From: cohen@ceremab.u-bordeaux.fr (Henri Cohen)
Newsgroups: sci.math.research
Subject: Re: Families of elliptic curves of positive rank
Date: 13 Dec 1993 13:06:45 GMT


In article <2eahva$n6i@wuecl.wustl.edu>, jbuddenh@papaya.wustl.edu (Jim
Buddenhagen) writes:
|> 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?  
|> Examples and/or references would be appreciated.
|> 
|> --Jim Buddenhagen (jb1556@daditz.sbc.com)
|> 
|> 
|> 

The best result is due to Mestre:
J.-F. Mestre, Courbes elliptiques de rang >=12 sur Q(t),
C.R. Acad. Sci. Paris (1991), pp 171-174.

He builds a 4 parameter family of elliptic curves over Q(t) (which makes 5
parameters in all if you count t) all of rank >=11 if certain conditions
are satisfied. By specializing, he obtains families of rank >=12 over
Q(t) (this is the current record), but by doing a computer search in his
initial family, several people (Tunnell, Fermigier, Nagao) have found
individual curves over Q of much higher rank, the present record is
of rank 20, held by Nagao.

The idea of the construction is very elementary and in fact dates back to
Neron, although Neron does not make it explicit (and Neron claimed only
rank 9 I think, not 12). You can read Mestre's paper of course, but the
basic idea is this (in TeX notation)

Choose $r_1$, $r_2$, $r_3$ and $r_4$ four rational numbers, let $t$ be a 
parameter, and consider the polynomial 
$$P(X)=\prod_{1\le i,j\le 4, i\neq j} (X-(r_i+tr_j)$$ which is of degree 12.
By considering the Laurent series expansion of $P^(1/3)$, you can find a monic
polynomial $g$ of degree 4 such that $\deg(P(X)-g^3(X))\le 7$. Now the crucial
point is that with our special choice of $P$, we have in fact
$\deg(P(X)-g^3(X))\le 6$. This can be proved very elegantly by an elementary
argument involving the geometry of a certain variety, but can of course be
checked brutally by hand or (better) with your favorite computer algebra
system.

Knowing this, write $P(X)=g^3(X)+q(X)g(X)+r(X)$ where $q$ and $r$ are quotient
and remainder of the Euclidean division of $P-g^3$ by $g$. Then $\deg(q)\le 2$
and $\deg(r)\le 3$. It follows that the equation

	$$Y^3+q(X)Y+r(X)$$

defines a CUBIC curve, on which 12 rational points are known, i.e. the points
$(r_i+tr_j,g(r_i+tr_j))$ for $1\le i,j\le 4$. One of these points can then
be sent to infinity, and then one must show that generically the 11
other points
are independent over Q(t), using formulas of Shioda for elliptic surfaces.

	Henri Cohen