of ellipic curves of positive rank over the rationals >are known. Usually these are parameterized by one, two or sometimes >three rational parameters. Sometimes certain relations between the >parameters must be avoided to insure positive rank. > >My question: Are similar families known for rank > 1? For rank > 2? I would appreciate a copy of any leads you get. However, I thought elliptic curves over Q had rank at most 2? More precisely, if you reduce mod p for any prime p and then pass to the alg. closure, I thought the subgroup of elements of order p was Z/p x Z/p. (I'm no expert; my knowledge is mostly limited to what came out of a survey article, umm, Tate? 1973? Math. Z? ) Again, let me know what you find. Both for FLT and for factoring big integers, elliptic curves have been really useful lately! dave rusin@math.niu.edu ============================================================================== Subject: Re: Families of elliptic curves of positive rank To: rusin@mp.cs.niu.edu (David Rusin) Date: Fri, 10 Dec 93 23:30:57 CST From: Jim Buddenhagen Reply to: jb1556@daditz.sbc.com (Jim Buddenhagen) X-Mailer: ELM [version 2.3 PL11] > I would appreciate a copy of any leads you get. However, I thought I'll forward any interesting replies I get. > elliptic curves over Q had rank at most 2? More precisely, if you reduce > mod p for any prime p and then pass to the alg. closure, I thought the > subgroup of elements of order p was Z/p x Z/p. (I'm no expert; my > knowledge is mostly limited to what came out of a survey article, umm, > Tate? 1973? Math. Z? ) I don't know about mod p, but over Q there are examples of rank up to 11 in the literature, but rank higher than 3 or 4 is rare. It is not known if there is an upper bound, but some think not. -- Jim Buddenhagen Southwestern Bell Telephone Co. jb1556@daditz.sbc.com Statistical Research 314-235-5183 One Bell Center, Room 12-V-8 St. Louis, MO 63101-3099 ============================================================================== Subject: high rank families of elliptic curves (fwd) To: rusin@mp.cs.niu.edu Date: Sat, 11 Dec 93 6:17:07 CST From: Jim Buddenhagen Forwarded message: > From larsen@math.upenn.edu Fri Dec 10 22:41:14 1993 > Date: Fri, 10 Dec 93 23:43:41 -0500 > From: larsen@math.upenn.edu > Posted-Date: Fri, 10 Dec 93 23:43:41 -0500 > Message-Id: <9312110443.AA05636@diophantus.math.upenn.edu> > To: jbuddenh@papaya.wustl.edu > Subject: high rank families of elliptic curves > > > N\'eron's original construction of elliptic curves of high rank > made use of high-rank families of curves over function fields. > The idea is to specify n <= 8 points in the plane, and consider > the linear system of cubic curves passing through them. By > throwing out a Zariski-closed set, you can assume that the curve in > the family is non-singular. Each curve in the resulting family is > endowed with n rational points. Of course, when you specialize to a fixed > curve in the family, there may be unexpected relations among these > points on the curve, but this will not happen on a "Hilbert set" > in the parameter space. > > -Michael Larsen -- Jim Buddenhagen Southwestern Bell Telephone Co. jb1556@daditz.sbc.com Statistical Research 314-235-5183 One Bell Center, Room 12-V-8 St. Louis, MO 63101-3099